Schaum's Outline of Mathematical Handbook of Formulas and Tables, 3ed (Schaum's Outline Series) ( PDFDrive )-1

SCHAUM'S
outlines
Mathematical Handbook of
Formulas and Tables
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SCHAUM'S
outlines
Mathematical Handbook of
Formulas and Tables
Third Edition
Murray R. Spiegel, PhD
Former Professor and Chairman
Mathematics Department
Rensselaer Polytechnic Institute
Hartford Graduate Center
Seymour Lipschutz, PhD
Mathematics Department
Temple University
John Liu, PhD
Mathematics Department
University of Maryland
Schaum’s Outline Series
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DOI: 10.1036/0071548556
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Preface
This handbook supplies a collection of mathematical formulas and tables which will be valuable to students
and research workers in the fields of mathematics, physics, engineering, and other sciences. Care has been
taken to include only those formulas and tables which are most likely to be needed in practice, rather than
highly specialized results which are rarely used. It is a “user-friendly” handbook with material mostly rooted
in university mathematics and scientific courses. In fact, the first edition can already be found in many libraries
and offices, and it most likely has moved with the owners from office to office since their college times. Thus,
this handbook has survived the test of time (while most other college texts have been thrown away).
This new edition maintains the same spirit as the second edition, with the following changes. First of all,
we have deleted some out-of-date tables which can now be easily obtained from a simple calculator, and
we have deleted some rarely used formulas. The main change is that sections on Probability and Random
Variables have been expanded with new material. These sections appear in both the physical and social
sciences, including education.
Topics covered range from elementary to advanced. Elementary topics include those from algebra,
geometry, trigonometry, analytic geometry, probability and statistics, and calculus. Advanced topics include
those from differential equations, numerical analysis, and vector analysis, such as Fourier series, gamma
and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, and elliptic and other
special functions of importance. This wide coverage of topics has been adopted to provide, within a single
volume, most of the important mathematical results needed by student and research workers, regardless of
their particular field of interest or level of attainment.
The book is divided into two main parts. Part A presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application
of the formulas. Part B presents the numerical tables. These tables include basic statistical distributions
(normal, Student’s t, chi-square, etc.), advanced functions (Bessel, Legendre, elliptic, etc.), and financial
functions (compound and present value of an amount, and annuity).
McGraw-Hill wishes to thank the various authors and publishers—for example, the Literary Executor
of the late Sir Ronald A. Fisher, F.R.S., Dr. Frank Yates, F.R.S., and Oliver and Boyd Ltd., Edinburgh, for
Table III of their book Statistical Tables for Biological, Agricultural and Medical Research—who gave their
permission to adapt data from their books for use in several tables in this handbook. Appropriate references
to such sources are given below the corresponding tables.
Finally, I wish to thank the staff of the McGraw-Hill Schaum’s Outline Series, especially Charles Wall,
for their unfailing cooperation.
SEYMOUR LIPSCHUTZ
Temple University
v
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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Contents
Part A
FORMULAS
1
Section I
Elementary Constants, Products, Formulas
3
1.
2.
3.
4.
5.
6.
Section II
Geometric Formulas
Formulas from Plane Analytic Geometry
Special Plane Curves
Formulas from Solid Analytic Geometry
Special Moments of Inertia
Derivatives
Indefinite Integrals
Tables of Special Indefinite Integrals
Definite Integrals
43
43
53
56
62
62
67
71
108
Differential Equations and Vector Analysis
19. Basic Differential Equations and Solutions
20. Formulas from Vector Analysis
Section VI
16
22
28
34
41
Calculus
15.
16.
17.
18.
Section V
16
Elementary Transcendental Functions
12. Trigonometric Functions
13. Exponential and Logarithmic Functions
14. Hyperbolic Functions
Section IV
3
5
7
10
13
15
Geometry
7.
8.
9.
10.
11.
Section III
Greek Alphabet and Special Constants
Special Products and Factors
The Binomial Formula and Binomial Coefficients
Complex Numbers
Solutions of Algebraic Equations
Conversion Factors
116
116
119
Series
21.
22.
23.
24.
Series of Constants
Taylor Series
Bernoulli and Euler Numbers
Fourier Series
134
134
138
142
144
vii
CONTENTS
viii
Section VII
Special Functions and Polynomials
25.
26.
27.
28.
29.
30.
31.
32.
Section VIII
The Gamma Function
The Beta Function
Bessel Functions
Legendre and Associated Legendre Functions
Hermite Polynomials
Laguerre and Associated Laguerre Polynomials
Chebyshev Polynomials
Hypergeometric Functions
198
198
203
205
205
207
Probability and Statistics
39. Descriptive Statistics
40. Probability
41. Random Variables
Section XII
180
193
Inequalities and Infinite Products
37. Inequalities
38. Infinite Products
Section XI
180
Elliptic and Miscellaneous Special Functions
35. Elliptic Functions
36. Miscellaneous and Riemann Zeta Functions
Section X
149
152
153
164
169
171
175
178
Laplace and Fourier Transforms
33. Laplace Transforms
34. Fourier Transforms
Section IX
149
208
208
217
223
Numerical Methods
42.
43.
44.
45.
46.
47.
Interpolation
Quadrature
Solution of Nonlinear Equations
Numerical Methods for Ordinary Differential Equations
Numerical Methods for Partial Differential Equations
Iteration Methods for Linear Systems
Part B
TABLES
Section I
Logarithmic,Trigonometric, Exponential Functions
1.
2.
3.
4.
Four Place Common Logarithms log1 0 N o r log N
Sin x (x in degrees and minutes)
Cos x (x in degrees and minutes)
Tan x (x in degrees and minutes)
227
227
231
233
235
237
240
243
245
245
247
248
249
CONTENTS
ix
5. Conversion of Radians to Degrees, Minutes, and Seconds
or Fractions of Degrees
6. Conversion of Degrees, Minutes, and Seconds to Radians
7. Natural or Napierian Logarithms loge x or ln x
8. Exponential Functions ex
9. Exponential Functions eⴚx
10. Exponential, Sine, and Cosine Integrals
Section II
Factorial and Gamma Function, Binomial Coefficients
11. Factorial n
12. Gamma Function
13. Binomial coefficients
Section III
Bessel Functions J0(x)
Bessel Functions J1(x)
Bessel Functions Y0(x)
Bessel Functions Y1(x)
Bessel Functions I0(x)
Bessel Functions I1(x)
Bessel Functions K0(x)
Bessel Functions K1(x)
Bessel Functions Ber(x)
Bessel Functions Bei(x)
Bessel Functions Ker(x)
Bessel Functions Kei(x)
Values for Approximate Zeros of Bessel Functions
261
261
261
262
262
263
263
264
264
265
265
266
266
267
268
268
269
Elliptic Integrals
29. Complete Elliptic Integrals of First and Second Kinds
30. Incomplete Elliptic Integral of the First Kind
31. Incomplete Elliptic Integral of the Second Kind
Section VI
257
258
259
Legendre Polynomials
27. Legendre Polynomials Pn(x)
28. Legendre Polynomials Pn(cos ␪)
Section V
257
Bessel Functions
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
Section IV
250
251
252
254
255
256
270
270
271
271
Financial Tables
32. Compound amount: (1 + r)n
33. Present Value of an Amount: (1 + r)n
(1+ r )n – 1
34. Amount of an Annuity:
r
–n
1 – (1+ r )
35. Present Value of an Annuity:
r
272
272
273
274
275
CONTENTS
x
Section VII
Probability and Statistics
36.
37.
38.
39.
40.
41.
42.
Areas Under the Standard Normal Curve
Ordinates of the Standard Normal curve
Percentile Values (tp) for Student's t Distribution
Percentile Values (2p) for 2 (Chi-Square) Distribution
95th Percentile Values for the F distribution
99th Percentile Values for the F distribution
Random Numbers
276
276
277
278
279
280
281
282
Index of Special Symbols and Notations
283
Index
285
PART A
FORMULAS
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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Section I: Elementary Constants, Products, Formulas
1
GREEK ALPHABET and SPECIAL CONSTANTS
Greek Alphabet
Greek
name
Greek letter
Lower case
Capital
Alpha
a
A
Beta
b
Gamma
Greek
name
Greek letter
Lower case
Capital
Nu
n
N
B
Xi
j
g
Omicron
o
Delta
d
Pi
p
Epsilon
E
Rho
r
Zeta
z
Z
Sigma
s
Eta
h
H
Tau
t
Theta
u
Upsilon
y
Iota
i
I
Phi
f
Kappa
k
K
Chi
x
X
Lambda
l
Psi
c
Mu
m
M
Omega
v
O
P
T
Special Constants
1.1. p = 3.14159 26535 89793 …
1
1.2. e = 2.71828 18284 59045 … = lim ⎛1 + ⎞
n⎠
n→∞ ⎝
n
= natural base of logarithms
1.3. γ = 0.57721 56649 01532 86060 6512 … = Euler’s constant
⎛ 1 1
1
⎞
= lim ⎜1 + + + + − ln n⎟
n→∞ ⎝
2 3
n
⎠
1.4. eγ = 1.78107 24179 90197 9852 … [see 1.3]
3
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
GREEK ALPHABET AND SPECIAL CONSTANTS
4
1.5.
e = 1.64872 12707 00128 1468 …
1.6.
π = Γ( 12 ) = 1.77245 38509 05516 02729 8167 …
where Γ is the gamma function [see 25.1].
1.7.
Γ( 13 ) = 2.67893 85347 07748 …
1.8.
Γ( 14 ) = 3.62560 99082 21908 …
1.9.
1 radian = 180°/p = 57.29577 95130 8232 …°
1.10.
1° = p/180 radians = 0.01745 32925 19943 29576 92 … radians
2
SPECIAL PRODUCTS and FACTORS
2.1.
( x + y)2 = x 2 + 2 xy + y 2
2.2.
( x − y)2 = x 2 − 2 xy + y 2
2.3.
( x + y)3 = x 3 + 3x 2 y + 3xy 2 + y 3
2.4.
( x − y)3 = x 3 − 3x 2 y + 3xy 2 − y 3
2.5.
( x + y)4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 xy 3 + y 4
2.6.
( x − y)4 = x 4 − 4 x 3 y + 6 x 2 y 2 − 4 xy 3 + y 4
2.7.
( x + y)5 = x 5 + 5x 4 y + 10 x 3 y 2 + 10 x 2 y3 + 5xy 4 + y5
2.8.
( x − y)5 = x 5 − 5x 4 y + 10 x 3 y 2 − 10 x 2 y 3 + 5xy 4 − y 5
2.9.
( x + y)6 = x 6 + 6 x 5 y + 15x 4 y 2 + 20 x 3 y 3 + 15x 2 y 4 + 6 xy 5 + y 6
2.10.
( x − y)6 = x 6 − 6 x 5 y + 15x 4 y 2 − 20 x 3 y3 + 15x 2 y 4 − 6 xy5 + y 6
The results 2.1 to 2.10 above are special cases of the binomial formula [see 3.3].
2.11.
x 2 − y 2 = ( x − y)( x + y)
2.12.
x 3 − y 3 = ( x − y)( x 2 + xy + y 2 )
2.13.
x 3 + y 3 = ( x + y)( x 2 − xy + y 2 )
2.14.
x 4 − y 4 = ( x − y)( x + y)( x 2 + y 2 )
2.15.
x 5 − y 5 = ( x − y)( x 4 + x 3 y + x 2 y 2 + xy 3 + y 4 )
2.16.
x 5 + y 5 = ( x + y)( x 4 − x 3 y + x 2 y 2 − xy 3 + y 4 )
2.17.
x 6 − y 6 = ( x − y)( x + y)( x 2 + xy + y 2 )( x 2 − xy + y 2 )
5
SPECIAL PRODUCTS AND FACTORS
6
2.18.
x 4 + x 2 y 2 + y 4 = ( x 2 + xy + y 2 )( x 2 − xy + y 2 )
2.19.
x 4 + 4 y 4 = ( x 2 + 2 xy + 2 y 2 )( x 2 − 2 xy + 2 y 2 )
Some generalizations of the above are given by the following results where n is a positive integer.
2.20.
x 2 n+1 − y 2 n+1 = ( x − y)( x 2 n + x 2 n −1 y + x 2 n − 2 y 2 + + y 2 n )
⎛
⎞⎛
⎞
2π
4π
= ( x − y) ⎜ x 2 − 2 xy cos
+ y 2⎟ ⎜ x 2 − 2 xy cos
+ y 2⎟
2n + 1
2n + 1
⎝
⎠⎝
⎠
⎛
⎞
2nπ
⎜ x 2 − 2 xy cos
+ y 2⎟
2n + 1
⎝
⎠
2.21.
x 2 n+1 + y 2 n+1 = ( x + y)( x 2 n − x 2 n −1 y + x 2 n − 2 y 2 − + y 2 n )
⎛
⎞⎛
⎞
2π
4π
= ( x + y) ⎜ x 2 + 2 xy cos
+ y 2⎟ ⎜ x 2 + 2 xy cos
+ y 2⎟
2n + 1
2n + 1
⎝
⎠⎝
⎠
⎛
⎞
2nπ
⎜ x 2 + 2 xy cos
+ y 2⎟
n
+
2
1
⎝
⎠
2.22.
x 2 n − y 2 n = ( x − y)( x + y)( x n −1 + x n − 2 y + x n −3 y 2 + )( x n −1 − x n − 2 y + x n −3 y 2 − )
⎛
⎞⎛
⎞
2π
π
= ( x − y)( x + y) ⎜ x 2 − 2 xy cos + y 2⎟ ⎜ x 2 − 2 xy cos
+ y 2⎟
n
n
⎝
⎠⎝
⎠
⎛
⎞
(n − 1)π
+ y 2⎟
⎜ x 2 − 2 xy cos
n
⎝
⎠
2.23.
⎛
⎞⎛
⎞
3π
π
x 2 n + y 2 n = ⎜ x 2 + 2 xy cos + y 2⎟ ⎜ x 2 + 2 xy cos
+ y 2⎟
2n
2n
⎝
⎠⎝
⎠
⎛
⎞
(2n − 1)π
+ y 2⎟
⎜ x 2 + 2 xy cos
2n
⎝
⎠
3
THE BINOMIAL FORMULA and BINOMIAL
COEFFICIENTS
Factorial n
For n = 1, 2, 3, …, factorial n or n factorial is denoted and defined by
3.1.
n! = n(n − 1) ⋅ ⋅ 3 ⋅ 2 ⋅1
Zero factorial is defined by
3.2.
0! = 1
Alternately, n factorial can be defined recursively by
0! = 1
EXAMPLE:
and
n! = n ⋅ (n – 1)!
4! = 4 ⋅ 3 ⋅ 2 ⋅ 1 = 24,
5! = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 5 ⋅ 4! = 5(24) = 120,
6! = 6 ⋅ 5! = 6(120) = 720
Binomial Formula for Positive Integral n
For n = 1, 2, 3, …,
3.3.
( x + y)n = x n + nx n−1 y +
n(n − 1) n− 2 2 n(n − 1)(n − 2) n−3 3
x y +
x y + + yn
2!
3!
This is called the binomial formula. It can be extended to other values of n, and also to an infinite series
[see 22.4].
EXAMPLE:
(a)
(a − 2b)4 = a 4 + 4 a 3 (−2b) + 6a 2 (−2b)2 + 4 a(−2b)3 + (−2b)4 = a 4 − 8a 3 b + 24 a 2 b 2 − 32ab 3 + 16b 4
Here x = a and y = −2b.
(b)
See Fig. 3-1a.
Binomial Coefficients
Formula 3.3 can be rewritten in the form
3.4.
⎛ n⎞
⎛ n⎞
⎛ n⎞
⎛ n⎞
( x + y)n = x n + ⎜ ⎟ x n−1 y + ⎜ ⎟ x n− 2 y 2 + ⎜ ⎟ x n−3 y 3 + + ⎜ ⎟ y n
⎝1⎠
⎝ n⎠
⎝ 2⎠
⎝ 3⎠
7
THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS
8
where the coefficients, called binomial coefficients, are given by
3.5.
n!
⎛ n⎞ n(n − 1)(n − 2) (n − k + 1)
⎛ n ⎞
=
=
⎜⎝ k⎟⎠ =
k!
k !(n − k )! ⎜⎝ n − k⎟⎠
EXAMPLE:
⎛ 9⎞ 9 ⋅ 8 ⋅ 7 ⋅ 6
⎜⎝ 4⎟⎠ = 1 ⋅ 2 ⋅ 3 ⋅ 4 = 126,
⎛12⎞ 12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8
⎜⎝ 5 ⎟⎠ = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 792,
⎛10⎞ ⎛10⎞ 10 ⋅ 9 ⋅ 8
⎜⎝ 7 ⎟⎠ = ⎜⎝ 3 ⎟⎠ = 1 ⋅ 2 ⋅ 3 = 120
n
Note that ⎛⎜ ⎞⎟ has exactly r factors in both the numerator and the denominator.
⎝ r⎠
The binomial coefficients may be arranged in a triangular array of numbers, called Pascal’s triangle, as
shown in Fig. 3.1b. The triangle has the following two properties:
(1) The first and last number in each row is 1.
(2) Every other number in the array can be obtained by adding the two numbers appearing directly above
it. For example
10 = 4 + 6,
15 = 5 + 10,
20 = 10 + 10
Property (2) may be stated as follows:
3.6.
⎛ n⎞ ⎛ n ⎞ ⎛ n + 1⎞
⎜⎝ k⎟⎠ + ⎜⎝ k + 1⎟⎠ = ⎜⎝ k + 1⎟⎠
Fig. 3-1
Properties of Binomial Coefficients
The following lists additional properties of the binomial coefficients:
3.7.
⎛ n⎞ ⎛ n⎞ ⎛ n⎞
⎛ n⎞
n
⎜⎝ 0⎟⎠ + ⎜⎝1⎟⎠ + ⎜⎝ 2⎟⎠ + + ⎜⎝ n⎟⎠ = 2
3.8.
n
⎛ n⎞ ⎛ n⎞ ⎛ n⎞
n ⎛ ⎞
⎜⎝ 0⎟⎠ − ⎜⎝1⎟⎠ + ⎜⎝ 2⎟⎠ − (−1) ⎜⎝ n⎟⎠ = 0
3.9.
⎛ n⎞ ⎛ n + 1⎞ ⎛ n + 2⎞
⎛n + m⎞ ⎛ n + m + 1⎞
⎜⎝ n⎟⎠ + ⎜⎝ n ⎟⎠ + ⎜⎝ n ⎟⎠ + + ⎜⎝ n ⎟⎠ = ⎜⎝ n + 1 ⎟⎠
THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS
3.10.
⎛ n⎞ ⎛ n⎞ ⎛ n⎞
n −1
⎜⎝ 0⎟⎠ + ⎜⎝ 2⎟⎠ + ⎜⎝ 4⎟⎠ + = 2
3.11.
⎛ n⎞ ⎛ n⎞ ⎛ n⎞
n −1
⎜⎝1⎟⎠ + ⎜⎝ 3⎟⎠ + ⎜⎝ 5⎟⎠ + = 2
3.12.
⎛ n⎞ ⎛ n⎞ ⎛ n⎞
⎛ n⎞
⎛ 2n⎞
⎜⎝ 0⎟⎠ + ⎜⎝1⎟⎠ + ⎜⎝ 2⎟⎠ + + ⎜⎝ n⎟⎠ = ⎜⎝ n ⎟⎠
3.13.
⎛ m⎞ ⎛ n⎞ ⎛ m⎞ ⎛ n ⎞
⎛ m⎞ ⎛ n⎞ ⎛ m + n⎞
⎜⎝ 0 ⎟⎠ ⎜⎝ p⎟⎠ + ⎜⎝ 1 ⎟⎠ ⎜⎝ p − 1⎟⎠ + + ⎜⎝ p⎟⎠ ⎜⎝ 0⎟⎠ = ⎜⎝ p ⎟⎠
3.14.
⎛n
⎛n
⎛n
⎛n
(1) ⎜ ⎞⎟ + (2) ⎜ ⎞⎟ + (3) ⎜ ⎞⎟ + + (n) ⎜ ⎞⎟ = n 2n−1
⎝1⎠
⎝ 2⎠
⎝ 3⎠
⎝ n⎠
3.15.
⎛ n⎞
⎛n
⎛n
⎛n
(1) ⎜ ⎟ − (2) ⎜ ⎞⎟ + (3) ⎜ ⎞⎟ − (−1)n+1 (n) ⎜ ⎞⎟ = 0
⎝1⎠
⎝ n⎠
⎝ 2⎠
⎝ 3⎠
2
2
2
9
2
Multinomial Formula
Let n1, n2, …, nr be nonnegative integers such that n1 + n2 + + nr = n. Then the following expression, called
a multinomial coefficient, is defined as follows:
3.16.
⎛
n!
n
⎞
⎜⎝ n , n , … , n ⎟⎠ = n ! n ! n !
1
2
r
1
2
r
EXAMPLE:
7!
⎛ 7 ⎞
⎜⎝2, 3, 2⎟⎠ = 2! 3! 2! = 210,
8
8!
⎛
⎞
⎜⎝4, 2, 2, 0⎟⎠ = 4! 2! 2! 0! = 420
The name multinomial coefficient comes from the following formula:
3.17.
⎛
n
⎞ n n
( x1 + x 2 + + x p )n = ∑ ⎜
x x xrn
⎝ n1, n2 , … , nr ⎟⎠ 1 2
1
2
r
where the sum, denoted by Σ, is taken over all possible multinomial coefficients.
4
COMPLEX NUMBERS
Definitions Involving Complex Numbers
A complex number z is generally written in the form
z = a + bi
where a and b are real numbers and i, called the imaginary unit, has the property that i2 = −1. The real numbers a and b are called the real and imaginary parts of z = a + bi, respectively.
The complex conjugate of z is denoted by z; it is defined by
a + bi = a − bi
Thus, a + bi and a – bi are conjugates of each other.
Equality of Complex Numbers
4.1.
a + bi = c + di
if and only if
a = c and b = d
Arithmetic of Complex Numbers
Formulas for the addition, subtraction, multiplication, and division of complex numbers follow:
4.2.
(a + bi) + (c + di) = (a + c) + (b + d )i
4.3.
(a + bi) − (c + di) = (a − c) + (b − d )i
4.4.
(a + bi)(c + di) = (ac − bd ) + (ad + bc)i
4.5.
a + bi a + bi c − di ac + bd ⎛ bc − ad ⎞
=
i
=
+
i
c + di c + di c − di c 2 + d 2 ⎝ c 2 + d 2 ⎠
Note that the above operations are obtained by using the ordinary rules of algebra and replacing i2 by −1
wherever it occurs.
EXAMPLE:
Suppose z = 2 + 3i and w = 5 − 2i. Then
z + w = (2 + 3i) + (5 − 2i) = 2 + 5 + 3i − 2i = 7 + i
zw = (2 + 3i)(5 − 2i) = 10 + 15i − 4i − 6i 2 = 16 + 11i
z = 2 + 3i = 2 − 3i and w = 5 − 2i = 5 + 2i
w 5 − 2i (5 − 2i)(2 − 3i) 4 − 19i 4 19
=
=
=
=
− i
z 2 + 3i (2 + 3i)(2 − 3i)
13
13 13
10
COMPLEX NUMBERS
11
Complex Plane
Real numbers can be represented by the points on a line, called the real line, and, similarly, complex numbers can be represented by points in the plane, called the Argand diagram or Gaussian plane or, simply, the
complex plane. Specifically, we let the point (a, b) in the plane represent the complex number z = a + bi. For
example, the point P in Fig. 4-1 represents the complex number z = −3 + 4i. The complex number can also
be interpreted as a vector from the origin O to the point P.
The absolute value of a complex number z = a + bi, written | z |, is defined as follows:
4.6.
| z | = a 2 + b 2 = zz
We note | z | is the distance from the origin O to the point z in the complex plane.
Fig. 4-1
Fig. 4-2
Polar Form of Complex Numbers
The point P in Fig. 4-2 with coordinates (x, y) represents the complex number z = x + iy. The point P can
also be represented by polar coordinates (r, q ). Since x = r cos q and y = r sin q , we have
4.7.
z = x + iy = r (cos θ + i sin θ )
called the polar form of the complex number. We often call r = | z | = x 2 + y 2 the modulus and q the
amplitude of z = x + iy.
Multiplication and Division of Complex Numbers in Polar Form
4.8.
[r1 (cos θ1 + i sin θ1 )][r2 (cos θ 2 + i sin θ 2 )] = r1r2 [cos(θ1 + θ 2 ) + i sin(θ1 + θ 2 )]
4.9.
r1 (cos θ1 + i sin θ1 ) r1
= [cos (θ1 − θ 2 ) + i sin (θ1 − θ 2 )]
r2 (cos θ 2 + i sin θ 2 ) r2
De Moivre’s Theorem
For any real number p, De Moivre’s theorem states that
4.10.
[r (cos θ + i sin θ )] p = r p (cos pθ + i sin pθ )
COMPLEX NUMBERS
12
Roots of Complex Numbers
Let p = 1/n where n is any positive integer. Then 4.10 can be written
4.11.
θ + 2kπ
θ + 2kπ ⎞
⎛
[r (cos θ + i sin θ )]1/n = r 1/n ⎜ cos
+ i sin
n
n ⎟⎠
⎝
where k is any integer. From this formula, all the nth roots of a complex number can be obtained by putting
k = 0, 1, 2, …, n – 1.
5
SOLUTIONS of ALGEBRAIC EQUATIONS
Quadratic Equation: ax 2 + bx + c = 0
5.1.
x=
Solutions:
− b ± b 2 − 4 ac
2a
If a, b, c are real and if D = b2 − 4ac is the discriminant, then the roots are
(i) real and unequal if D > 0
(ii) real and equal if D = 0
(iii) complex conjugate if D < 0
5.2.
If x1, x2 are the roots, then x1 + x2 = −b/a and x1x2 = c/a.
Cubic Equation: x 3 + a1 x 2 + a2 x + a3 = 0
Let
Q=
9a a − 27a3 − 2a13
3a2 − a12
, R= 1 2
,
9
54
S = 3 R + Q3 + R2 ,
T = 3 R − Q3 + R2
where ST = – Q.
5.3. Solutions:
⎧x1 = S + T − 13 a1
⎪
⎨x 2 = − 12 (S + T ) − 13 a1 + 12 i 3 (S − T )
⎪x = − 1 (S + T ) − 1 a − 1 i 3 (S − T )
2
3 1
2
⎩ 3
If a1, a2, a3, are real and if D = Q3 + R2 is the discriminant, then
(i) one root is real and two are complex conjugate if D > 0
(ii) all roots are real and at least two are equal if D = 0
(iii) all roots are real and unequal if D < 0.
If D < 0, computation is simplified by use of trigonometry.
5.4. Solutions:
if D < 0 :
⎧x = 2 −Q cos( 13 θ ) − 13 a
1
⎪1
⎨x 2 = 2 −Q cos( 13 θ + 120° ) − 13 a1
⎪x = 2 −Q cos( 1 θ + 240° ) − 1 a
3
3
⎩ 3
where cos θ = R/ −Q 3
13
SOLUTIONS OF ALGEBRAIC EQUATIONS
14
5.5.
x1 + x 2 + x3 = − a1 , x1 x 2 + x 2 x3 + x3 x1 = a2 , x1 x 2 x3 = − a3
where x1, x2, x3 are the three roots.
Quartic Equation: x 4 + a1 x 3 + a2 x 2 + a3 x + a4 = 0
Let y1 be a real root of the following cubic equation:
5.6.
y 3 − a2 y 2 + (a1a3 − 4 a4 ) y + (4 a2 a4 − a32 − a12 a4 ) = 0
The four roots of the quartic equation are the four roots of the following equation:
5.7.
(
)
(
)
z 2 + 12 a1 ± a12 − 4 a2 + 4 y1 z + 12 y1 ∓ y12 − 4 a4 = 0
Suppose that all roots of 5.6 are real; then computation is simplified by using the particular real root that
produces all real coefficients in the quadratic equation 5.7.
5.8.
⎧x1 + x 2 + x3 + x 4 = − a1
⎪⎪x1 x 2 + x 2 x3 + x3 x 4 + x 4 x1 + x1 x3 + x 2 x 4 = a2
⎨x x x + x x x + x x x + x x x = − a
2 3 4
1 2 4
1 3 4
3
⎪1 2 3
⎪⎩x1 x 2 x3 x 4 = x 4
where x1, x2, x3, x4 are the four roots.
6
CONVERSION FACTORS
= 1000 meters (m)
= 100 centimeters (cm)
= 10−2 m
= 10−3 m
= 10−6 m
= 10−9 m
= 10−10 m
Length
1 kilometer (km)
1 meter (m)
1 centimeter (cm)
1 millimeter (mm)
1 micron (m)
1 millimicron (mm)
1 angstrom (Å)
Area
1 square meter (m2) = 10.76 ft2
1 square foot (ft2) = 929 cm2
1 inch (in)
1 foot (ft)
1 mile (mi)
1 millimeter
1 centimeter
1 meter
1 kilometer
= 2.540 cm
= 30.48 cm
= 1.609 km
= 10−3 in
= 0.3937 in
= 39.37 in
= 0.6214 mi
1 square mile (mi2) = 640 acres
1 acre
= 43,560 ft2
Volume 1 liter (l) = 1000 cm3 = 1.057 quart (qt) = 61.02 in3 = 0.03532 ft3
1 cubic meter (m3) = 1000 l = 35.32 ft3
1 cubic foot (ft3) = 7.481 U.S. gal = 0.02832 m3 = 28.32 l
1 U.S. gallon (gal) = 231 in3 = 3.785 l; 1 British gallon = 1.201 U.S. gallon = 277.4 in3
Mass
1 kilogram (kg) = 2.2046 pounds (lb) = 0.06852 slug; 1 lb = 453.6 gm = 0.03108 slug
1 slug = 32.174 lb = 14.59 kg
Speed
1 km/hr = 0.2778 m/sec = 0.6214 mi/hr = 0.9113 ft/sec
1 mi/hr = 1.467 ft/sec = 1.609 km/hr = 0.4470 m/sec
Density 1 gm/cm3 = 103 kg/m3 = 62.43 lb/ft3 = 1.940 slug/ft3
1 lb/ft3 = 0.01602 gm/cm3; 1 slug/ft3 = 0.5154 gm/cm3
Force
1 newton (nt) = 105 dynes = 0.1020 kgwt = 0.2248 lbwt
1 pound weight (lbwt) = 4.448 nt = 0.4536 kgwt = 32.17 poundals
1 kilogram weight (kgwt) = 2.205 lbwt = 9.807 nt
1 U.S. short ton = 2000 lbwt; 1 long ton = 2240 lbwt; 1 metric ton = 2205 lbwt
Energy
1 joule = 1 nt m = 107 ergs = 0.7376 ft lbwt = 0.2389 cal = 9.481 × 10−4 Btu
1 ft lbwt = 1.356 joules = 0.3239 cal = 1.285 × 10–3 Btu
1 calorie (cal) = 4.186 joules = 3.087 ft lbwt = 3.968 × 10–3 Btu
1 Btu (British thermal unit) = 778 ft lbwt = 1055 joules = 0.293 watt hr
1 kilowatt hour (kw hr) = 3.60 × 106 joules = 860.0 kcal = 3413 Btu
1 electron volt (ev) = 1.602 × 10−19 joule
Power
1 watt = 1 joule/sec = 107 ergs/sec = 0.2389 cal/sec
1 horsepower (hp) = 550 ft lbwt/sec = 33,000 ft lbwt/min = 745.7 watts
1 kilowatt (kw) = 1.341 hp = 737.6 ft lbwt/sec = 0.9483 Btu/sec
Pressure 1 nt/m2 = 10 dynes/cm2 = 9.869 × 10−6 atmosphere = 2.089 × 10−2 lbwt/ft2
1 lbwt/in2 = 6895 nt/m2 = 5.171 cm mercury = 27.68 in water
1 atm = 1.013 × 105 nt/m2 = 1.013 × 106 dynes/cm2 = 14.70 lbwt/in2
= 76 cm mercury = 406.8 in water
15
Section II: Geometry
7
GEOMETRIC FORMULAS
Rectangle of Length b and Width a
7.1. Area = ab
7.2. Perimeter = 2a + 2b
Fig. 7-1
Parallelogram of Altitude h and Base b
7.3. Area = bh = ab sin u
7.4. Perimeter = 2a + 2b
Fig. 7-2
Triangle of Altitude h and Base b
7.5. Area = 12 bh = 12 ab sin θ
= s(s − a)(s − b)(s − c)
where s = 12 (a + b + c) = semiperimeter
7.6. Perimeter = a + b + c
Fig. 7-3
Trapezoid of Altitude h and Parallel Sides a and b
7.7. Area = 12 h (a + b)
⎛ 1
1 ⎞
7.8. Perimeter = a + b + h ⎜
+
⎝ sin θ sin φ ⎟⎠
= a + b + h (csc θ + csc φ )
Fig. 7-4
16
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
GEOMETRIC FORMULAS
17
Regular Polygon of n Sides Each of Length b
7.9. Area = 14 nb 2 cot
π
cos(π / n)
= 14 nb 2
n
sin (π / n)
7.10. Perimeter = nb
Fig. 7-5
Circle of Radius r
7.11. Area = pr2
7.12. Perimeter = 2pr
Fig. 7-6
Sector of Circle of Radius r
7.13. Area = 12 r 2θ [q in radians]
7.14. Arc length s = rq
Fig. 7-7
Radius of Circle Inscribed in a Triangle of Sides a, b, c
7.15.
r=
s(s − a)(s − b)(s − c)
s
where s = 12 (a + b + c) = semiperimeter.
Fig. 7-8
Radius of Circle Circumscribing a Triangle of Sides a, b, c
7.16.
R=
abc
4 s(s − a)(s − b)(s − c)
where s = 12 (a + b + c) = semiperimeter.
Fig. 7-9
GEOMETRIC FORMULAS
18
Regular Polygon of n Sides Inscribed in Circle of Radius r
7.17. Area = 12 nr 2 sin
2π 1 2
360°
= 2 nr sin
n
n
7.18. Perimeter = 2nr sin
π
180°
= 2nr sin
n
n
Fig. 7-10
Regular Polygon of n Sides Circumscribing a Circle of Radius r
7.19. Area = nr 2 tan
π
180°
= nr 2 tan
n
n
7.20. Perimeter = 2nr tan
π
180°
= 2nr tan
n
n
Fig. 7-11
Segment of Circle of Radius r
7.21. Area of shaded part = 12 r 2 (θ − sin θ )
Fig. 7-12
Ellipse of Semi-major Axis a and Semi-minor Axis b
7.22. Area = pab
7.23. Perimeter = 4 a ∫
= 2π
π/ 2
1 − k 2 sin 2 θ dθ
0
1
2
(a 2 + b 2 ) [approximately]
where k = a 2 − b 2 /a. See Table 29 for numerical values.
Fig. 7-13
Segment of a Parabola
7.24. Area = 23 ab
7.25. Arc length ABC =
1
2
b 2 + 16a 2 +
⎛ 4 a + b 2 + 16a 2 ⎞
b2
ln ⎜
⎟
b
8a
⎝
⎠
Fig. 7-14
GEOMETRIC FORMULAS
19
Rectangular Parallelepiped of Length a, Height b, Width c
7.26. Volume = abc
7.27. Surface area = 2(ab + ac + bc)
Fig. 7-15
Parallelepiped of Cross-sectional Area A and Height h
7.28. Volume = Ah = abc sinq
Fig. 7-16
Sphere of Radius r
7.29. Volume =
4 3
πr
3
7.30. Surface area = 4πr2
Fig. 7-17
Right Circular Cylinder of Radius r and Height h
7.31. Volume = pr2h
7.32. Lateral surface area = 2prh
Fig. 7-18
Circular Cylinder of Radius r and Slant Height l
7.33. Volume = pr2h = pr2l sin u
7.34. Lateral surface area = 2π rl =
2π rh
= 2π rh csc θ
sin θ
Fig. 7-19
GEOMETRIC FORMULAS
20
Cylinder of Cross-sectional Area A and Slant Height l
7.35. Volume = Ah = Al sinq
7.36. Lateral surface area = ph = pl sinq
Note that formulas 7.31 to 7.34 are special cases of formulas 7.35 and 7.36.
Fig. 7-20
Right Circular Cone of Radius r and Height h
7.37. Volume = 13 πr 2 h
7.38. Lateral surface area = π r r 2 + h 2 = π rl
Fig. 7-21
Pyramid of Base Area A and Height h
7.39. Volume = 13 Ah
Fig. 7-22
Spherical Cap of Radius r and Height h
7.40. Volume (shaded in figure) = 13 π h 2 (3r − h)
7.41. Surface area = 2p rh
Fig. 7-23
Frustum of Right Circular Cone of Radii a, b and Height h
7.42. Volume = 13 π h(a 2 + ab + b 2 )
7.43. Lateral surface area = π (a + b) h 2 + (b − a)2
= p (a + b)l
Fig. 7-24
GEOMETRIC FORMULAS
21
Spherical Triangle of Angles A, B, C on Sphere of Radius r
7.44. Area of triangle ABC = (A + B + C − p)r 2
Fig. 7-25
Torus of Inner Radius a and Outer Radius b
7.45. Volume = 14 π 2 (a + b)(b − a)2
7.46. Surface area = p 2(b2 − a2)
Fig. 7-26
Ellipsoid of Semi-axes a, b, c
7.47. Volume = 43 π abc
Fig. 7-27
Paraboloid of Revolution
7.48. Volume = 12 π b 2 a
Fig. 7-28
FORMULAS from PLANE ANALYTIC
GEOMETRY
8
Distance d Between Two Points P1(x1, y1) and P2(x2, y2)
8.1. d = ( x 2 − x1 )2 + ( y2 − y1 )2
Fig. 8-1
Slope m of Line Joining Two Points P1(x1, y1) and P2(x2, y2)
8.2.
m=
y2 − y1
= tan θ
x 2 − x1
Equation of Line Joining Two Points P1(x1, y1) and P2(x2, y2)
8.3.
y − y1 y2 − y1
=
= m or
x − x1 x 2 − x1
y − y1 = m( x − x1 )
8.4. y = mx + b
where b = y1 − mx1 =
x 2 y1 − x1 y2
is the intercept on the y axis, i.e., the y intercept.
x 2 − x1
Equation of Line in Terms of x Intercept a ≠ 0 and y Intercept b ≠ 0
8.5.
x
y
+ =1
a b
Fig. 8-2
22
FORMULAS FROM PLANE ANALYTIC GEOMETRY
23
Normal Form for Equation of Line
8.6. x cos a + y sin a = p
where p = perpendicular distance from origin O to line
and a = angle of inclination of perpendicular
with positive x axis.
Fig. 8-3
General Equation of Line
8.7. Ax + By + C = 0
Distance from Point (x1, y1) to Line Ax + By + C = 0
8.8.
Ax1 + By1 + C
± A2 + B 2
where the sign is chosen so that the distance is nonnegative.
Angle x Between Two Lines Having Slopes m1 and m2
8.9.
tan ψ =
m2 − m1
1 + m1m2
Lines are parallel or coincident if and only if m1 = m2.
Lines are perpendicular if and only if m2 = −1/m1.
Fig. 8-4
Area of Triangle with Vertices at (x1, y1), (x2, y2), (x3, y3)
8.10. Area = ±
x
1 1
x2
2
x3
y1 1
y2 1
y3 1
1
= ± ( x1 y2 + y1 x3 + y3 x 2 − y2 x3 − y1 x 2 − x1 y3 )
2
where the sign is chosen so that the area is nonnegative.
If the area is zero, the points all lie on a line.
Fig. 8-5
FORMULAS FROM PLANE ANALYTIC GEOMETRY
24
Transformation of Coordinates Involving Pure Translation
8.11.
⎧x = x ′ + x 0
⎨y = y ′ + y
0
⎩
or
⎧x ′ = x − x 0
⎨y ′ = y − y
0
⎩
where (x, y) are old coordinates (i.e., coordinates relative to xy
system), (x′, y′) are new coordinates (relative to x′, y′ system),
and (x0, y0) are the coordinates of the new origin O′ relative
to the old xy coordinate system.
Fig. 8-6
Transformation of Coordinates Involving Pure Rotation
8.12.
{
x = x ′ cos α − y ′ sin α
y = x ′ sin α + y ′ cos α
or
{
x ′ = x cos α + y sin α
y ′ = y cos α − x sin α
where the origins of the old [xy] and new [x′y′] coordinate
systems are the same but the x′ axis makes an angle a with
the positive x axis.
Fig. 8-7
Transformation of Coordinates Involving Translation and Rotation
⎧x = x ′ cos α − y ′ sin α + x 0
⎨ y = x ′ sin α + y ′ cos α + y
0
⎩
8.13.
or
⎧x ′ = ( x − x 0 ) cos α + ( y − y0 ) sin α
⎨ y ′ = ( y − y ) cos α − ( x − x ) sin α
0
0
⎩
where the new origin O′ of x′y′ coordinate system has
coordinates (x0, y0) relative to the old xy coordinate
system and the x′ axis makes an angle a with the
positive x axis.
Fig. 8-8
Polar Coordinates (r, p )
A point P can be located by rectangular coordinates (x, y) or polar
coordinates (r, u). The transformation between these coordinates is
as follows:
8.14.
{
x = r cos θ
y = r sin θ
or
⎧ r = x 2 + y2
⎨
−1
⎩ θ = tan ( y / x)
Fig. 8-9
FORMULAS FROM PLANE ANALYTIC GEOMETRY
25
Equation of Circle of Radius R, Center at (x0, y0)
8.15.
(x − x0)2 + (y − y0)2 = R2
Fig. 8-10
Equation of Circle of Radius R Passing Through Origin
8.16. r = 2R cos(u − a)
where (r, u) are polar coordinates of any point on the
circle and (R, a) are polar coordinates of the center of
the circle.
Fig. 8-11
Conics (Ellipse, Parabola, or Hyperbola)
If a point P moves so that its distance from a fixed point
(called the focus) divided by its distance from a fixed line
(called the directrix) is a constant ⑀ (called the eccentricity),
then the curve described by P is called a conic (so-called
because such curves can be obtained by intersecting a
plane and a cone at different angles).
If the focus is chosen at origin O, the equation of a conic in
polar coordinates (r, u) is, if OQ = p and LM = D (see Fig. 8-12),
8.17.
r =
D
p
=
1 − cos θ 1 − cos θ
The conic is
(i) an ellipse if ⑀ < 1
(ii) a parabola if ⑀ = 1
(iii) a hyperbola if ⑀ > 1
Fig. 8-12
FORMULAS FROM PLANE ANALYTIC GEOMETRY
26
Ellipse with Center C(x0, y0) and Major Axis Parallel to x Axis
8.18. Length of major axis A′A = 2a
8.19. Length of minor axis B′B = 2b
8.20. Distance from center C to focus F or F′ is
c = a2 − b2
8.21. Eccentricity = =
c
a2 − b2
=
a
a
Fig. 8-13
8.22. Equation in rectangular coordinates:
( x − x 0 )2 ( y − y0 )2
+
=1
a2
b2
8.23. Equation in polar coordinates if C is at O: r 2 =
a2b2
a sin θ + b 2 cos 2 θ
2
2
8-24. Equation in polar coordinates if C is on x axis and F′ is at O: r =
a (1 − 2 )
1 − cos θ
8.25. If P is any point on the ellipse, PF + PF′ = 2a
If the major axis is parallel to the y axis, interchange x and y in the above or replace u by 12 π − θ (or 90° − u).
Parabola with Axis Parallel to x Axis
If vertex is at A (x0, y0) and the distance from A to focus F is a > 0, the equation of the parabola is
8.26.
(y − y0)2 = 4a(x − x0)
if parabola opens to right (Fig. 8-14)
8.27.
(y − y0)2 = −4a(x − x0)
if parabola opens to left (Fig. 8-15)
If focus is at the origin (Fig. 8-16), the equation in polar coordinates is
8.28.
r =
2a
1 − cos θ
Fig. 8-14
Fig. 8-15
In case the axis is parallel to the y axis, interchange x and y or replace u by
Fig. 8-16
1
2
π − θ (or 90° − u).
FORMULAS FROM PLANE ANALYTIC GEOMETRY
27
Hyperbola with Center C(x0, y0) and Major Axis Parallel to x Axis
Fig. 8-17
8.29. Length of major axis A′A = 2a
8.30. Length of minor axis B′B = 2b
8.31. Distance from center C to focus F or F ′ = c = a 2 + b 2
8.32. Eccentricity =
c
=
a
a2 + b2
a
8.33. Equation in rectangular coordinates:
( x − x 0 )2 ( y − y0 )2
−
=1
a2
b2
8.34. Slopes of asymptotes G′H and GH ′ = ±
b
a
8.35. Equation in polar coordinates if C is at O: r 2 =
a2b2
b cos θ − a 2 sin 2 θ
2
2
8.36. Equation in polar coordinates if C is on x axis and F′ is at O: r =
a( 2 − 1)
1 − cos θ
8.37. If P is any point on the hyperbola, PF − PF′ = ±2a (depending on branch)
If the major axis is parallel to the y axis, interchange x and y in the above or replace u by
(or 90° − u).
1
2
π −θ
9
SPECIAL PLANE CURVES
Lemniscate
9.1. Equation in polar coordinates:
r2 = a2 cos 2u
9.2. Equation in rectangular coordinates:
(x2 + y2)2 = a2(x2 − y2)
9.3. Angle between AB′ or A′B and x axis = 45°
Fig. 9-1
9.4. Area of one loop = a
2
Cycloid
9.5. Equations in parametric form:
{
x = a(φ − sin φ )
y = a(1 − cos φ )
9.6. Area of one arch = 3πa2
9.7. Arc length of one arch = 8a
This is a curve described by a point P on a circle of radius a
rolling along x axis.
Fig. 9-2
Hypocycloid with Four Cusps
9.8. Equation in rectangular coordinates:
x2/3 + y2/3 = a2/3
9.9. Equations in parametric form:
⎧x = a cos3 θ
⎨ y = a sin 3 θ
⎩
9.10. Area bounded by curve = 83 π a 2
9.11. Arc length of entire curve = 6a
This is a curve described by a point P on a circle of radius
a/4 as it rolls on the inside of a circle of radius a.
28
Fig. 9-3
SPECIAL PLANE CURVES
29
Cardioid
9.12. Equation: r = 2a(1 + cos u)
9.13. Area bounded by curve = 6pa2
9.14. Arc length of curve = 16a
This is the curve described by a point P of a circle of radius a
as it rolls on the outside of a fixed circle of radius a. The curve
is also a special case of the limacon of Pascal (see 9.32).
Fig. 9-4
Catenary
9.15. Equation: y =
a x /a − x /a
x
(e + e ) = a cos h
2
a
This is the curve in which a heavy uniform chain would hang if
suspended vertically from fixed points A and B.
Fig. 9-5
Three-Leaved Rose
9.16. Equation: r = a cos 3u
The equation r = a sin 3u is a similar curve obtained by
rotating the curve of Fig. 9-6 counterclockwise through 30°
or p/6 radians.
In general, r = a cos nu or r = a sin nu has n leaves if
n is odd.
Fig. 9-6
Four-Leaved Rose
9.17. Equation: r = a cos 2u
The equation r = a sin 2u is a similar curve obtained by
rotating the curve of Fig. 9-7 counterclockwise through 45°
or p/4 radians.
In general, r = a cos nu or r = a sin nu has 2n leaves if n is even.
Fig. 9-7
SPECIAL PLANE CURVES
30
Epicycloid
9.18. Parametric equations:
⎧
⎛ a + b⎞
⎪⎪x = (a + b) cosθ − b cos ⎜⎝ b ⎟⎠ θ
⎨
⎪y = (a + b)sin θ − b sin ⎛⎜ a + b⎞⎟ θ
⎝ b ⎠
⎪⎩
This is the curve described by a point P on a circle of radius b
as it rolls on the outside of a circle of radius a.
The cardioid (Fig. 9-4) is a special case of an epicycloid.
Fig. 9-8
General Hypocycloid
9.19. Parametric equations:
⎧
⎛ a − b⎞
φ
⎪x = (a − b) cos φ + b cos ⎜
⎝ b ⎟⎠
⎪
⎨
⎛ a − b⎞
⎪
⎪y = (a − b) sin φ − b sin ⎜⎝ b ⎟⎠ φ
⎩
This is the curve described by a point P on a circle of radius b
as it rolls on the inside of a circle of radius a.
If b = a/4, the curve is that of Fig. 9-3.
Fig. 9-9
Trochoid
9.20. Parametric equations:
{
x = aφ − b sin φ
y = a − b cos φ
This is the curve described by a point P at distance b from the center of a circle of radius a as the circle
rolls on the x axis.
If b < a, the curve is as shown in Fig. 9-10 and is called a curtate cycloid.
If b > a, the curve is as shown in Fig. 9-11 and is called a prolate cycloid.
If b = a, the curve is the cycloid of Fig. 9-2.
Fig. 9-10
Fig. 9-11
SPECIAL PLANE CURVES
31
Tractrix
⎧x = a(ln cot 12 φ − cos φ )
9.21. Parametric equations: ⎨
⎩y = a sin φ
This is the curve described by endpoint P of a taut string
PQ of length a as the other end Q is moved along the x axis.
Fig. 9-12
Witch of Agnesi
9.22. Equation in rectangular coordinates: y =
8a 3
x 2 + 4a 2
⎧x = 2a cot θ
9.23. Parametric equations: ⎨
⎩y = a(1 − cos 2θ )
In Fig. 9-13 the variable line QA intersects y = 2a and the
circle of radius a with center (0, a) at A and B, respectively.
Any point P on the “witch” is located by constructing lines
parallel to the x and y axes through B and A, respectively, and
determining the point P of intersection.
Fig. 9-13
Folium of Descartes
9.24. Equation in rectangular coordinates:
x3 + y3 = 3axy
9.25. Parametric equations:
⎧
3at
⎪⎪x = 1 + t 3
⎨
2
⎪y = 3at
1 + t3
⎪⎩
9.26. Area of loop =
3 2
a
2
Fig. 9-14
9.27. Equation of asymptote: x + y + a = 0
Involute of a Circle
9.28. Parametric equations:
⎧x = a(cos φ + φ sin φ )
⎨
⎩y = a(sin φ − φ cos φ )
This is the curve described by the endpoint P of a string
as it unwinds from a circle of radius a while held taut.
Fig. 9-15
SPECIAL PLANE CURVES
32
Evolute of an Ellipse
9.29. Equation in rectangular coordinates:
(ax)2/3 + (by)2/3 = (a2 − b2)2/3
9.30. Parametric equations:
⎧ax = (a 2 − b 2 ) cos3 θ
⎨
2
2
3
⎩by = (a − b ) sin θ
This curve is the envelope of the normals to the ellipse
x2/a2 + y2/b2 = 1 shown dashed in Fig. 9-16.
Fig. 9-16
Ovals of Cassini
9.31. Polar equation: r4 + a4 − 2a2r2 cos 2u = b4
This is the curve described by a point P such that the product of its distance from two fixed points
(distance 2a apart) is a constant b2.
The curve is as in Fig. 9-17 or Fig. 9-18 according as b < a or b > a, respectively.
If b = a, the curve is a lemniscate (Fig. 9-1).
Fig. 9-17
Fig. 9-18
Limacon of Pascal
9.32. Polar equation: r = b + a cos u
Let OQ be a line joining origin O to any point Q on a circle of diameter a passing through O. Then the
curve is the locus of all points P such that PQ = b.
The curve is as in Fig. 9-19 or Fig. 9-20 according as 2a > b > a or b < a, respectively. If b = a, the curve
is a cardioid (Fig. 9-4). If b 2a, the curve is convex.
Fig. 9-19
Fig. 9-20
SPECIAL PLANE CURVES
33
Cissoid of Diocles
9.33. Equation in rectangular coordinates:
y2 =
x2
2a − x
9.34. Parametric equations:
⎧x = 2a sin 2 θ
⎪
2a sin 3 θ
⎨
y
=
⎪⎩
cos θ
This is the curve described by a point P such that the
distance OP = distance RS. It is used in the problem of
duplication of a cube, i.e., finding the side of a cube
which has twice the volume of a given cube.
Fig. 9-21
Spiral of Archimedes
9.35. Polar equation: r = au
Fig. 9-22
FORMULAS from SOLID ANALYTIC
GEOMETRY
10
Distance d Between Two Points P1(x1, y1, z1) and P2(x2, y2, z2)
10.1.
d = ( x 2 − x1 )2 + ( y2 − y1 )2 + (z 2 − z1 )2
z
d
γ
P2 (x2, y2, z2)
β
P1 (x1, y1, z1)
α
y
O
x
Fig. 10-1
Direction Cosines of Line Joining Points P1(x1, y1, z1) and P2(x2, y2, z2)
10.2.
l = cos α =
x 2 − x1
y − y1
z − z1
, m = cos β = 2
, n = cos γ = 2
d
d
d
where a, b, g are the angles that line P1P2 makes with the positive x, y, z axes, respectively, and d
is given by 10.1 (see Fig. 10-1).
Relationship Between Direction Cosines
10.3.
cos 2 α + cos 2 β + cos 2 γ = 1 or l 2 + m 2 + n 2 = 1
Direction Numbers
Numbers L, M, N, which are proportional to the direction cosines l, m, n, are called direction numbers. The
relationship between them is given by
10.4.
l=
L
L +M +N
2
2
2
, m=
M
L +M +N
2
2
2
, n=
N
L + M2 + N2
2
Equations of Line Joining P1(x1, y1, z1) and P2(x2, y2, z2) in Standard Form
10.5.
x − x1
y − y1
z − z1
=
=
x 2 − x1
y2 − y1
z 2 − z1
or
x − x1
y − y1
z − z1
=
=
l
m
n
These are also valid if l, m, n are replaced by L, M, N, respectively.
34
FORMULAS FROM SOLID ANALYTIC GEOMETRY
35
Equations of Line Joining P1(x1, y1, z1) and P2(x2, y2, z2) in Parametric Form
x = x1 + lt , y = y1 + mt , z = z1 + nt
10.6.
These are also valid if l, m, n are replaced by L, M, N, respectively.
Angle e Between Two Lines with Direction Cosines l1, m1, n1 and l2, m2, n2
cos φ = l1l2 + m1m2 + n1n2
10.7.
General Equation of a Plane
Ax + By + Cz + D = 0
10.8.
(A, B, C, D are constants)
Equation of Plane Passing Through Points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3)
x − x1 y − y1 z − z1
x 2 − x1 y2 − y1 z2 − z1 = 0
x3 − x1 y3 − y1 z3 − z1
10.9.
or
10.10.
y2 − y1
y3 − y1
z 2 − z1
z −z
( x − x1 ) + 2 1
z3 − z1
z3 − z1
x 2 − x1
x − x1
( y − y1 ) + 2
x3 − x1
x3 − x1
y2 − y1
(z − z1 ) = 0
y3 − y1
Equation of Plane in Intercept Form
10.11.
x
y z
+ + =1
a b c
z
where a, b, c are the intercepts on the x, y, z axes, respectively.
c
b
a
O
x
Fig. 10-2
Equations of Line Through (x0, y0, z0) and Perpendicular
to Plane Ax + By + Cz + D = 0
10.12.
x − x0
y − y0
z − z0
=
=
A
B
C
or
x = x 0 + At , y = y0 + Bt , z = z0 + Ct
Note that the direction numbers for a line perpendicular to the plane Ax + By + Cz + D = 0 are
A, B, C.
y
FORMULAS FROM SOLID ANALYTIC GEOMETRY
36
Distance from Point (x0, y0, z0) to Plane Ax + By + Cz + D = 0
10.13.
Ax 0 + By0 + Cz 0 + D
± A2 + B 2 + C 2
where the sign is chosen so that the distance is nonnegative.
Normal Form for Equation of Plane
10.14.
x cos α + y cos β + z cos γ = p
z
where p = perpendicular distance from O to plane at P
and a, b, g are angles between OP and positive x, y, z axes.
γ
α
P
p
β
y
O
x
Fig. 10-3
Transformation of Coordinates Involving Pure Translation
10.15.
⎧x = x ′ + x 0
⎪
⎨y = y ′ + y0
⎪z = z′ + z
0
⎩
or
⎧x ′ = x − x 0
⎪
⎨ y ′ = y − y0
⎪z′ = z − z
0
⎩
z
z'
(x0, y0, z0)
y'
O'
where (x, y, z) are old coordinates (i.e., coordinates relative
to xyz system), (x′, y′, z′) are new coordinates (relative to
x′y′z′ system) and (x0, y0, z0) are the coordinates of the
new origin O′ relative to the old xyz coordinate system.
x'
y
O
x
Fig. 10-4
Transformation of Coordinates Involving Pure Rotation
10.16.
⎧x = l1x ′ + l2 y′ + l3 z ′
⎪
⎨y = m1x ′ + m2 y′ + m3 z ′
⎪⎩z = n1x ′ + n2 y ′ + n3 z ′
z
z'
y'
⎧x ′ = l1x + m1 y + n1z
⎪
or ⎨y′ = l2 x + m2 y + n2 z
⎪⎩z ′ = l3 x + m3 y + n3 z
where the origins of the xyz and x′y′z′ systems are
the same and l1, m1, n1; l2, m2, n2; l3, m3, n3 are the direction
cosines of the x′,y′,z′ axes relative to the x, y, z axes,
respectively.
y
O
x
x'
Fig. 10-5
FORMULAS FROM SOLID ANALYTIC GEOMETRY
37
Transformation of Coordinates Involving Translation and Rotation
10.17.
⎧x = l1x ′ + l2 y′ + l3 z ′ + x 0
⎪
⎨y = m1x ′ + m2 y′ + m3 z ′ + y0
⎪⎩z = n1x ′ + n2 y′ + n3 z ′ + z 0
z
⎧x ′ = l1 ( x − x 0 ) + m1 ( y − y0 ) + n1 (z − z0 )
⎪
or ⎨y′ = l2 ( x − x 0 ) + m2 ( y − y0 ) + n2 (z − z 0 )
⎪⎩z ′ = l3 ( x − x 0 ) + m3 ( y − y0 ) + n3 (z − z 0 )
y'
O' (x0 , y0 , z0)
y
O
x'
where the origin O′ of the x′y′z′ system has coordinates
(x0, y0, z0) relative to the xyz system and
l1 , m1 , n1; l2 , m2 , n2 ; l3 , m3 , n3
z'
x
Fig. 10-6
are the direction cosines of the x′, y′, z′ axes relative to
the x, y, z axes, respectively.
Cylindrical Coordinates (r, p, z)
A point P can be located by cylindrical coordinates (r, u, z)
(see Fig. 10-7) as well as rectangular coordinates (x, y, z).
The transformation between these coordinates is
10.18.
⎧r = x 2 + y 2
⎧x = r cos θ
⎪
⎪
−1
⎨y = r sin θ or ⎨θ = tan ( y / x )
⎪⎩z = z
⎪z = z
⎩
Fig. 10-7
Spherical Coordinates (r, p, e )
A point P can be located by spherical coordinates (r, u, f)
(see Fig. 10-8) as well as rectangular coordinates (x, y, z).
The transformation between those coordinates is
10.19.
⎧x = r sin θ cos φ
⎪
⎨y = r sin θ sin φ
⎪⎩z = r cos θ
⎧r = x 2 + y 2 + z 2
⎪
or ⎨φ = tan −1 ( y / x )
⎪θ = cos −1 (z / x 2 + y 2 + z 2 )
⎩
Fig. 10-8
FORMULAS FROM SOLID ANALYTIC GEOMETRY
38
Equation of Sphere in Rectangular Coordinates
10.20.
( x − x 0 )2 + ( y − y0 )2 + (z − z0 )2 = R 2
where the sphere has center (x0, y0, z0) and radius R.
Fig. 10-9
Equation of Sphere in Cylindrical Coordinates
10.21.
r 2 − 2r0 r cos(θ − θ 0 ) + r02 + (z − z 0 )2 = R 2
where the sphere has center (r0, u0, z0) in cylindrical coordinates and radius R.
If the center is at the origin the equation is
10.22.
r 2 + z 2 = R2
Equation of Sphere in Spherical Coordinates
10.23.
r 2 + r02 − 2r0 r sin θ sin θ 0 cos(φ − φ0 ) = R 2
where the sphere has center (r0, u0, f0) in spherical coordinates and radius R.
If the center is at the origin the equation is
10.24. r = R
Equation of Ellipsoid with Center (x0, y0, z0) and Semi-axes a, b, c
10.25.
( x − x 0 )2 ( y − y0 )2 (z − z 0 )2
+
+
=1
a2
b2
c2
Fig. 10-10
FORMULAS FROM SOLID ANALYTIC GEOMETRY
39
Elliptic Cylinder with Axis as z Axis
10.26.
x 2 y2
+
=1
a2 b2
where a, b are semi-axes of elliptic cross-section.
If b = a it becomes a circular cylinder of radius a.
Fig. 10-11
Elliptic Cone with Axis as z Axis
10.27.
x 2 y2 z 2
+
=
a2 b2 c2
Fig. 10-12
Hyperboloid of One Sheet
10.28.
x 2 y2 z 2
+ − =1
a2 b2 c2
Fig. 10-13
Hyperboloid of Two Sheets
10.29.
x 2 y2 z 2
− − =1
a2 b2 c2
Note orientation of axes in Fig. 10-14.
Fig. 10-14
FORMULAS FROM SOLID ANALYTIC GEOMETRY
40
Elliptic Paraboloid
10.30.
x 2 y2 z
+ =
a2 b2 c
Fig. 10-15
Hyperbolic Paraboloid
10.31.
x 2 y2 z
− =
a2 b2 c
Note orientation of axes in Fig. 10-16.
Fig. 10-16
11
SPECIAL MOMENTS of INERTIA
The table below shows the moments of inertia of various rigid bodies of mass M. In all cases it is assumed
the body has uniform (i.e., constant) density.
TYPE OF RIGID BODY
MOMENT OF INERTIA
11.1. Thin rod of length a
(a)
about axis perpendicular to the rod through the center
of mass
1
12
Ma 2
(b)
about axis perpendicular to the rod through one end
1
3
Ma 2
11.2.
Rectangular parallelepiped with sides a, b, c
(a)
about axis parallel to c and through center of face ab
(b)
about axis through center of face bc and parallel to c
1
12
M (a 2 + b 2 )
M (4 a 2 + b 2 )
1
12
11.3. Thin rectangular plate with sides a, b
M (a 2 + b 2 )
(a)
about axis perpendicular to the plate through center
(b)
about axis parallel to side b through center
11.4.
Circular cylinder of radius a and height h
(a)
about axis of cylinder
(b)
about axis through center of mass and perpendicular to
cylindrical axis
(c)
about axis coinciding with diameter at one end
11.5.
Hollow circular cylinder of outer radius a, inner radius
b and height h
(a)
about axis of cylinder
(b)
about axis through center of mass and perpendicular to
cylindrical axis
(c)
about axis coinciding with diameter at one end
11.6.
Circular plate of radius a
(a)
about axis perpendicular to plate through center
1
2
Ma 2
(b)
about axis coinciding with a diameter
1
4
Ma 2
1
12
1
12
1
2
Ma 2
M (4 h 2 + 3a 2 )
1
2
1
12
Ma 2
M (h 2 + 3a 2 )
1
12
1
12
1
12
M (a 2 + b 2 )
M (3a 2 + 3b 2 + h 2 )
M (3a 2 + 3b 2 + 4 h 2 )
41
SPECIAL MOMENTS OF INERTIA
42
11.7.
Hollow circular plate or ring with outer radius a and
inner radius b
(a)
about axis perpendicular to plane of plate through center
1
2
M (a 2 + b 2 )
(b)
about axis coinciding with a diameter
1
4
M (a 2 + b 2 )
11.8.
Thin circular ring of radius a
(a)
about axis perpendicular to plane of ring through center
(b)
about axis coinciding with diameter
11.9.
Sphere of radius a
(a)
Ma2
1
2
Ma 2
about axis coinciding with a diameter
2
5
Ma 2
(b)
about axis tangent to the surface
7
5
Ma 2
11.10.
Hollow sphere of outer radius a and inner radius b
(a)
about axis coinciding with a diameter
(b)
about axis tangent to the surface
11.11.
Hollow spherical shell of radius a
(a)
about axis coinciding with a diameter
2
3
Ma 2
(b)
about axis tangent to the surface
5
3
Ma 2
11.12.
Ellipsoid with semi-axes a, b, c
(a)
about axis coinciding with semi-axis c
(b)
about axis tangent to surface, parallel to semi-axis c
and at distance a from center
M (a 5 − b 5 )/(a 3 − b 3 )
5
5
3
3
2
2
5 M (a − b )/(a − b ) + Ma
2
5
1
5
1
5
M (a 2 + b 2 )
M (6a 2 + b 2 )
11.13. Circular cone of radius a and height h
3
10
Ma 2
(a)
about axis of cone
(b)
about axis through vertex and perpendicular to axis
3
20
M (a 2 + 4 h 2 )
(c)
about axis through center of mass and perpendicular
to axis
3
80
M (4 a 2 + h 2 )
11.14. Torus with outer radius a and inner radius b
(a)
about axis through center of mass and perpendicular
to the plane of torus
(b)
about axis through center of mass and in the plane of
torus
1
4
1
4
M (7a 2 − 6ab + 3b 2 )
M (9a 2 − 10 ab + 5b 2 )
Section III: Elementary Transcendental Functions
12
TRIGONOMETRIC FUNCTIONS
Definition of Trigonometric Functions for a Right Triangle
Triangle ABC has a right angle (90°) at C and sides of length a, b, c. The trigonometric functions of angle
A are defined as follows:
12.1. sine of A = sin A =
a
opposite
=
c hypotenuse
12.2. cosine of A = cos A = b = adjacent
c hypotenuse
12.3. tangent of A = tan A = a = opposite
b adjacent
12.4. cotangent of A = cot A =
12.5. secant of A = sec A =
b adjacent
=
a opposite
c hypotenuse
=
b
adjacent
12.6. cosecant of A = csc A =
Fig. 12-1
c hypotenuse
=
a
opposite
Extensions to Angles Which May be Greater Than 90°
Consider an xy coordinate system (see Figs. 12-2 and 12-3). A point P in the xy plane has coordinates (x, y) where
x is considered as positive along OX and negative along OX′ while y is positive along OY and negative along OY′.
The distance from origin O to point P is positive and denoted by r = x 2 + y 2 . The angle A described counterclockwise from OX is considered positive. If it is described clockwise from OX it is considered negative. We call
X′OX and Y′OY the x and y axis, respectively.
The various quadrants are denoted by I, II, III, and IV called the first, second, third, and fourth quadrants,
respectively. In Fig. 12-2, for example, angle A is in the second quadrant while in Fig. 12-3 angle A is in the
third quadrant.
Fig. 12-2
Fig. 12-3
43
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
TRIGONOMETRIC FUNCTIONS
44
For an angle A in any quadrant, the trigonometric functions of A are defined as follows.
12.7. sin A = y/r
12.8. cos A = x/r
12.9. tan A = y/x
12.10. cot A = x/y
12.11. sec A = r/x
12.12. csc A = r/y
Relationship Between Degrees and Radians
A radian is that angle q subtended at center O of a circle by an arc
MN equal to the radius r.
Since 2p radians = 360° we have
12.13.
1 radian = 180°/p = 57.29577 95130 8232 …°
12.14.
1° = p/180 radians = 0.01745 32925 19943 29576 92 … radians
Fig. 12-4
Relationships Among Trigonometric Functions
12.15.
tan A =
sin A
cos A
12.19.
sin 2 A + cos 2 A = 1
12.16.
cot A =
1
cos A
=
tan A sin A
12.20.
sec 2 A − tan 2 A = 1
12.17.
sec A =
1
cos A
12.21.
csc 2 A − cot 2 A = 1
12.18.
csc A =
1
sin A
Signs and Variations of Trigonometric Functions
Quadrant
I
II
III
IV
sin A
+
0 to 1
+
1 to 0
–
0 to –1
–
–1 to 0
cos A
+
1 to 0
–
0 to –1
–
–1 to 0
+
0 to 1
tan A
+
0 to ∞
–
–∞ to 0
+
0 to ∞
–
–∞ to 0
cot A
+
∞ to 0
–
0 to –∞
+
∞ to 0
–
0 to –∞
sec A
+
1 to ∞
–
–∞ to –1
–
–1 to –∞
+
∞ to 1
csc A
+
∞ to 1
+
1 to ∞
–
–∞ to –1
–
–1 to –∞
TRIGONOMETRIC FUNCTIONS
45
Exact Values for Trigonometric Functions of Various Angles
Angle A
in degrees
Angle A
in radians
sin A
cos A
tan A
cot A
sec A
csc A
0°
0
0
1
0
∞
1
∞
15°
p/12
2− 3
2+ 3
6− 2
6+ 2
30°
p/6
45°
p/4
1
2
2
60°
p/3
1
2
3
75°
5p/12
90°
p/2
105°
7p/12
120°
2p/3
1
2
3
− 12
− 3
− 13 3
–2
135°
3p/4
1
2
2
− 12 2
–1
–1
− 2
2
150°
5p/6
− 12 3
− 13 3
− 3
− 23 3
2
165°
11p/12
180°
p
195°
13p/12
210°
7p/6
− 12
− 12 3
225°
5p/4
− 12 2
− 12 2
1
240°
4p/3
− 12 3
− 12
3
255°
17p/12
270°
3p/2
–1
285°
19p/12
− 14 ( 6 + 2 )
300°
5p/3
− 12 3
315°
7p/4
− 12 2
1
2
330°
11p/6
− 12
1
2
345°
23p/12
− 14 ( 6 − 2 )
360°
2p
0
1
4
( 6 − 2)
1
4
1
2
1
4
( 6 + 2)
( 6 + 2)
1
2
3
1
2
2
1
4
( 6 − 2)
3
2
3
1
3
0
1
3
3
3
− 14 ( 6 + 2 ) − 14 ( 6 − 2 )
For other angles see Tables 2, 3, and 4.
0
1
4
2− 3
6+ 2
6− 2
±∞
0
±∞
1
0
−
+∞
2− 3
2+ 3
1
3
3
2
3
3
6+ 2
±∞
− ( 6 − 2) − ( 6 + 2)
− 23 3
–2
1
− 2
− 2
–2
− 23 3
1
3
3
2+ 3
2− 3
±∞
0
− ( 6 + 2) − ( 6 − 2)
−
+∞
–1
6+ 2
− ( 6 − 2)
− 3
− 13 3
2
− 23 3
2
–1
–1
2
− 2
3
− 13 3
− 3
( 6 + 2 ) − (2 − 3 ) − (2 + 3 )
1
–1
6− 2
3
( 6 − 2 ) − (2 + 3 ) − (2 − 3 )
1
2
1
4
3
2+ 3
–1
− 14 ( 6 − 2 ) − 14 ( 6 + 2 )
2
2
3
2
( 6 − 2 ) − 14 ( 6 + 2 ) − (2 − 3 ) − (2 + 3 ) − ( 6 − 2 )
0
2
2
( 6 + 2 ) − 14 ( 6 − 2 ) − (2 + 3 ) − (2 − 3 ) − ( 6 + 2 )
1
2
1
4
3
1
1
2
1
1
4
1
3
0
−
+∞
2
3
3
–2
6− 2
− ( 6 + 2)
1
−
+∞
TRIGONOMETRIC FUNCTIONS
46
Graphs of Trigonometric Functions
In each graph x is in radians.
12.22. y = sin x
12.23.
y = cos x
Fig. 12-5
Fig. 12-6
12.24. y = tan x
12.25.
y = cot x
Fig. 12-7
Fig. 12-8
12.26. y = sec x
12.27. y = csc x
Fig. 12-9
Fig. 12-10
Functions of Negative Angles
12.28. sin(–A) = – sin A
12.29.
cos(–A) = cos A
12.30.
tan(–A) = – tan A
12.31. csc(–A) = – csc A
12.32.
sec(–A) = sec A
12.33.
cot(–A) = – cot A
TRIGONOMETRIC FUNCTIONS
47
Addition Formulas
12.34. sin (A ± B) = sin A cos B ± cos A sin B
12.35. cos (A ± B) = cos A cos B −
+ sin A sin B
12.36.
tan ( A ± B) =
12.37.
cot ( A ± B) =
tan A ± tan B
− tan A tan B
1+
cot A cot B −
+1
cot B ± cot A
Functions of Angles in All Quadrants in Terms of Those in Quadrant I
–A
sin
– sin A
cos
cos A
tan
– tan A
csc
– csc A
sec
sec A
cot
– cot A
90° ± A
π
±A
2
180° ± A
p±A
cos A
sin A
−
+ sin A
−
+ cot A
– cos A
± tan A
−
+ csc A
sec A
−
+ csc A
−
+ tan A
– sec A
± cot A
270° ± A
3π
±A
2
k(360°) ± A
2kp ± A
k = integer
– cos A
−
+ sin A
± sin A
−
+ cot A
± tan A
– sec A
± csc A
± csc A
−
+ tan A
sec A
cos A
± cot A
Relationships Among Functions of Angles in Quadrant I
sin A = u
sin A
u
cos A = u
tan A = u
cot A = u
sec A = u
csc A = u
1 − u2
u/ 1 + u 2
1/ 1 + u 2
u 2 − 1/u
1/u
u
1/ 1 + u 2
u/ 1 + u 2
1/u
u2 − 1
1/ u 2 − 1
1/ u 2 − 1
u2 − 1
cos A
1 − u2
tan A
u/ 1 − u 2
1 − u 2 /u
u
1/u
cot A
1 − u 2 /u
u/ 1 − u 2
1/u
u
sec A
1/ 1 − u 2
1/u
1+ u2
1 + u 2 /u
csc A
1/u
1/ 1 − u 2
1 + u 2 /u
1+ u2
u
u/ u 2 − 1
For extensions to other quadrants use appropriate signs as given in the preceding table.
u 2 − 1/u
u/ u 2 − 1
u
TRIGONOMETRIC FUNCTIONS
48
Double Angle Formulas
12.38. sin 2A = 2 sin A cos A
12.39. cos 2A = cos2 A – sin2 A = 1 – 2 sin2 A = 2 cos2 A – 1
12.40.
tan 2 A =
2 tan A
1 − tan 2 A
Half Angle Formulas
12.41.
sin
A
1 − cos A
=±
2
2
⎡+ if A / 2 is in quadrant I or II ⎤
⎢
⎥
⎢⎣− if A / 2 is in quadrant III or IV⎥⎦
12.42.
cos
A
1 + cos A
=±
2
2
⎡+ if A / 2 is in quadrant I or IV ⎤
⎢
⎥
⎢⎣− if A / 2 is in quadrant II or III⎥⎦
12.43.
tan
A
1 − cos A ⎡+ if A / 2 is in quadrant I orr III ⎤
=±
2
1 + cos A ⎢⎢− if A / 2 is in quadrant II or IV⎥⎥
⎣
⎦
=
sin A
1 − cos A
=
= csc A − cot A
1 + cos A
sin A
Multiple Angle Formulas
12.44. sin 3A = 3 sin A – 4 sin3 A
12.45. cos 3A = 4 cos3 A –3 cos A
12.46.
tan 3A =
3 tan A − tan 3 A
1 − 3 tan 2 A
12.47. sin 4A = 4 sin A cos A – 8 sin3 A cos A
12.48. cos 4A = 8 cos4 A – 8 cos2 A + 1
12.49.
tan 4 A =
4 tan A − 4 tan 3 A
1 − 6 tan 2 A + tan 4 A
12.50. sin 5A = 5 sin A – 20 sin3 A + 16 sin5 A
12.51. cos 5A = 16 cos5 A – 20 cos3 A + 5 cos A
12.52.
tan 5 A =
tan 5 A − 10 tan 3 A + 5 tan A
1 − 10 tan 2 A + 5 tan 4 A
See also formulas 12.68 and 12.69.
Powers of Trignometric Functions
12.53.
sin 2 A = 12 − 12 cos 2 A
12.57.
sin 4 A = 83 − 12 cos 2 A + 18 cos 4 A
12.54.
cos 2 A = 12 + 12 cos 2 A
12.58.
cos 4 A = 83 + 12 cos 2 A + 18 cos 4 A
12.55.
sin 3 A =
12.59.
sin 5 A = 58 sin A − 165 sin 3A + 161 sin 5 A
12.56.
cos 3 A = 43 cos A + 14 cos 3A
12.60.
cos 5 A = 58 cos A + 165 cos 3A + 161 cos 5 A
3
4
sin A − 14 sin 3A
See also formulas 12.70 through 12.73.
TRIGONOMETRIC FUNCTIONS
49
Sum, Difference, and Product of Trignometric Functions
12.61.
sin A + sin B = 2 sin 12 ( A + B) cos 12 ( A − B)
12.62.
sin A − sin B = 2 cos 12 ( A + B)sin 12 ( A − B)
12.63.
cos A + cos B = 2 cos 12 ( A + B) cos 12 ( A − B)
12.64.
cos A − cos B = 2 sin 12 ( A + B)sin 12 ( B − A)
12.65.
sin A sin B = 12 {cos( A − B) − cos ( A − B)}
12.66.
cos A cos B = 12 {cos( A − B) + cos ( A + B)}
12.67.
sin A cos B = 12 {sin( A − B) + sin( A + B)}
General Formulas
12.68.
⎫
⎧
⎛ n − 3⎞
⎛ n − 2⎞
n−3
n−5
sin nA = sin A ⎨(2 cos A)n−1 − ⎜
⎟⎠ (2 cos A) + ⎜⎝ 2 ⎟⎠ (2 cos A) − ⋅⋅⋅⎬
1
⎝
⎭
⎩
12.69.
cos nA =
1⎧
n
n ⎛ n − 3⎞
(2 cos A)n − (2 cos A)n− 2 + ⎜
(2 cos A)n− 4
2 ⎨⎩
1
2 ⎝ 1 ⎟⎠
n ⎛ n − 4⎞
⎫
(2 cos A)n−6 + ⋅⋅⋅⎬
− ⎜
3 ⎝ 2 ⎟⎠
⎭
12.70.
sin 2 n−1 A =
⎫
(−1)n−1 ⎧
⎛ 2n − 1⎞
⎛ 2n − 1⎞
sin (2n − 1) A − ⎜
sin (2n − 3) A + ⋅⋅⋅ (−1)n−1 ⎜
sin A⎬
22 n− 2 ⎨⎩
⎝ 1 ⎟⎠
⎝ n − 1 ⎟⎠
⎭
12.71.
cos 2 n−1 A =
1 ⎧
⎫
⎛ 2n − 1⎞
⎛ 2n − 1⎞
cos (2n − 1) A + ⎜
cos (2n − 3) A + ⋅⋅⋅ + ⎜
cos A⎬
22 n− 2 ⎨⎩
⎝ 1 ⎟⎠
⎝ n − 1 ⎟⎠
⎭
12.72.
sin 2 n A =
1 ⎛ 2n⎞
(−1)n
+ 2 n−1
⎟
2n ⎜
2
2 ⎝ n⎠
12.73.
cos 2 n A =
⎫
1 ⎧
1 ⎛ 2n⎞
⎛ 2n ⎞
⎛ 2n⎞
cos 2 A⎬
+ 2 n−1 ⎨cos 2nA + ⎜ ⎟ cos (2n − 2) A + ⋅⋅⋅ + ⎜
⎟
⎟
2n ⎜
2 ⎝ n⎠ 2
⎝ n − 1⎠
⎝ 1⎠
⎭
⎩
⎫
⎧
⎛ 2n⎞
n −1 ⎛ 2n ⎞
⎨cos 2nA − ⎜ 1 ⎟ cos (2n − 2) A + ⋅⋅⋅ (−1) ⎜ n − 1⎟ cos 2 A⎬
⎝
⎝ ⎠
⎠
⎭
⎩
Inverse Trigonometric Functions
If x = sin y, then y = sin–1x, i.e. the angle whose sine is x or inverse sine of x is a many-valued function of x
which is a collection of single-valued functions called branches. Similarly, the other inverse trigonometric
functions are multiple-valued.
For many purposes a particular branch is required. This is called the principal branch and the values for
this branch are called principal values.
TRIGONOMETRIC FUNCTIONS
50
Principal Values for Inverse Trigonometric Functions
Principal values for x 0
Principal values for x < 0
0 sin–1 x p/2
–p/2 sin–1 x < 0
0 cos–1 x p/2
p/2 < cos–1 x p
0 tan–1 x < p/2
–p/2 < tan–1 x < 0
0 < cot–1 x p/2
p/2 < cot–1 x < p
0 sec–1 x < p/2
p/2 < sec–1 x p
0 < csc–1 x p/2
–p/2 csc–1 x < 0
Relations Between Inverse Trigonometric Functions
In all cases it is assumed that principal values are used.
12.74.
sin −1 x + cos−1 x = π /2
12.80.
sin −1 (− x ) = − sin −1 x
12.75.
tan −1 x + cot −1 x = π / 2
12.81.
cos−1 (− x ) = π − cos−1 x
12.76.
sec−1 x + csc−1 x = π /2
12.82.
tan −1 (− x ) = − tan −1 x
12.77.
csc−1 x = sin −1 (1/x )
12.83.
cot −1 (− x ) = π − cot −1 x
12.78.
sec−1 x = cos−1 (1/x )
12.84.
sec−1 (− x ) = π − sec −1 x
12.79.
cot −1 x = tan −1 (1/x )
12.85.
csc −1 (− x ) = − csc −1 x
Graphs of Inverse Trigonometric Functions
In each graph y is in radians. Solid portions of curves correspond to principal values.
12.86.
y = sin −1 x
Fig. 12-11
12.87.
y = cos−1 x
Fig. 12-12
12.88.
y = tan −1 x
Fig. 12-13
TRIGONOMETRIC FUNCTIONS
12.89.
y = cot −1 x
Fig. 12-14
12.90.
51
y = sec−1 x
12.91.
y = csc −1 x
Fig. 12-15
Fig. 12-16
Relationships Between Sides and Angles of a Plane Triangle
The following results hold for any plane triangle ABC with sides a, b, c and angles A, B, C.
12.92. Law of Sines:
a
b
c
=
=
sin A sin B sin C
12.93. Law of Cosines:
c 2 = a 2 + b 2 − 2 ab cos C
with similar relations involving the other sides and angles.
12.94. Law of Tangents:
a + b tan 12 ( A + B)
=
a − b tan 12 ( A − B)
with similar relations involving the other sides and angles.
Fig. 12-17
2
s(s − a )(s − b )(s − c)
bc
where s = 12 (a + b + c) is the semiperimeter of the triangle. Similar relations involving angles B and C can
be obtained.
12.95.
sin A =
See also formula 7.5.
Relationships Between Sides and Angles of a Spherical Triangle
Spherical triangle ABC is on the surface of a sphere as shown
in Fig. 12-18. Sides a, b, c (which are arcs of great circles) are
measured by their angles subtended at center O of the sphere.
A, B, C are the angles opposite sides a, b, c, respectively. Then the
following results hold.
12.96. Law of Sines:
sin a sin b sin c
=
=
sin A sin B sin C
12.97. Law of Cosines:
cos a = cos b cos c + sin b sin c cos A
cos A = –cos B cos C + sin B sin C cos a
with similar results involving other sides and angles.
Fig. 12-18
TRIGONOMETRIC FUNCTIONS
52
12.98. Law of Tangents:
tan 12 ( A + B) tan 12 (a + b )
=
tan 12 ( A − B) tan 12 (a − b )
with similar results involving other sides and angles.
12.99.
cos
A
=
2
sin s sin (s − c)
sin b sin c
where s = 12 (a + b + c). Similar results hold for other sides and angles.
12.100.
cos
a
cos(S − B) cos(S − C )
=
2
sin B sin C
where S = 12 ( A + B + C ). Similar results hold for other sides and angles.
See also formula 7.44.
Napier’s Rules for Right Angled Spherical Triangles
Except for right angle C, there are five parts of spherical triangle ABC which, if arranged in the order as given
in Fig. 12-19, would be a, b, A, c, B.
Fig. 12-19
Fig. 12-20
Suppose these quantities are arranged in a circle as in Fig. 12-20 where we attach the prefix “co” (indicating complement) to hypotenuse c and angles A and B.
Any one of the parts of this circle is called a middle part, the two neighboring parts are called adjacent
parts, and the two remaining parts are called opposite parts. Then Napier’s rules are
12.101. The sine of any middle part equals the product of the tangents of the adjacent parts.
12.102. The sine of any middle part equals the product of the cosines of the opposite parts.
EXAMPLE: Since co-A = 90° – A, co-B = 90° – B, we have
sin a = tan b (co-B)
sin (co-A) = cos a cos (co-B)
or
or
sin a = tan b cot B
cos A = cos a sin B
These can of course be obtained also from the results of 12.97.
13
EXPONENTIAL and LOGARITHMIC
FUNCTIONS
Laws of Exponents
In the following p, q are real numbers, a, b are positive numbers, and m, n are positive integers.
13.1.
a p ⋅ a q = a p+ q
13.2.
a p /a q = a p−q
13.3.
(a p )q = a pq
13.4.
a 0 = 1, a ≠ 0
13.5.
a − p = 1/a p
13.6.
(ab ) p = a p b p
13.7.
n
13.8.
n
13.9.
n
a = a1/n
a m = a m/n
a/b = n a / n b
In ap, p is called the exponent, a is the base, and ap is called the pth power of a. The function y = ax is
called an exponential function.
Logarithms and Antilogarithms
If ap = N where a ≠ 0 or 1, then p = loga N is called the logarithm of N to the base a. The number N = ap is
called the antilogarithm of p to the base a, written antiloga p.
Example: Since 32 = 9 we have log3 9 = 2. antilog3 2 = 9.
The function y = loga x is called a logarithmic function.
Laws of Logarithms
13.10. loga MN = loga M + loga N
13.11.
log a
M
= log a M − log a N
N
13.12. loga Mp = p loga M
Common Logarithms and Antilogarithms
Common logarithms and antilogarithms (also called Briggsian) are those in which the base a = 10. The
common logarithm of N is denoted by log10 N or briefly log N. For numerical values of common logarithms,
see Table 1.
Natural Logarithms and Antilogarithms
Natural logarithms and antilogarithms (also called Napierian) are those in which the base a = e =
2.71828 18 … [see page 3]. The natural logarithm of N is denoted by loge N or In N. For numerical
values of natural logarithms see Table 7. For values of natural antilogarithms (i.e., a table giving ex for
values of x) see Table 8.
53
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
54
Change of Base of Logarithms
The relationship between logarithms of a number N to different bases a and b is given by
13.13.
log a N =
log b N
log b a
In particular,
13.14. loge N = ln N = 2.30258 50929 94 … log10 N
13.15. log10 N = log N = 0.43429 44819 03 … loge N
Relationship Between Exponential and Trigonometric Functions
13.16. eiθ = cos θ + i sin θ,
e–iθ = cos θ – i sin θ
These are called Euler’s identities. Here i is the imaginary unit [see page 10].
13.17.
sin θ =
eiθ − e−iθ
2i
13.18.
cos θ =
eiθ + e−iθ
2
13.19.
tan θ =
⎛ eiθ − e−iθ ⎞
eiθ − e−iθ
= −i ⎜ iθ −iθ ⎟
iθ
− iθ
i (e + e )
⎝e + e ⎠
13.20.
⎛ eiθ + e−iθ ⎞
cot θ = i ⎜ iθ −iθ ⎟
⎝e − e ⎠
13.21.
sec θ =
2
e + e−iθ
13.22.
csc θ =
2i
eiθ − e−iθ
iθ
Periodicity of Exponential Functions
13.23. ei(θ + 2kp) = eiθ
k = integer
From this it is seen that ex has period 2pi.
Polar Form of Complex Numbers Expressed as an Exponential
The polar form (see 4.7) of a complex number z = x + iy can be written in terms of exponentials as follows:
13.24.
z = x + iy = r (cos θ + i sin θ ) = reiθ
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Operations with Complex Numbers in Polar Form
Formulas 4.8 to 4.11 are equivalent to the following:
13.25.
(r1eiθ )(r2 eiθ ) = r1r2 ei (θ +θ )
13.26.
r1eiθ
r
= 1 ei (θ −θ )
r2
r2 eiθ
1
2
1
2
1
1
2
2
13.27.
(reiθ ) p = r p eipθ
13.28.
(reiθ )1/n = [ rei (θ +2 kπ ) ]1/n = r1/n ei (θ +2 kπ )/n
(De Moivre’s theorem)
Logarithm of a Complex Number
13.29.
ln (re iθ ) = ln r + iθ + 2kπ i
k = integer
55
14
HYPERBOLIC FUNCTIONS
Definition of Hyperbolic Functions
14.1. Hyperbolic sine of x
= sinh x =
e x − e− x
2
14.2. Hyperbolic cosine of x
= cosh x =
e x + e− x
2
14.3. Hyperbolic tangent of x
= tanh x =
e x − e− x
e x + e− x
14.4. Hyperbolic cotangent of x = coth x =
e x + e− x
e x − e− x
14.5. Hyperbolic secant of x
= sech x =
2
e x + e− x
14.6. Hyperbolic cosecant of x
= csch x =
2
e x − e− x
Relationships Among Hyperbolic Functions
14.7.
tanh x =
sinh x
cosh x
14.8.
coth x =
1
cosh x
=
tanh x sinh x
14.9.
sech x =
1
cosh x
14.10.
csch x =
1
sinh x
14.11.
cosh 2 x − sinh 2x = 1
14.12.
sech 2 x + tanh 2 x = 1
14.13.
coth 2 x − csc h 2 x = 1
Functions of Negative Arguments
14.14. sinh (–x) = – sinh x
14.15. cosh (–x) = cosh x
14.16. tanh (–x) = – tanh x
14.17. csch (–x) = – csch x
14.18. sech (–x) = sech x
14.19. coth (–x) = – coth x
56
HYPERBOLIC FUNCTIONS
Addition Formulas
14.20.
sinh ( x ± y) = sinh x cosh y ± cosh x sinh y
14.21.
cosh ( x ± y) = cosh x cosh y ± sinh x sinh y
14.22.
tanh ( x ± y) =
tanh x ± tanh y
1 ± tanh x tanh y
14.23.
coth ( x ± y) =
coth x coth y ± 1
coth y ± coth x
Double Angle Formulas
14.24.
sinh 2 x = 2 sinh x cosh x
14.25.
cosh 2 x = cos h 2 x + sin h 2 x = 2 cos h 2 x − 1 = 1 + 2 sin h 2 x
14.26.
tanh 2 x =
2 tan h x
1 + tanh 2 x
Half Angle Formulas
cosh x − 1
[+ if x > 0, − if x < 0 ]
2
14.27.
sinh
x
=±
2
14.28.
cosh
x
cosh x + 1
=
2
2
14.29.
tanh
x
=±
2
=
cosh x − 1
[+ if x > 0, − if x < 0 ]
cosh x + 1
cosh x − 1
sinh x
=
cosh x + 1
sinh x
Multiple Angle Formulas
14.30.
sinh 3x = 3 sinh x + 4 sinh 3 x
14.31.
cosh 3x = 4 cosh 3 x − 3 cosh x
14.32.
tanh 3x =
14.33.
sinh 4 x = 8 sinh 3 x cosh x + 4 sinh x cosh x
14.34.
cosh 4 x = 8 cosh 4 x − 8 cosh 2 x + 1
14.35.
tanh 4 x =
3 tanh x + tanh 3 x
1 + 3 tanh 2 x
4 tanh x + 4 tanh 3 x
1 + 6 tanh 2 x + tanh 4 x
57
HYPERBOLIC FUNCTIONS
58
Powers of Hyperbolic Functions
14.36.
sinh 2 x = 12 cosh 2 x − 12
14.37.
cosh 2 x = 12 cosh 2 x + 12
14.38.
sinh 3 x = 14 sinh 3x − 43 sinh x
14.39.
cosh 3 x = 14 cosh 3x + 43 cosh x
14.40.
sinh 4 x = 83 − 12 cosh 2 x + 18 cosh 4 x
14.41.
cosh 4 x = 83 + 12 cosh 2 x + 18 cosh 4 x
Sum, Difference, and Product of Hyperbolic Functions
14.42.
sinh x + sinh y = 2 sinh 12 ( x + y) cosh 12 ( x − y )
14.43.
sinh x − sinh y = 2 cosh 12 ( x + y)sinh 12 ( x − y)
14.44.
cosh x + cosh y = 2 cosh 12 ( x + y) cosh 12 ( x − y)
14.45.
cosh x − cosh y = 2 sinh 12 ( x + y)sinh 12 ( x − y )
14.46.
sinh x sinh y = 12 {cosh ( x + y) − cosh ( x − y)}
14.47.
cosh x cosh y = 12 {cosh ( x + y) + cosh ( x − y)}
14.48.
sinh x cosh y = 12 {sinh ( x + y) + sinh ( x − y)}
Expression of Hyperbolic Functions in Terms of Others
In the following we assume x > 0. If x < 0, use the appropriate sign as indicated by formulas 14.14 to
14.19.
sinh x = u
sinh x
u
cosh x
1+ u2
cosh x = u
tanh x = u
coth x = u
sech x = u
csch x = u
u/ 1 − u 2
1/ u 2 − 1
1 − u 2 /u
1/u
u
1/ 1 − u 2
u/ u 2 − 1
1/u
1 + u 2 /u
u
1/u
1 − u2
1/ 1 + u 2
1+ u2
u2 − 1
tanh x
u/ 1 + u 2
u 2 − 1/u
coth x
u 2 + 1/u
u/ u 2 − 1
1/u
u
1/ 1 − u 2
sech x
1/ 1 + u 2
1/u
1 − u2
u 2 − 1/u
u
u/ 1 + u 2
csch x
1/u
1/ u 2 − 1
1 − u 2 /u
u2 − 1
u/ 1 − u 2
u
HYPERBOLIC FUNCTIONS
59
Graphs of Hyperbolic Functions
14.49. y = sinh x
14.50.
y = cosh x
Fig. 14-1
14.51. y = tanh x
Fig. 14-2
14.52. y = coth x
14.53.
y = sech x
Fig. 14-4
Fig. 14-3
14.54. y = csch x
Fig. 14-5
Fig. 14-6
Inverse Hyperbolic Functions
If x = sinh y, then y = sinh–1 x is called the inverse hyperbolic sine of x. Similarly we define the other inverse
hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigonometric functions [see page 49] we restrict ourselves to principal values for which they can be considered
as single-valued.
The following list shows the principal values (unless otherwise indicated) of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued.
14.55.
sinh −1 x = ln ( x + x 2 + 1)
− < x < 14.56.
cosh −1 x = ln ( x + x 2 − 1)
x 1
14.57.
tanh −1 x =
1 ⎛1 + x ⎞
ln ⎜
⎟
2 ⎝1 − x ⎠
−1 < x < 1
14.58.
coth −1 x =
1 ⎛ x + 1⎞
ln
2 ⎝ x − 1⎠
x > 1 or x < −1
14.59.
⎛1
sech −1 x = ln ⎜ +
⎝x
⎞
1
2 − 1⎟
x
⎠
0 < x 1
14.60.
⎛1
csch −1 x = ln ⎜ +
⎝x
⎞
1
+ 1⎟
x2
⎠
x≠0
(cosh −1 x > 0 is prinncipal value)
(sech −1 x > 0 is principal value)
HYPERBOLIC FUNCTIONS
60
Relations Between Inverse Hyperbolic Functions
14.61.
csch −1 x = sinh −1 (1/x )
14.62.
sech −1 x = cosh −1 (1/x )
14.63.
coth −1 x = tanh −1 (1/x )
14.64.
sinh −1 (− x ) = − sinh −1 x
14.65.
tanh −1 (− x ) = − tanh −1 x
14.66.
coth −1 (− x ) = − coth −1 x
14.67.
csch −1 (− x ) = −csch −1 x
Graphs of Inverse Hyperbolic Functions
14.68.
y = sinh −1 x
14.69.
Fig. 14-7
14.71.
y = coth −1 x
Fig. 14-10
y = cosh −1 x
14.70.
Fig. 14-8
14.72.
y = sech −1 x
Fig. 14-11
y = tanh −1 x
Fig. 14-9
14.73.
y = csch −1 x
Fig. 14-12
HYPERBOLIC FUNCTIONS
61
Relationship Between Hyperbolic and Trigonometric Functions
14.74.
sin (ix ) = i sinh x
14.75.
cos (ix ) = cosh x
14.76.
tan (ix ) = i tanh x
14.77.
csc (ix ) = −i csch x
14.78.
sec (ix ) = sech x
14.79.
cot (ix ) = −i coth x
14.80.
sinh (ix ) = i sin x
14.81.
cosh (ix ) = cos x
14.82.
tanh (ix ) = i tan x
14.83.
csch (ix ) = −i csc x
14.84.
sech (ix ) = sec x
14.85.
coth (ix ) = −i cot x
Periodicity of Hyperbolic Functions
In the following k is any integer.
14.86.
sinh ( x + 2 k π i ) = sinh x
14.87.
cosh ( x + 2kπ i) = cosh x
14.88.
tanh ( x + kπ i) = tanh x
14.89.
csch ( x + 2 k π i ) = csch x
14.90.
sech ( x + 2 k π i ) = sech x
14.91.
coth ( x + k π i ) = coth x
Relationship Between Inverse Hyperbolic and Inverse Trigonometric Functions
14.92.
sin −1 (ix ) = i sin −1 x
14.93.
sinh −1 (ix ) = i sin −1 x
14.94.
cos−1 x = ± i cosh −1 x
14.95.
cosh −1 x = ± i cos −1 x
14.96.
tan −1 (ix ) = i tanh −1 x
14.97.
tanh −1 (ix ) = i tan −1 x
14.98.
cot −1 (ix ) = i coth −1 x
14.99.
coth −1 (ix ) = −i cot −1 x
14.100.
sec−1 x = ± i sech −1 x
14.101.
sech −1 x = ± i sec −1 x
14.102.
csc −1 (ix ) = −i csch −1 x
14.103.
csch −1 (ix ) = −i csc −1 x
Section IV: Calculus
15
DERIVATIVES
Definition of a Derivative
Suppose y = f(x). The derivative of y or f(x) is defined as
15.1.
dy
f ( x + h) − f ( x )
f ( x + Δx ) − f ( x )
= lim
= lim
Δx
dx h→0
h
Δx → 0
where h = Δx. The derivative is also denoted by y′, df/dx or f ′(x). The process of taking a derivative is called
differentiation.
General Rules of Differentiation
In the following, u, , w are functions of x; a, b, c, n are constants (restricted if indicated); e = 2.71828 … is the
natural base of logarithms; ln u is the natural logarithm of u (i.e., the logarithm to the base e) where it is assumed
that u > 0 and all angles are in radians.
15.2.
d
(c) = 0
dx
15.3.
d
(cx ) = c
dx
15.4.
d
(cx n ) = ncx n−1
dx
15.5.
d
du d dw
(u ± ± w ± ) =
±
±
±
dx
dx dx dx
15.6.
d
du
(cu) = c
dx
dx
15.7.
d
d
du
+
(u ) = u
dx
dx
dx
15.8.
d
dw
d
du
+ uw
+ w
(u w) = u
dx
dx
dx
dx
15.9.
d ⎛ u ⎞ (du /dx ) − u(d /dx )
=
2
dx ⎜⎝ ⎟⎠
15.10.
d n
du
(u ) = nu n−1
dx
dx
15.11.
dy dy du
=
dx du dx
(Chain rule)
62
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
DERIVATIVES
15.12.
du
1
=
dx dx /du
15.13.
dy dy/du
=
dx dx/du
63
Derivatives of Trigonometric and Inverse Trigonometric Functions
15.14.
d
du
sin u = cos u
dx
dx
15.15.
d
du
cos u = − sin u
dx
dx
15.16.
d
du
tan u = sec 2 u
dx
dx
15.17.
d
du
cot u = − csc 2 u
dx
dx
15.18.
d
du
sec u = sec u tan u
dx
dx
15.19.
d
du
csc u = − csc u cot u
dx
dx
15.20.
d
1
du
sin −1 u =
2 dx
dx
1− u
π⎤
⎡ π
−1
⎢⎣− 2 < sin u < 2 ⎥⎦
15.21.
d
−1 du
cos −1 u =
dx
1 − u 2 dx
[0 < cos −1 u < π ]
15.22.
d
1 du
tan −1 u =
dx
1 + u 2 dx
π⎤
⎡ π
−1
⎢⎣− 2 < tan u < 2 ⎥⎦
15.23.
d
−1 du
cot −1 u =
dx
1 + u 2 dx
[0 < cot −1 u < π ]
15.24.
d
1
±1
du
du
=
sec −1 u =
2
2
dx
dx
u u − 1 dx
| u | u −1
⎡+ if 0 < sec −1 u < π / 2⎤
⎢⎣− if π / 2 < sec −1 u < π ⎥⎦
15.25.
d
du
∓1
−1
du
=
csc −1 u =
2
2
dx
dx
| u | u −1
u u − 1 dx
⎡− if 0 < csc −1 u < π / 2 ⎤
⎢+ if − π / 2 < csc −1 u < 0⎥
⎣
⎦
Derivatives of Exponential and Logarithmic Functions
15.26.
loga e du
d
loga u =
dx
u dx
15.27.
d
d
1 du
ln u =
loge u =
dx
dx
u dx
15.28.
d u
du
a = a u ln a
dx
dx
a ≠ 0, 1
DERIVATIVES
64
15.29.
d u
du
e = eu
dx
dx
15.30.
d d ln u
d
du
d
u =
e
= e ln u
[ ln u] = u −1
+ u ln u
dx
dx
dx
dx
dx
Derivatives of Hyperbolic and Inverse Hyperbolic Functions
15.31.
d
du
sinh u = cosh u
dx
dx
15.32.
d
du
cosh u = sinh u
dx
dx
15.33.
d
du
tanh u = sech 2 u
dx
dx
15.34.
d
du
coth u = −csch 2 u
dx
dx
15.35.
d
du
sech u = −sech u tanh u
dx
dx
15.36.
d
du
csch u = − csch u coth u
dx
dx
15.37.
d
sinh −1 u =
dx
du
u + 1 dx
15.38.
d
cosh −1 u =
dx
±1 du
u 2 − 1 dx
15.39.
d
1 du
tanh −1 u =
dx
1 − u 2 dx
[–1 < u < 1]
15.40.
d
1 du
coth −1 u =
dx
1 − u 2 dx
[u > 1 or u < –1]
15.41.
d
∓1
du
sech −1u =
2 dx
dx
u 1− u
⎡− if sech −1 u > 0, 0 < u < 1⎤
⎢+ if sech −1 u < 0, 0 < u < 1⎥
⎣
⎦
15.42.
d
du
∓1
−1
du
=
csch −1u =
2 dx
2 dx
dx
| u | 1+ u
u 1+ u
[– if u > 0, + if u < 0]
1
2
Higher Derivatives
The second, third, and higher derivatives are defined as follows.
15.43.
Second derivative =
d ⎛ dy⎞ d 2 y
=
= f ′′( x ) = y ′′
dx ⎜⎝ dx ⎟⎠ dx 2
2
3
15.44. Third derivative = d ⎛ d y⎞ = d y = f ′′′( x ) = y ′′′
⎜
2⎟
dx ⎝ dx ⎠ dx 3
15.45.
n −1
n
nth derivative = d ⎛ d y⎞ = d y = f ( n ) ( x ) = y ( n )
⎜
n −1 ⎟
dx ⎝ dx ⎠ dx n
⎡+ if cosh −1 u > 0, u > 1⎤
⎢⎣− if cosh −1 u < 0, u > 1⎥⎦
DERIVATIVES
65
Leibniz’s Rule for Higher Derivatives of Products
Let Dp stand for the operator
15.46.
dp
d pu
p
so
that
D
u
=
= the pth derivative of u. Then
dx p
dx p
⎛ n⎞
⎛ n⎞
D n (u ) = uD n + ⎜ ⎟ ( Du)( D n −1 ) + ⎜ ⎟ ( D 2 u)( D n − 2 ) + + D n u
⎝1⎠
⎝ 2⎠
⎛ n⎞ ⎛ n⎞
where ⎜ ⎟ , ⎜ ⎟ ,… are the binomial coefficients (see 3.5).
⎝1⎠ ⎝ 2⎠
As special cases we have
15.47.
d2
d 2
du d
d 2u
(
)
=
+
2
+
u
u
dx 2
dx 2
dx dx
dx 2
15.48.
d3
d 3u
d 3
du d 2 d 2 u d
(
)
=
+
3
+
3
+
u
u
dx 3
dx 3
dx 3
dx dx 2
dx 2 dx
Differentials
Let y = f(x) and Δy = f ( x + Δx ) − f ( x ). Then
15.49.
Δy f ( x + Δx ) − f ( x )
dy
=
= f ′( x ) + =
+
Δx
Δx
dx
where → 0 as Δx → 0. Thus,
15.50.
Δy = f ′( x )Δx + Δx
If we call Δx = dx the differential of x, then we define the differential of y to be
15.51.
dy = f ′( x ) dx
Rules for Differentials
The rules for differentials are exactly analogous to those for derivatives. As examples we observe that
15.52.
d (u ± ± w ± ) = du ± d ± dw ± 15.53.
d (u ) = u d + du
15.54.
⎛ u ⎞ du − u d
d⎜ ⎟ =
2
⎝⎠
15.55.
d (u n ) = nu n−1du
15.56.
d (sin u) = cos u du
15.57.
d (cos u) = − sin u du
DERIVATIVES
66
DERIVATIVES
Partial Derivatives
Let z = f (x, y) be a function of the two variables x and y. Then we define the partial derivative of z or f(x, y) with
respect to x, keeping y constant, to be
15.58.
∂f
f ( x + Δx , y) − f ( x , y)
= lim
Δx
∂x Δx →0
This partial derivative is also denoted by ∂z/∂x , f x , or z x .
Similarly the partial derivative of z = f (x, y) with respect to y, keeping x constant, is defined to be
15.59.
∂f
f ( x , y + Δy) − f ( x , y)
= lim
Δy
∂y Δy→0
This partial derivative is also denoted by ∂z/∂y, f y , or z y .
Partial derivatives of higher order can be defined as follows:
15.60.
∂2 f
∂ ⎛ ∂f ⎞
=
,
∂x 2 ∂x ⎜⎝ ∂x ⎟⎠
15.61.
∂2 f
∂ ⎛ ∂f ⎞
=
,
∂x ∂y ∂x ⎜⎝ ∂y⎟⎠
∂ ⎛ ∂f ⎞
∂2 f
=
∂y 2 ∂y ⎜⎝ ∂y⎟⎠
∂ ⎛ ∂f ⎞
∂2 f
=
∂y ∂x ∂y ⎜⎝ ∂x ⎟⎠
The results in 15.61 will be equal if the function and its partial derivatives are continuous; that is, in such
cases, the order of differentiation makes no difference.
Extensions to functions of more than two variables are exactly analogous.
Multivariable Differentials
The differential of z = f(x, y) is defined as
15.62.
dz = df =
∂f
∂f
dx + dy
∂x
∂y
where dx = Δx and dy = Δy. Note that dz is a function of four variables, namely x, y, dx, dy, and is linear in the
variables dx and dy.
Extensions to functions of more than two variables are exactly analogous.
EXAMPLE:
Let z = x2 + 5xy + 2y3. Then
zx = 2x + 5y
and
zy = 5x + 6y2
and hence
dz = (2x + 5y) dx + (5x + 6y2) dy
Suppose we want to find dz for dx = 2, dy = 3 and at the point P (4, 1), i.e., when x = 4 and y = 1. Substitution
yields
dz = (8 + 5)2 + (20 + 6)3 = 26 + 78 = 104
16
INDEFINITE INTEGRALS
Definition of an Indefinite Integral
dy
= f ( x ), then y is the function whose derivative is f(x) and is called the anti-derivative of f(x) or the indefidx
dy
nite integral of f(x), denoted by ∫ f ( x ) dx. Similarly if y = ∫ f (u) du, then
= f (u). Since the derivative of a
du
constant is zero, all indefinite integrals differ by an arbitrary constant.
For the definition of a definite integral, see 18.1. The process of finding an integral is called integration.
If
General Rules of Integration
In the following, u, , w are functions of x; a, b, p, q, n any constants, restricted if indicated; e = 2.71828 … is
the natural base of logarithms; ln u denotes the natural logarithm of u where it is assumed that u > 0 (in general,
to extend formulas to cases where u < 0 as well, replace ln u by ln |u|); all angles are in radians; all constants of
integration are omitted but implied.
16.1.
∫ a dx = ax
16.2.
∫ af ( x ) dx = a ∫ f ( x ) dx
16.3.
∫ (u ± ± w ± ) dx = ∫ u dx ± ∫ dx ± ∫ w dx ± 16.4.
∫ u d = u − ∫ du
(Integration by parts)
For generalized integration by parts, see 16.48.
1
16.5.
∫ f (ax ) dx = a ∫ f (u) du
16.6.
∫ F{ f ( x )} dx = ∫ F(u) du du = ∫
16.7.
∫ u n du =
16.8.
∫
dx
F (u)
du where u = f ( x )
f ′( x )
u n+1
, n ≠ −1 (For n = −1, see 16.8)
n +1
du
= ln u if u > 0 or ln(−u) if u < 0
u
= ln | u |
16.9.
16.10.
∫ e du = e
u
∫ au du =
u
∫ eu ln a du =
e u ln a
au
=
, a > 0, a ≠ 1
ln a ln a
67
INDEFINITE INTEGRALS
68
16.11.
∫ sin u du = − cos u
16.12.
∫ cos u du = sin u
16.13.
∫ tan u du = ln sec u = − ln cos u
16.14.
∫ cot u du = ln sin u
16.15.
∫ sec u du = ln (sec u + tan u) = ln tan ⎜⎝ 2 + 4 ⎟⎠
16.16.
∫ csc u du = ln(csc u − cot u) = ln tan 2
16.17.
∫ sec
16.18.
∫ csc
16.19.
∫ tan
16.20.
∫ cot
16.21.
∫ sin
16.22.
∫ cos
16.23.
∫ sec u tan u du = sec u
16.24.
∫ csc u cot u du = − csc u
16.25.
∫ sinh u du = cosh u
16.26.
∫ cosh u du = sinh u
16.27.
∫ tanh u du = ln cosh u
16.28.
∫ coth u du = ln sinh u
16.29.
∫ sech u du = sin
16.30.
∫ csch u du = ln tanh 2
16.31.
∫ sech u du = tanh u
⎛u
π⎞
u
2
u du = tan u
2
u du = − cot u
2
u du = tan u − u
2
u du = − cot u − u
2
u du =
u sin 2u 1
−
= (u − sin u cos u)
2
4
2
u du =
u sin 2u 1
+
= (u + sin u cos u)
2
4
2
2
−1
(tanh u) or 2 tan −1 eu
u
2
or − coth −1 e u
INDEFINITE INTEGRALS
16.32.
∫ csch u du = − coth u
16.33.
∫ tanh u du = u − tanh u
16.34.
∫ coth u du = u − coth u
16.35.
∫ sinh u du =
sinh 2u u 1
− = (sinh u cosh u − u)
4
2 2
16.36.
∫ cosh u du =
sinh 2u u 1
+ = (sinh u cosh u + u)
4
2 2
16.37.
∫ sech u tanh u du = −sech u
16.38.
∫ csch u coth u du = − csch u
16.39.
∫u
2
16.40.
∫u
2
16.41.
∫a
2
16.42.
∫
du
u
= sin −1
2
a
a −u
16.43.
∫
du
u
= ln(u + u 2 + a 2 ) or sinh −1
2
a
u +a
16.44.
∫
du
= ln (u + u 2 − a 2 )
u2 − a2
16.45.
∫u
16.46.
du
1 ⎛ a + u2 + a2 ⎞
=
−
⎟
∫ u u 2 + a 2 a ln ⎜⎝
u
⎠
16.47.
∫u
16.48.
∫f
69
2
2
2
2
2
u
du
1
= tan −1
a
+ a2 a
u
du
1 ⎛ u − a⎞
1
=
ln
= − coth −1 u 2 > a 2
a
a
− a 2 2a ⎜⎝ u + a⎟⎠
u
du
1 ⎛ a + u⎞ 1
=
ln
= tanh −1 u 2 < a 2
a
− u 2 2a ⎜⎝ a − u⎟⎠ a
2
2
du
u2 − a2
=
u
1
sec −1
a
a
du
1 ⎛ a + a2 − u2 ⎞
= − ln ⎜
⎟
2
2
a ⎝
u
a −u
⎠
(n)
g dx = f ( n −1) g − f ( n − 2) g ′ + f ( n −3) g ′′ − (−1)n
This is called generalized integration by parts.
∫ fg
(n)
dx
INDEFINITE INTEGRALS
70
Important Transformations
Often in practice an integral can be simplified by using an appropriate transformation or substitution together
with Formula 16.6. The following list gives some transformations and their effects.
1
16.49.
∫ F(ax + b)dx = a ∫ F(u) du
16.50.
∫ F(
16.51.
∫ F(
16.52.
∫ F(
a 2 − x 2 ) dx = a
∫ F(a cos u) cos u du
where x = a sin u
16.53.
∫ F(
x 2 + a 2 ) dx = a
∫ F(a sec u) sec
where x = a tan u
16.54.
∫ F(
x 2 − a 2 ) dx = a
∫ F(a tan u) sec u tan u du
16.55.
∫ F (e
16.56.
∫ F(ln x) dx = ∫ F(u) e du
16.57.
∫ F ⎜⎝sin
n
where u = ax + b
ax + b ) dx =
2
a
∫ u F(u) du
where u = ax + b
ax + b ) dx =
n
a
∫u
where u = n ax + b
ax
) dx =
1
a
∫
n −1
F(u) du
2
u du
F (u)
du
u
u
⎛
−1
x⎞
dx = a
a ⎟⎠
∫ F(u) cos u du
where x = a sec u
where u = eax
where u = ln x
where u = sin −1
Similar results apply for other inverse trigonometric functions.
16.58.
∫ F(sin x , cos x) dx = 2
⎛ 2u 1 − u 2 ⎞ du
F
∫ ⎜⎝1 + u 2 , 1 + u 2 ⎟⎠ 1 + u 2
where u = tan
x
2
x
a
17
TABLES of SPECIAL INDEFINITE
INTEGRALS
Here we provide tables of special indefinite integrals. As stated in the remarks on page 67, here a, b, p, q, n
are constants, restricted if indicated; e = 2.71828 . . . is the natural base of logarithms; ln u denotes the natural
logarithm of u, where it is assumed that u > 0 (in general, to extend formulas to cases where u < 0 as well, replace
ln u by ln |u|); all angles are in radians; and all constants of integration are omitted but implied. It is assumed in
all cases that division by zero is excluded.
Our integrals are divided into types which involve the following algebraic expressions and functions:
(1)
ax + b
ax + b
(2)
(3)
ax + b and px + q
(5)
(25)
eax
(14) x3 + a3
(26)
ln x
(27)
sinh ax
(16)
x 4 ± a4
x n ± an
(28)
cosh ax
(17)
sin ax
(29)
sinh ax and cosh ax
(15)
ax + b and px + q
ax + b and px + q
(4)
ax 2 + bx + c
(13)
(6)
x2 + a2
(18) cos ax
(30)
tanh ax
(7)
x2 – a2, with x2 > a2
(19)
sin ax and cos ax
(31)
coth ax
(8)
a2 – x2, with x2 < a2
(20)
tan ax
(32)
sech ax
csch ax
(9)
x 2 + a2
(21) cot ax
(33)
(10)
x 2 − a2
(22) sec ax
(34) inverse hyperbolic functions
a2 − x 2
(12) ax2 + bx + c
(11)
(23) csc ax
(24)
inverse trigonometric functions
Some integrals contain the Bernouilli numbers Bn and the Euler numbers En defined in Chapter 23.
(1)
Integrals Involving ax ⴙ b
dx
1
x dx
x
17.1.1.
∫ ax + b = a ln (ax + b)
17.1.2.
∫ ax + b = a − a
17.1.3.
17.1.4.
2
ln (ax + b)
x 2dx
(ax + b)2 2b(ax + b) b 2
=
−
+ 3 ln (ax + b)
∫ ax + b
a3
a
2a 3
dx
1 ⎛ x ⎞
∫ x (ax + b) = b ln ⎜⎝ ax + b⎟⎠
a ⎛ ax + b⎞
dx
1
=−
+ ln
bx b 2 ⎜⎝ x ⎟⎠
(ax + b)
17.1.5.
∫x
17.1.6.
∫ (ax + b)
17.1.7.
b
2
−1
a(ax + b)
x dx
b
1
∫ (ax + b)2 = a 2 (ax + b) + a 2 ln (ax + b)
dx
2
=
2
=
x 2 dx
17.1.8.
∫ (ax + b)
17.1.9.
∫ x (ax + b)
dx
2
ax + b
b2
2b
− 3
−
ln (ax + b)
3
a
a (ax + b) a 3
=
1
1 ⎛ x ⎞
+ ln
b(ax + b) b 2 ⎜⎝ ax + b⎟⎠
71
TABLES OF SPECIAL INDEFINITE INTEGRALS
72
dx
−a
1
2a ⎛ ax + b⎞
− 2 + 3 ln ⎜
2 =
2
(ax + b)
b (ax + b) b x b
⎝ x ⎟⎠
17.1.10.
∫x
17.1.11.
∫ (ax + b)
17.1.12.
∫ (ax + b)
17.1.13.
∫ (ax + b)
17.1.14.
∫ (ax + b)
17.1.15.
∫ x (ax + b) dx =
2
3
=
−1
2(ax + b)2
3
=
−1
b
+
a 2 (ax + b) 2a 2 (ax + b)2
3
=
2b
b2
1
− 3
+ ln (ax + b)
a (ax + b) 2a (ax + b)2 a 3
dx
x dx
x 2 dx
n
3
(ax + b)n+1
. If n = −1, see 17.1.1.
(n + 1)a
dx =
n
(ax + b)n+ 2 b(ax + b)n+1
, n ≠ −1, − 2
−
(n + 2)a 2
(n + 1))a 2
If n = –1, –2, see 17.1.2 and 17.1.7.
∫x
17.1.16.
2
(ax + b)n dx =
(ax + b)n+3 2b(ax + b)n+ 2 b 2 (ax + b)n+1
−
+
(n + 3)a 3
(n + 2)a 3
(n + 1)a 3
If n = –1, –2, –3, see 17.1.3, 17.1.8, and 17.1.13.
nb
⎧ x m +1 (ax + b)n
n −1
m
⎪ m + n + 1 + m + n + 1 ∫ x (ax + b) dx
⎪⎪ x m (ax + b)n+1
mb
m −1
m
n
n
x
(
ax
+
b
)
dx
=
⎨ (m + n + 1)a − (m + n + 1)a ∫ x (ax + b) dx
∫
⎪ m +1
n +1
⎪− x (ax + b) + m + n + 2 x m (ax + b)n+1 dx
(n + 1)b
(n + 1)b ∫
⎪⎩
17.1.17.
(2)
Integrals Involving
ax ⴙ b
17.2.1.
∫
dx
2 ax + b
=
a
ax + b
17.2.2.
∫
x dx
2(ax − 2b)
=
ax + b
3a 2
ax + b
17.2.3.
∫
x 2 dx
2(3a 2 x 2 − 4 abx + 8b 2 )
=
ax + b
15a 3
ax + b
17.2.4.
∫
⎧
⎪
dx
⎪
=⎨
x ax + b ⎪
⎪
⎩
17.2.5.
∫x
17.2.6.
∫
17.2.7.
∫x
dx
2
ax + b
=−
ax + b dx =
⎛ ax + b − b ⎞
ln ⎜
⎟
b ⎝ ax + b + b ⎠
1
2
−b
tan −1
ax + b
−b
ax + b
a
−
2b
bx
∫x
dx
ax + b
2 (ax + b)3
3a
ax + b dx =
2(3ax − 2b)
(ax + b)3
15a 2
(see 17.2.12.)
TABLES OF SPECIAL INDEFINITE INTEGRALS
2(15a 2 x 2 − 12abx + 8b 2 )
(ax + b)3
105a 3
17.2.8.
∫x
17.2.9.
∫
ax + b
dx = 2 ax + b + b
x
17.2.10.
∫
ax + b
ax + b a
dx = −
+
2
2
x
x
17.2.11.
∫
xm
2 x m ax + b
2mb
dx =
−
2
m
1
a
2
m
(
)
(
+
+ 1)a
ax + b
17.2.12.
∫x
17.2.13.
∫x
17.2.14.
∫
ax + b
ax + b
a
dx = −
+
xm
(m − 1) x m −1 2(m − 1)
∫x
17.2.15.
∫
ax + b
−(ax + b)3/ 2 (2m − 5)a
dx =
−
m
x
(m − 1)bx m −1 (2m − 2)b
∫
17.2.16.
∫ (ax + b)m / 2 dx =
17.2.17.
∫ x (ax + b)m / 2 dx =
17.2.18.
∫x
17.2.19.
(ax + b)m / 2
2(ax + b)m / 2
=
+b
dx
∫ x
m
17.2.20.
(ax + b)m / 2
(ax + b)( m + 2)/ 2 ma
dx
=
−
+
2
∫ x
2b
bx
17.2.21.
∫ x (ax + b)
(3)
ax + b dx =
2
m
m
2
∫x
∫x
dx
(See 17.2.12.)
ax + b
dx
ax + b dx =
=
x m −1
dx
ax + b
∫
∫x
m −1
dx
ax + b
2x m
2mb
(ax + b)3/ 2 −
(2m + 3)a
(2m + 3)a
m −1
∫x
m −1
ax + b dx
dx
ax + b
ax + b
dx
x m −1
2(ax + b)( m + 2)/ 2
a 2 (m + 2)
2(ax + b)( m + 4 )/ 2 2b(ax + b)( m + 2)/ 2
−
a 2 (m + 2)
a 2 (m + 4)
(ax + b)m / 2 dx =
m/2
(See 17.2.12.)
ax + b
dx
ax + b
(2m − 3)a
=−
m −1 −
(
2m − 2)b
(m − 1)bx
ax + b
dx
73
2(ax + b)( m +6)/ 2 4 b(ax + b)( m + 4 )/ 2 2b 2 (ax + b)( m + 2)/ 2
−
+
a 3 (m + 4)
a 3 (m + 2)
a 3 (m + 6)
(ax + b)( m − 2)/ 2
dx
∫
x
(ax + b)m / 2
∫ x dx
2
1
+
(m − 2)b(ax + b)( m − 2)/ 2 b
dx
∫ x(ax + b)
( m − 2 )/ 2
Integrals Involving ax ⴙ b and px ⴙ q
dx
1
x dx
1
⎛ px + q⎞
17.3.1.
∫ (ax + b)( px + q) = bp − aq ln ⎜⎝ ax + b ⎟⎠
17.3.2.
∫ (ax + b)( px + q) = bp − aq ⎨⎩a ln (ax + b) − p ln ( px + q)⎬⎭
⎧b
q
⎫
TABLES OF SPECIAL INDEFINITE INTEGRALS
74
dx
1
⎧ 1
1
⎧
⎛ px + q⎞ ⎫
p
17.3.3.
∫ (ax + b) ( px + q) = bp − aq ⎨⎩ax + b + bp − aq ln ⎜⎝ ax + b ⎟⎠ ⎬⎭
17.3.4.
∫ (ax + b) ( px + q) = bp − aq ⎨⎩bp − aq ln ⎜⎝ px + q⎟⎠ − a(ax + b) ⎬⎭
17.3.5.
∫ (ax + b) ( px + q) = (bp − aq)a (ax + b) + (bp − aq)
17.3.6.
∫ (ax + b)
2
x dx
⎛ ax + b ⎞
q
⎫
b
2
x 2 dx
b2
2
1
2
{
2
⎧q2
⎫
b(bp − 2aq)
ln (ax + b)⎬
⎨ ln ( px + q) +
2
p
a
⎩
⎭
dx
1
−1
=
( px + q)n (n − 1)(bp − aq) (ax + b)m −1 ( px + q)n−1
m
+ a(m + n − 2)
ax + b
∫ (ax + b)
dx
( px + q)n−1
m
ax bp − aq
ln( px + q)
+
p
p2
17.3.7.
∫ px + q dx =
17.3.8.
⎧
⎧(ax + b)m +1
⎫
(ax + b)m
−1
2
+
(
n
−
m
−
)
a
dx⎬
⎨
⎪
∫
n −1
n −1
( px + q)
⎭
⎪(n − 1)(bp − aq) ⎩ ( px + q)
m
m
m −1
⎪
⎧ (ax + b)
⎫
(ax + b)
(ax + b)
−1
∫ ( px + q)n dx = ⎨(n − m − 1) p ⎨⎩( px + q)n−1 + m(bp − aq) ∫ ( px + q)n dx⎬⎭
⎪
⎪ −1 ⎧ (ax + b)m
(ax + b)m −1 ⎫
a
m
dx⎬
−
⎨
⎪
∫
n −1
( px + q)n −1 ⎭
⎩(n − 1) p ⎩( px + q)
(4)
}
ax ⴙ b and px ⴙ q
Integrals Involving
px + q
2(apx + 3aq − 2bp)
dx =
3a 2
ax + b
17.4.1.
∫
17.4.2.
⎧
⎛ p(ax + b) − bp − aq ⎞
1
⎪
ln ⎜
⎟
⎪ bp − aq p ⎝ p(ax + b) + bp − aq ⎠
dx
=
∫ ( px + q) ax + b ⎨
2
p(ax + b)
⎪
tan −1
⎪
aq − bp
⎩ aq − bp p
ax + b
⎧ 2 ax + b
bp − aq ⎛ p(ax + b) − bp − aq ⎞
⎪
+
ln ⎜
⎟
p
⎪
p p
ax + b
⎝ p(ax + b) + bp − aq ⎠
dx = ⎨
px + q
p(ax + b)
⎪ 2 ax + b 2 aq − bp
tan −1
−
⎪
p
aq − bp
p p
⎩
17.4.3.
∫
17.4.4.
∫ ( px + q)n ax + b dx =
17.4.5.
∫ ( px + q)
17.4.6.
( px + q)n
2( px + q)n ax + b 2n(aq − bp)
dx
=
+
∫ ax + b
(2n + 1)a
(2n + 1)a
17.4.7.
∫ ( px + q)
dx
n
ax + b
n
ax + b
dx =
=
2( px + q)n+1 ax + b
bp − aq
+
(2n + 3) p
(2n + 3) p
( px + q)n
∫ ax + b
ax + b
(2n − 3)a
n −1 +
2
(
n
− 1)(aq − bp)
(n − 1)(aq − bp)( px + q)
− ax + b
a
+
(n − 1) p( px + q)n−1 2(n − 1) p
( px + q)n−1 dx
∫ ax + b
∫ ( px + q)
dx
n −1
ax + b
∫ ( px + q)
dx
n −1
ax + b
TABLES OF SPECIAL INDEFINITE INTEGRALS
(5)
ax ⴙ b and
Integrals Involving
∫
17.5.1.
⎧
⎪
dx
⎪
=⎨
(ax + b)( px + q) ⎪
⎪
⎩
2
ln
ap
2
− ap
(
75
px + q
a( px + q ) + p(ax + b)
tan −1
)
− p(ax + b)
a( px + q)
17.5.2.
∫
x dx
=
(ax + b)( px + q)
(ax + b)( px + q) bp + aq
−
ap
2ap
17.5.3.
∫
(ax + b)( px + q) dx =
2apx + bp + aq
(bp − aq)2
(ax + b)( px + q) −
8ap
4ap
17.5.4.
∫
px + q
dx =
ax + b
17.5.5.
∫ ( px + q)
(6)
(ax + b)( px + q) aq − bp
+
a
2a
dx
2 ax + b
=
(ax + b)( px + q) (aq − bp) px + q
Integrals Involving x2 ⴙ a2
x
dx
1
= tan −1
a
+ a2 a
17.6.1.
∫x
17.6.2.
∫x
17.6.3.
∫x
x 2 dx
−1 x
2
2 = x − a tan
a
+a
17.6.4.
∫x
x 3 dx
x 2 a2
− ln ( x 2 + a 2 )
2
2 =
2
2
+a
17.6.5.
∫ x(x
17.6.6.
∫x
17.6.7.
∫ x (x
17.6.8.
∫ (x
2
17.6.9.
∫ (x
2
17.6.10.
2
x dx
1
= ln ( x 2 + a 2 )
2
+a
2
2
2
⎛ x2 ⎞
dx
1
=
ln
⎜
⎟
2
+ a 2 ) 2a 2 ⎝ x 2 + a 2 ⎠
dx
1
1
x
= − 2 − 3 tan −1
a
(x 2 + a2 )
a x a
⎛ x2 ⎞
dx
1
1
=
−
−
ln
⎜
⎟
2
+ a2 )
2a 2 x 2 2a 4 ⎝ x 2 + a 2 ⎠
3
dx
x
x
1
tan −1
=
+
a
+ a 2 ) 2 2a 2 ( x 2 + a 2 ) 2a 3
x dx
−1
2 2 =
2
+a )
2( x + a 2 )
∫ (x
x
x 2 dx
1
−x
=
+
tan −1
a
+ a 2 )2 2( x 2 + a 2 ) 2a
2
∫
∫
dx
(ax + b)( px + q)
dx
(ax + b)( px + q)
∫
dx
(ax + b)( px + q)
TABLES OF SPECIAL INDEFINITE INTEGRALS
76
x 3dx
a2
1
=
∫ ( x 2 + a 2 )2 2( x 2 + a 2 ) + 2 ln ( x 2 + a 2 )
17.6.11.
dx
1
1
⎛ x2 ⎞
=
+
ln
x ( x 2 + a 2 )2 2a 2 ( x 2 + a 2 ) 2a 4 ⎜⎝ x 2 + a 2 ⎟⎠
17.6.12.
∫
17.6.13.
∫x
2
dx
1
x
3
x
=− 4 − 4 2
−
tan −1
a
( x 2 + a 2 )2
a x 2a ( x + a 2 ) 2a 5
dx
1
1
1 ⎛ x2 ⎞
2 2 = −
4 2 −
4
2
2 − 6 ln ⎜ 2
x (x + a )
2a x
2a ( x + a ) a
⎝ x + a 2 ⎟⎠
17.6.14.
∫
17.6.15.
∫ (x
2
17.6.16.
∫ (x
2
17.6.17.
∫ x(x
17.6.18.
∫ (x
17.6.19.
∫x
(7)
3
2
dx
x
2n − 3
=
+
+ a 2 )n 2(n − 1)a 2 ( x 2 + a 2 )n−1 (2n − 2)a 2
2
dx
1
1
=
+
+ a 2 )n 2(n − 1)a 2 ( x 2 + a 2 )n−1 a 2
x m dx
=
+ a 2 )n
∫ (x
x m − 2 dx
− a2
+ a 2 )n−1
2
∫x
2
17.7.2.
∫x
2
dx
1
=
( x 2 + a 2 )n a 2
∫x
m
dx
1 ⎛ x − a⎞
=
ln
− a 2 2a ⎜⎝ x + a⎟⎠
dx
1
−
( x 2 + a 2 )n−1 a 2
or
−
∫
17.7.4.
∫x
x
1
coth −1
a
a
x dx
1
= ln ( x 2 − a 2 )
− a2 2
x 3dx
x 2 a2
=
+ ln ( x 2 − a 2 )
2
2
−a
2
2
dx
1
⎛ x 2 − a2 ⎞
=
ln
x ( x 2 − a 2 ) 2a 2 ⎜⎝ x 2 ⎟⎠
17.7.5.
∫
17.7.6.
∫ x (x
dx
1
1
⎛ x − a⎞
=
+
ln
2
− a 2 ) a 2 x 2a 3 ⎜⎝ x + a⎟⎠
2
dx
1
1
⎛ x2 ⎞
=
−
ln
x 3( x 2 − a 2 ) 2a 2 x 2 2a 4 ⎜⎝ x 2 − a 2 ⎟⎠
17.7.7.
∫
17.7.8.
∫ (x
2
∫ x(x
2
dx
+ a 2 )n−1
x m − 2 dx
2
+ a 2 )n
x 2 dx
a ⎛ x − a⎞
ln
2
2 = x +
2 ⎜⎝ x + a⎟⎠
x −a
17.7.3.
dx
+ a 2 )n−1
∫ (x
Integrals Involving x2 ⴚ a2, x2 > a2
17.7.1.
2
x dx
−1
=
+ a 2 )n 2(n − 1)( x 2 + a 2 )n−1
2
m
∫ (x
dx
−x
1
⎛ x − a⎞
ln
=
−
− a 2 )2 2a 2 ( x 2 − a 2 ) 4 a 3 ⎜⎝ x + a⎟⎠
∫x
m−2
dx
( x 2 + a 2 )n
TABLES OF SPECIAL INDEFINITE INTEGRALS
∫ (x
17.7.9.
x dx
−1
=
− a 2 )2 2( x 2 − a 2 )
2
17.7.10.
x 2 dx
−x
1 ⎛ x − a⎞
∫ ( x − a 2 )2 = 2( x 2 − a 2 ) + 4a ln ⎜⎝ x + a⎟⎠
17.7.11.
∫ (x
2
x 3dx
−a2
1
=
+ ln ( x 2 − a 2 )
2 2
−a )
2( x 2 − a 2 ) 2
2
dx
−1
1
⎛ x2 ⎞
=
+
ln
x ( x 2 − a 2 )2 2a 2 ( x 2 − a 2 ) 2a 4 ⎜⎝ x 2 − a 2 ⎟⎠
17.7.12.
∫
17.7.13.
∫ x (x
2
dx
1
x
3
⎛ x − a⎞
=− 4 − 4 2
−
ln
− a 2 )2
a x 2a ( x − a 2 ) 4 a 5 ⎜⎝ x + a⎟⎠
2
dx
1
1
1 ⎛ x2 ⎞
2 2 = −
4 2 −
4
2
2 + 6 ln ⎜ 2
x (x − a )
2a x
2a ( x − a ) a
⎝ x − a 2 ⎟⎠
17.7.14.
∫
17.7.15.
∫ (x
2
17.7.16.
∫ (x
2
17.7.17.
∫ x(x
17.7.18.
x m dx
∫ ( x − a 2 )n =
17.7.19.
∫x
(8)
77
3
2
dx
−x
2n − 3
=
−
− a 2 )n 2(n − 1)a 2 ( x 2 − a 2 )n−1 (2n − 2)a 2
2
dx
−1
1
=
−
− a 2 )n 2(n − 1)a 2 ( x 2 − a 2 )n−1 a 2
x m − 2 dx
∫ ( x − a 2 )n−1 + a 2
2
dx
1
2 n =
a2
(x − a )
2
∫x
m−2
dx
1 ⎛ a + x⎞
ln
2 =
2
a ⎜⎝ a − x ⎟⎠
−x
17.8.1.
∫a
2
17.8.2.
∫a
2
17.8.3.
∫a
x 2 dx
a ⎛ a + x⎞
= − x + ln ⎜
2
2 ⎝ a − x ⎟⎠
− x2
17.8.4.
∫a
dx
1
2
2 n − 2
a
(x − a )
or
1
x
tanh −1
a
a
x dx
1
= − ln (a 2 − x 2 )
2
− x2
x 3dx
x 2 a2
= − − ln (a 2 − x 2 )
2
2
−x
2
2
dx
1
⎛ x2 ⎞
2 =
2 ln ⎜ 2
x (a − x ) 2a
⎝ a − x 2 ⎟⎠
∫
17.8.6.
∫x
2
2
dx
− a 2 )n−1
∫ x(x
2
dx
− a 2 )n−1
x m − 2 dx
∫ ( x 2 − a 2 )n
Integrals Involving x2 ⴚ a2, x2 < a2
17.8.5.
2
x dx
−1
=
− a 2 )n 2(n − 1)( x 2 − a 2 )n−1
2
m
∫ (x
dx
1
1
⎛ a + x⎞
= − 2 + 3 ln ⎜
(a 2 − x 2 )
a x 2a
⎝ a − x ⎟⎠
∫x
m
dx
( x − a 2 )n−1
2
TABLES OF SPECIAL INDEFINITE INTEGRALS
78
dx
1
1
⎛ x2 ⎞
2
2 = −
2 2 +
4 ln ⎜ 2
x (a − x )
2a x
2a
⎝ a − x 2 ⎟⎠
17.8.7.
∫
17.8.8.
∫ (a
2
17.8.9.
∫ (a
2
3
dx
x
1
⎛ a + x⎞
=
+
ln
− x 2 )2 2a 2 (a 2 − x 2 ) 4a 3 ⎜⎝ a − x ⎟⎠
x dx
1
=
− x 2 )2 2(a 2 − x 2 )
17.8.10.
x 2 dx
x
1 ⎛ a + x⎞
∫ (a 2 − x 2 )2 = 2(a 2 − x 2 ) − 4a ln ⎜⎝ a − x ⎟⎠
17.8.11.
x 3dx
a2
1
=
∫ (a 2 − x 2 )2 2(a 2 − x 2 ) + 2 ln (a 2 − x 2 )
dx
1
1
⎛ x2 ⎞
=
+
ln
x (a 2 − x 2 )2 2a 2 (a 2 − x 2 ) 2a 4 ⎜⎝ a 2 − x 2 ⎟⎠
17.8.12.
∫
17.8.13.
∫ x (a
2
17.8.14.
∫ x (a
2
17.8.15.
∫ (a
2
17.8.16.
∫ (a
2
17.8.17.
∫ x (a
17.8.18.
∫ (a
2
17.8.19.
∫x
dx
1
=
(a 2 − x 2 ) n a 2
(9)
2
3
dx
−1
x
3
⎛ a + x⎞
+ 4 2
2 2 =
4
2 +
5 ln ⎜
−x )
a x 2a (a − x ) 4 a
⎝ a − x ⎟⎠
dx
−1
1
1 ⎛ x2 ⎞
2 2 =
4 2 +
4
2
2 + 6 ln ⎜ 2
−x )
2a x
2a (a − x ) a
⎝ a − x 2 ⎟⎠
dx
x
2n − 3
2 n =
2
2
2 n −1 +
−x )
2(n − 1)a (a − x )
(2n − 2)a 2
2
dx
1
1
=
+
− x 2 )n 2(n − 1)a 2 (a 2 − x 2 )n−1 a 2
dx
x m − 2 dx
−
2
− x 2 )n
∫ (a
Integrals Involving
∫x
m
∫ (a
∫ x (a
dx
− x 2 )n−1
dx
− x 2 )n−1
x m − 2 dx
− x 2 )n−1
dx
1
+
(a 2 − x 2 )n−1 a 2
∫x
x 2 + a2
x
a
∫
x +a
17.9.2.
∫
x dx
= x 2 + a2
x 2 + a2
17.9.3.
∫
x 2 dx
x x 2 + a2 a2
=
− ln( x + x 2 + a 2 )
2
2
x 2 + a2
17.9.4.
∫
x 3 dx
( x 2 + a 2 )3 / 2
=
− a2 x 2 + a2
3
x 2 + a2
2
2
2
17.9.1.
2
2
x dx
1
=
− x 2 )n 2(n − 1)(a 2 − x 2 )n−1
x m dx
= a2
− x 2 )n
m
∫ (a
= ln ( x + x 2 + a 2 ) or sinh −1
m−2
dx
(a 2 − x 2 ) n
TABLES OF SPECIAL INDEFINITE INTEGRALS
17.9.5.
∫x
17.9.6.
∫x
17.9.7.
∫x
17.9.8.
17.9.9.
dx
1 ⎛ a + x 2 + a2 ⎞
=
−
ln
⎟
a ⎜⎝
x
x 2 + a2
⎠
dx
x 2 + a2
=−
2
2
a2 x
x +a
2
⎛ a + x 2 + a2 ⎞
dx
x 2 + a2
1
=−
+ 3 ln ⎜
2 2
⎟
2
2
x
2a x
2a
x +a
⎝
⎠
2
2
2
x
x
a
a
+
∫ x 2 + a 2 dx = 2 + 2 ln ( x + x 2 + a 2 )
( x 2 + a 2 )3 / 2
∫ x x 2 + a 2 dx =
3
3
17.9.10.
∫ x 2 x 2 + a 2 dx =
x ( x 2 + a 2 )3 / 2 a 2 x x 2 + a 2 a 4
−
− ln ( x + x 2 + a 2 )
4
8
8
17.9.11.
∫x
( x 2 + a 2 )5 / 2 a 2 ( x 2 + a 2 )3 / 2
−
5
3
x 2 + a 2 dx =
3
∫
⎛ a + x 2 + a2 ⎞
x 2 + a2
dx = x 2 + a 2 − a ln ⎜
⎟
x
x
⎝
⎠
∫
x 2 + a2
x 2 + a2
dx
=
−
+ ln ( x + x 2 + a 2 )
x2
x
17.9.14.
∫
x 2 + a2
x 2 + a2
1 ⎛ a + x 2 + a2 ⎞
dx = −
ln
−
3
2
⎟
x
2a ⎜⎝
x
2x
⎠
17.9.15.
∫ (x
2
17.9.16.
∫ (x
2
17.9.17.
x 2 dx
∫ ( x 2 + a 2 )3 / 2 =
17.9.18.
∫ (x
17.9.12.
17.9.13.
2
dx
x
=
+ a 2 )3 / 2 a 2 x 2 + a 2
x dx
=
+ a 2 )3 / 2
−1
x 2 + a2
−x
+ ln ( x + x 2 + a 2 )
x + a2
2
x 3 dx
= x 2 + a2 +
+ a 2 )3 / 2
a2
x + a2
2
∫
dx
1
1 ⎛ a + x 2 + a2 ⎞
−
2 3/ 2 =
3 ln ⎜
⎟
x
x(x + a )
a2 x 2 + a2 a
⎝
⎠
∫
dx
x 2 + a2
x
=
−
− 4 2
2
2
2 3/ 2
4
x (x + a )
a x
a x + a2
17.9.21.
∫
⎛ a + x 2 + a2 ⎞
dx
3
3
−1
l
=
−
+
n
⎟
x
x 3( x 2 + a 2 )3/ 2 2a 2 x 2 x 2 + a 2 2a 4 x 2 + a 2 2a 5 ⎜⎝
⎠
17.9.22.
∫ (x
17.9.23.
∫ x ( x 2 + a 2 )3/ 2 dx =
17.9.19.
17.9.20.
2
2
+ a 2 )3/ 2 dx =
x ( x 2 + a 2 )3/ 2 3a 2 x x 2 + a 2 3 4
+
+ a ln ( x + x 2 + a 2 )
4
8
8
( x 2 + a 2 )5 / 2
5
79
TABLES OF SPECIAL INDEFINITE INTEGRALS
80
2
+ a 2 )3/ 2 dx =
x ( x 2 + a 2 )5 / 2 a 2 x ( x 2 + a 2 )3 / 2 a 4 x x 2 + a 2 a 6
−
−
− ln ( x + x 2 + a 2 )
6
24
16
16
2
+ a 2 )3/ 2 dx =
( x 2 + a 2 ) 7 / 2 a 2 ( x 2 + a 2 )5 / 2
−
7
5
17.9.24.
∫ x (x
17.9.25.
∫ x (x
17.9.26.
∫
⎛ a + x 2 + a2 ⎞
( x 2 + a 2 )3 / 2
( x 2 + a 2 )3 / 2
+ a 2 x 2 + a 2 − a 3 ln ⎜
dx =
⎟
3
x
x
⎝
⎠
17.9.27.
∫
( x 2 + a 2 )3 / 2
( x 2 + a 2 )3 / 2 3 x x 2 + a 2 3 2
+
+ a ln ( x + x 2 + a 2 )
dx = −
2
x
2
2
x
17.9.28.
∫
⎛ a + x 2 + a2 ⎞
( x 2 + a 2 )3 / 2
( x 2 + a 2 )3 / 2 3 2
3
dx = −
x + a 2 − a ln ⎜
+
3
2
⎟
x
2
2
x
2x
⎝
⎠
(10)
2
3
Integrals Involving
dx
x 2 − a2
∫
x −a
17.10.2.
∫
x 2 dx
x x 2 − a2 a2
=
+ ln ( x + x 2 − a 2 )
2
2
2
2
x −a
17.10.3.
∫
x 3 dx
( x 2 − a 2 )3 / 2
=
+ a2 x 2 − a2
3
x 2 − a2
17.10.4.
∫x
2
17.10.5.
∫x
17.10.6.
∫x
17.10.7.
∫
17.10.8.
∫x
17.10.9.
∫x
2
= ln ( x + x 2 − a 2 ),
dx
x 2 − a2
2
2
= x 2 − a2
x
1
sec −1
a
a
x 2 − a2
a2 x
dx
1
x 2 − a2
x
+ 3 sec −1
2a 2 x 2
2a
a
=
x 2 − a 2 dx =
x x 2 − a2 a2
− ln ( x + x 2 − a 2 )
2
2
x 2 − a 2 dx =
( x 2 − a 2 )3 / 2
3
x 2 − a 2 dx =
2
x −a
2
dx
=
x 2 − a2
x 2 − a2
3
=
∫
x dx
17.10.1.
x ( x 2 − a 2 )3 / 2 a 2 x x 2 − a 2 a 4
+
−
ln ( x + x 2 − a 2 )
4
8
8
( x 2 − a 2 )5 / 2 a 2 ( x 2 − a 2 )3 / 2
+
5
3
17.10.10.
∫x
17.10.11.
∫
x 2 − a2
x
dx = x 2 − a 2 − a sec −1
x
a
17.10.12.
∫
x 2 − a2
x 2 − a2
dx = −
+ ln ( x + x 2 − a 2 )
2
x
x
3
x 2 − a 2 dx =
TABLES OF SPECIAL INDEFINITE INTEGRALS
1
x 2 − a2
x 2 − a2
x
+
dx = −
sec −1
3
2x 2
2a
x
a
17.10.13.
∫
17.10.14.
∫ (x
2
17.10.15.
∫ (x
2
17.10.16.
x 2dx
x
∫ ( x 2 − a 2 )3/ 2 = − x 2 − a 2 + ln ( x + x 2 − a 2 )
17.10.17.
∫ (x
17.10.18.
∫ x(x
17.10.19.
∫x
17.10.20.
∫x
17.10.21.
∫ (x
17.10.22.
∫ x ( x 2 − a 2 )3/ 2 dx =
17.10.23.
∫ x 2 ( x 2 − a 2 )3/ 2 dx =
x ( x 2 − a 2 )5 / 2 a 2 x ( x 2 − a 2 )3 / 2 a 4 x x 2 − a 2 a 6
+
−
+ ln ( x + x 2 − a 2 )
6
24
16
16
17.10.24.
∫x
( x 2 − a 2 ) 7 / 2 a 2 ( x 2 − a 2 )5 / 2
+
7
5
17.10.25.
∫
( x 2 − a 2 )3 / 2
( x 2 − a 2 )3 / 2
x
dx =
− a 2 x 2 − a 2 + a 3 sec −1
x
3
a
17.10.26.
∫
( x 2 − a 2 )3 / 2
( x 2 − a 2 )3 / 2 3 x x 2 − a 2 3 2
dx = −
+
− a ln ( x + x 2 − a 2 )
2
x
2
2
x
17.10.27.
( x 2 − a 2 )3 / 2
( x 2 − a 2 )3 / 2 3 x 2 − a 2 3
x
=
−
+
− a sec −1
dx
∫ x3
2
2x
2
2
a
(11)
2
2
3
x dx
=
− a 2 )3 / 2
−1
x 2 − a2
x 3 dx
= x 2 − a2 −
− a 2 )3 / 2
2
a2
x 2 − a2
dx
1
x
−1
=
− 3 sec −1
2 3/ 2
2
2
2
−a )
a
a
a x −a
dx
x 2 − a2
x
− 4 2
2 3/ 2 = −
a4 x
(x − a )
a x − a2
2
dx
1
3
3
x
=
−
− 5 sec −1
2 3/ 2
2
2
2
2
4
2
2
(x − a )
2a
a
2a x x − a
2a x − a
2
3
dx
x
=− 2 2
− a 2 )3 / 2
a x − a2
2
− a 2 )3/ 2 dx =
x ( x 2 − a 2 )3/ 2 3a 2 x x 2 − a 2 3 4
−
+ a ln ( x + x 2 − a 2 )
4
8
8
( x 2 − a 2 )5 / 2
5
( x 2 − a 2 )3/ 2 dx =
Integrals Involving
17.11.1.
∫
dx
x
= sin −1
2
a
a −x
17.11.2.
∫
x dx
= − a2 − x 2
a2 − x 2
2
a2 − x 2
81
TABLES OF SPECIAL INDEFINITE INTEGRALS
82
17.11.3.
∫
x 2 dx
x a2 − x 2 a2
x
=
−
+ sin −1
2
2
a
2
2
a −x
17.11.4.
∫
x 3 dx
(a 2 − x 2 )3 / 2
=
− a2 a2 − x 2
2
2
3
a −x
17.11.5.
∫
dx
1 ⎛ a + a2 − x 2 ⎞
=
−
ln
⎟
a ⎜⎝
x
x a2 − x 2
⎠
dx
a2 − x 2
=
−
a2 x
a2 − x 2
17.11.6.
∫x
17.11.7.
∫x
17.11.8.
∫
17.11.9.
∫x
17.11.10.
∫x
17.11.11.
∫x
17.11.12.
∫
⎛ a + a2 − x 2 ⎞
a2 − x 2
dx = a 2 − x 2 − a ln ⎜
⎟
x
x
⎝
⎠
∫
a2 − x 2
a2 − x 2
x
dx
=
−
− sin −1
2
x
a
x
17.11.14.
∫
a2 − x 2
a2 − x 2
1 ⎛ a + a2 − x 2 ⎞
dx = −
+
ln
3
2
⎟
x
2a ⎜⎝
x
2x
⎠
17.11.15.
∫ (a
2
17.11.16.
∫ (a
2
17.11.17.
∫ (a
2
17.11.18.
x 3 dx
a2
2
2
a
x
=
−
+
/
2
3
2
∫ (a − x )
a2 − x 2
17.11.19.
∫ x (a
17.11.13.
2
⎛ a + a2 − x 2 ⎞
dx
a2 − x 2
1
=
−
−
ln
2
2
3
⎜
⎟
x
2a x
2a
a2 − x 2
⎝
⎠
3
a 2 − x 2 dx =
x a2 − x 2 a2
x
+ sin −1
2
2
a
a 2 − x 2 dx = −
(a 2 − x 2 ) 3 / 2
3
2
a 2 − x 2 dx = −
3
a 2 − x 2 dx =
x
x (a 2 − x 2 ) 3 / 2 a 2 x a 2 − x 2 a 4
+
+ sin −1
4
8
8
a
(a 2 − x 2 ) 5 / 2 a 2 (a 2 − x 2 ) 3 / 2
−
5
3
dx
x
2 3/ 2 =
2
−x )
a a2 − x 2
x dx
=
− x 2 )3 / 2
1
a − x2
x 2 dx
=
− x 2 )3 / 2
x
x
− sin −1
2
a
a −x
2
2
2
2
dx
1
1 ⎛ a + a2 − x 2 ⎞
− 3 ln ⎜
2 3/ 2 =
⎟
2
2
2
x
−x )
a
a a −x
⎝
⎠
dx
a2 − x 2
x
=
−
+ 4 2
2
2
2 3/ 2
4
x (a − x )
a x
a a − x2
17.11.20.
∫
17.11.21.
∫x
3
⎛ a + a2 − x 2 ⎞
dx
3
3
−1
+
−
l
n
2 3/ 2 =
5
⎜
⎟
x
(a − x )
2a 2 x 2 a 2 − x 2 2a 4 a 2 − x 2 2a
⎝
⎠
2
TABLES OF SPECIAL INDEFINITE INTEGRALS
83
x (a 2 − x 2 )3/ 2 3a 2 x a 2 − x 2 3 4 −1 x
+
+ a sin
a
4
8
8
17.11.22.
∫ (a
17.11.23.
∫ x (a
17.11.24.
∫ x 2 (a 2 − x 2 )3/ 2 dx = −
17.11.25.
∫x
17.11.26.
⎛ a + a2 − x 2 ⎞
(a 2 − x 2 )3 / 2
(a 2 − x 2 )3 / 2
2
2
2
3
dx
a
a
x
a
ln
=
+
−
−
⎟
⎜
∫ x
x
3
⎠
⎝
17.11.27.
∫
(a 2 − x 2 )3 / 2
(a 2 − x 2 )3/ 2 3x a 2 − x 2 3 2 −1 x
dx = −
−
− a sin
2
x
x
2
2
a
17.11.28.
∫
⎛ a + a2 − x 2 ⎞
(a 2 − x 2 )3 / 2
(a 2 − x 2 )3 / 2 3 a 2 − x 2 3
dx = −
−
+ a ln ⎜
3
2
⎟
x
2
2
x
2x
⎝
⎠
(12)
17.12.1.
− x 2 )3/ 2 dx =
2
3
2
− x 2 )3/ 2 dx = −
(a 2 − x 2 )3/ 2 dx =
(a 2 − x 2 )5 / 2
5
x
x (a 2 − x 2 ) 5 / 2 a 2 x (a 2 − x 2 ) 3 / 2 a 4 x a 2 − x 2 a 6
+
+
+ sin −1
6
24
16
16
a
(a 2 − x 2 ) 7 / 2 a 2 (a 2 − x 2 ) 5 / 2
−
7
5
Integrals Involving ax2 ⴙ bx ⴙ c
⎧
2
2ax + b
tan −1
⎪
2
4 ac − b 2
dx
⎪ 4 ac − b
=
∫ ax 2 + bx + c ⎨ 1
⎛ 2ax + b − b 2 − 4 ac ⎞
⎪
ln ⎜
⎪ b 2 − 4 ac ⎝ 2ax + b + b 2 − 4 ac ⎟⎠
⎩
If b 2 = 4 ac, ax 2 + bx + c = a( x + b / 2a)2 and the results 17.1.6 to 17.1.10 and 17.1.14 to 17.1.17 can be used.
If b = 0 use results on page 75. If a or c = 0 use results on pages 71–72.
x dx
1
b
=
ln (ax 2 + bx + c) −
2a
+ bx + c 2a
∫ ax
17.12.3.
x 2 dx
x
b
b 2 − 2ac
2
ax
bx
c
=
−
+
+
+
ln
(
)
∫ ax 2 + bx + c a 2a 2
2a 2
17.12.4.
x m dx
x m −1
c
=
∫ ax + bx + c (m − 1)a − a
17.12.5.
∫ x (ax
17.12.6.
∫x
2
17.12.7.
∫x
n
17.12.8.
∫ (ax
2
17.12.9.
∫ (ax
2
2
2
2
∫ ax
dx
+ bx + c
17.12.2.
2
x m − 2 dx
b
∫ ax + bx + c − a
∫ ax
2
dx
+ bx + c
dx
b
⎛ ax 2 + bx + c⎞ 1 b 2 − 2ac
= 2 ln ⎜
⎟⎠ − cx + 2c 2
x2
(ax + bx + c) 2c
⎝
dx
1
b
=−
n −1 −
c
(ax + bx + c)
(n − 1)cx
∫x
n −1
dx
+ bx + c
2
2
2
2
x m−1dx
∫ ax + bx + c
2
x2
dx
1 ⎛
⎞ b
ln ⎜ 2
=
−
+ bx + c) 2c ⎝ ax + bx + c⎟⎠ 2c
∫ ax
∫ ax
dx
a
−
(ax + bx + c) c
2
dx
2ax + b
2a
+
2 =
2
2
+ bx + c)
(4 ac − b )(ax + bx + c) 4 ac − b 2
x dx
bx + 2c
b
=−
−
+ bx + c)2
(4 ac − b 2 )(ax 2 + bx + c) 4 ac − b 2
∫ ax
2
2
dx
+ bx + c
∫x
n− 2
dx
(ax + bx + c)
dx
+ bx + c
∫ ax
2
dx
+ bx + c
2
TABLES OF SPECIAL INDEFINITE INTEGRALS
84
17.12.10.
∫ (ax
2
17.12.11.
∫ (ax
2
x 2 dx
(b 2 − 2ac) x + bc
2c
+
2 =
+ bx + c)
a(4 ac − b 2 )(ax 2 + bx + c) 4 ac − b 2
2
dx
+ bx + c
x m dx
x m −1
(m − 1)c
+
n = −
+ bx + c)
(2n − m − 1)a(ax 2 + bx + c)n−1 (2n − m − 1)a
−
17.12.12.
x 2 n−1dx
1
∫ (ax 2 + bx + c)n = a
17.12.13.
∫ x (ax
17.12.14.
∫x
2
17.12.15.
∫x
m
2
(n − m )b
(2n − m − 1)a
∫ (ax
x m −1dx
∫ (ax + bx + c)n
dx
1
b
=
−
+ bx + c)2 2c(ax 2 + bx + c) 2c
x 2 n−3 dx
b
∫ (ax 2 + bx + c)n − a
∫ (ax
dx
1
3a
=−
−
(ax 2 + bx + c)2
cx (ax 2 + bx + c) c
2
dx
1
+
+ bx + c)2 c
∫ (ax
2
x 2 n− 2 dx
∫ (ax 2 + bx + c)n
∫ x (ax
dx
2b
−
c
+ bx + c)2
dx
1
(m + 2n − 3)a
n = −
m −1
2
n −1 −
(m − 1)c
(ax + bx + c)
(m − 1)cx (ax + bx + c)
2
Integrals Involving
(m + n − 2)b
(m − 1)c
x m − 2 dx
+ bx + c)n
2
2
x 2 n−3 dx
c
∫ (ax 2 + bx + c)n−1 − a
−
(13)
∫ ax
∫x
m −1
dx
+ bx + c)
2
∫ x (ax
∫x
m−2
2
dx
+ bx + c)2
dx
(ax + bx + c)n
2
dx
(ax + bx + c)n
2
ax 2 ⴙ bx ⴙ c
In the following results if b 2 = 4 ac, ax 2 + bx + c = a ( x + b / 2a) and the results 17.1 can be used. If b = 0
use the results 17.9. If a = 0 or c = 0 use the results 17.2 and 17.5.
17.13.1.
17.13.2.
17.13.3.
∫
⎧ 1
2
⎪⎪ a ln (2 a ax + bx + c + 2ax + b)
dx
=⎨
ax 2 + bx + c ⎪− 1 sin −1 ⎛ 2ax + b ⎞ or 1 sinh −1 ⎛ 2ax + b ⎞
⎜⎝ b 2 − 4 ac ⎟⎠
⎜⎝ 4 ac − b 2 ⎟⎠
a
⎪⎩ − a
∫
x dx
=
2
ax + bx + c
∫
x 2 dx
2ax − 3b
3b 2 − 4 ac
2
=
ax
+
bx
+
c
+
4a 2
8a 2
ax 2 + bx + c
ax 2 + bx + c
b
−
2a
a
∫
dx
ax + bx + c
2
dx
ax + bx + c
∫
2
⎧ 1
⎛ 2 c ax 2 + bx + c + bx + 2c⎞
ln ⎜
⎪−
⎟
x
dx
⎪ c ⎝
⎠
=
⎨
2
x ax + bx + c ⎪ 1
⎛
⎛
1
bx + 2c ⎞
bx + 2c ⎞
−1
−1
⎪ −c sin ⎜⎝ | x | b 2 − 4 ac ⎟⎠ or − c sinh ⎜⎝ | x | 4 ac − b 2 ⎟⎠
⎩
17.13.4.
∫
17.13.5.
∫x
17.13.6.
∫
2
dx
ax 2 + bx + c b
=−
−
2c
cx
ax + bx + c
2
ax 2 + bx + c dx =
∫x
dx
ax 2 + bx + c
(2ax + b) ax 2 + bx + c 4 ac − b 2
+
4a
8a
∫
dx
ax 2 + bx + c
TABLES OF SPECIAL INDEFINITE INTEGRALS
17.13.7.
∫x
ax 2 + bx + c dx =
(ax 2 + bx + c)3/ 2 b(2ax + b)
−
ax 2 + bx + c
3a
8a 2
−
17.13.8.
∫ x 2 ax 2 + bx + c dx =
b(4 ac − b 2 )
16a 2
∫
∫
ax 2 + bx + c
b
dx = ax 2 + bx + c +
x
2
17.13.10.
∫
ax 2 + bx + c
ax 2 + bx + c
dx = −
+a
2
x
x
17.13.11.
∫ (ax
2
17.13.12.
∫ (ax
2
17.13.13.
∫ (ax
2
17.13.14.
∫ x (ax
∫
dx
∫
ax + bx + c
2
dx
∫
+
ax + bx + c
2
∫x
b
2
ax 2 + bx + c dx
dx
ax + bx + c
2
∫x
dx
ax + bx + c
2
x dx
2(bx + 2c)
=
+ bx + c)3/ 2 (b 2 − 4 ac) ax 2 + bx + c
x 2 dx
(2b 2 − 4 ac) x + 2bc
1
+
3/ 2 =
+ bx + c)
a(4 ac − b 2 ) ax 2 + bx + c a
2
dx
1
1
=
+
+ bx + c)3/ 2 c ax 2 + bx + c c
∫x
∫
dx
b
−
2
c
ax + bx + c
2
∫ (ax
17.13.17.
∫ x (ax 2 + bx + c)n+1/ 2 dx =
17.13.18.
∫ (ax
2
2
2
dx
+ bx + c)3/ 2
dx
+ bx + c)3/ 2
dx
ax 2 + bx + c
b
(ax 2 + bx + c)n+3/ 2
−
2a
a(2n + 3)
∫ (ax
2
+ bx + c)n+1/ 2 dx
dx
2(2ax + b)
=
+ bx + c)n+1/ 2 (2n − 1)(4 ac − b 2 )(ax 2 + bx + c)n−1/ 2
+
∫ x (ax
∫x
∫ (ax
∫ (ax
(2ax + b)(ax 2 + bx + c)n+1/ 2 (2n + 1)(4 ac − b 2 )
+
∫ (ax 2 + bx + c)n−1/ 2 dx
4 a(n + 1)
8a(n + 1)
17.13.16.
+ bx + c)n+1/ 2 dx =
3b
2c 2
dx
ax 2 + bx + c
2
dx
ax 2 + 2bx + c
b 2 − 2ac
+
3/ 2 = − 2
x (ax + bx + c)
2c 2
c x ax 2 + bx + c
2
2
+c
∫
dx
2(2ax + b)
=
+ bx + c)3/ 2 (4 ac − b 2 ) ax 2 + bx + c
−
17.13.19.
dx
ax 2 + bx + c
6ax − 5b
5b 2 − 4 ac
2
3/ 2
(
+
+
)
+
ax
bx
c
24 a 2
16a 2
17.13.9.
17.13.15.
85
2
8a(n − 1)
(2n − 1)(4 ac − b 2 )
∫ (ax
2
dx
+ bx + c)n−1/ 2
dx
1
=
+ bx + c)n+1/ 2 (2n − 1)c(ax 2 + bx + c)n−1/ 2
+
1
c
∫ x (ax
2
dx
b
−
+ bx + c)n−1/ 2 2c
∫ (ax
2
dx
+ bx + c)n+1/22
TABLES OF SPECIAL INDEFINITE INTEGRALS
86
(14)
Integrals Involving x3 ⴙ a3
Note that for formulas involving x3 – a3 replace a with –a.
∫
⎛ ( x + a) 2 ⎞
dx
1
1
2x − a
=
ln
tan −1
+
⎜⎝ 2
3
3
2
2⎟
⎠
2
x +a
6a
x − ax + a
a 3
a 3
∫
x dx
1 ⎛ x 2 − ax + a 2 ⎞
1
2x − a
=
ln ⎜
tan −1
+
⎟
3
3
2
⎝
⎠
( x + a)
x +a
6a
a 3
a 3
17.14.3.
∫
x 2 dx
1
= ln ( x 3 + a 3 )
x 3 + a3 3
17.14.4.
∫ x(x
17.14.1.
17.14.2.
dx
1
⎛ x3 ⎞
3 =
3 ln ⎜ 3
+ a ) 3a
⎝ x + a 3 ⎟⎠
3
⎛ x 2 − ax + a 2 ⎞
dx
1
1
1
2x − a
−
tan −1
=
−
−
ln
⎟
⎜
2
3
3
3
4
2
4
⎝ ( x + a) ⎠ a 3
x (x + a )
a x 6a
a 3
17.14.5.
∫
17.14.6.
∫ (x
17.14.7.
∫ (x
17.14.8.
x 2 dx
1
∫ ( x 3 + a3 )2 = − 3( x 3 + a3 )
17.14.9.
∫ x(x
3
⎛ ( x + a) 2 ⎞
1
dx
x
2
2x − a
tan −1
= 3 3
+ 5 ln ⎜ 2
+
3 2
3
⎝ x − ax + a 2 ⎟⎠ 3a 5 3
+a )
3a ( x + a ) 9a
a 3
⎛ x 2 − ax + a 2 ⎞
1
x dx
x2
1
2x − a
= 3 3
+
tan −1
ln ⎜
+
3 2
3
4
⎝ ( x + a)2 ⎟⎠ 3a 4 3
+a )
3a ( x + a ) 18a
a 3
3
3
dx
1
1
⎛ x3 ⎞
3 2 =
3
3
3 +
6 ln ⎜ 3
+a )
3a ( x + a ) 3a
⎝ x + a 3 ⎟⎠
∫
dx
1
x2
4
=
−
−
− 6
2
3
3 2
6
6
3
3
x (x + a )
a x 3a ( x + a ) 3a
17.14.11.
∫
x m dx
x m−2
x m −3dx
3
a
=
−
∫ x 3 + a3
x 3 + a3 m − 2
17.14.12.
∫x
17.14.10.
(15)
dx
−1
1
=
−
( x 3 + a 3 ) a 3 (n − 1) x n−1 a 3
∫x
n−3
x dx
+ a3
3
(See 17.14.2.)
dx
( x 3 + a3 )
Integrals Involving x4 ⴞ a4
17.15.1.
∫x
4
17.15.2.
∫x
4
17.15.3.
n
∫x
∫
⎛ x 2 + ax 2 + a 2 ⎞
⎛ x 2⎞⎤
dx
1
1 ⎡ −1 ⎛ x 2 ⎞
− 3
ln ⎜ 2
− tan −1 ⎜1 +
tan ⎜1 −
⎢
4 =
⎟
⎟
3
2
a ⎠
a ⎟⎠ ⎥⎥
+a
4 a 2 ⎝ x − ax 2 + a ⎠ 2a 2 ⎣⎢
⎝
⎝
⎦
2
x dx
1
−1 x
4 =
2 tan
2
+a
2a
a
⎛ x 2 − ax 2 + a 2 ⎞
⎛ x 2⎞⎤
x 2 dx
1
1 ⎡ −1 ⎛ x 2 ⎞
−
ln ⎜ 2
⎢tan ⎜1 − a ⎟ − tan −1 ⎜1 + a ⎟ ⎥
4
4 =
2⎟
x +a
4 a 2 ⎝ x + ax 2 + a ⎠ 2a 2 ⎢⎣
⎝
⎠
⎝
⎠ ⎥⎦
TABLES OF SPECIAL INDEFINITE INTEGRALS
x 3 dx
1
= ln ( x 4 + a 4 )
4
+ a4 4
17.15.4.
∫x
17.15.5.
∫ x(x
∫
17.15.6.
dx
1
⎛ x4 ⎞
4 =
4 ln ⎜ 4
+ a ) 4a
⎝ x + a 4 ⎟⎠
4
⎛ x 2 − ax 2 + a 2 ⎞
dx
1
1
− 5
ln ⎜ 2
4
4 = − 4
x (x + a )
a x 4 a 2 ⎝ x + ax 2 + a 2 ⎟⎠
2
+
17.15.7.
∫x
3
17.15.8.
∫x
4
17.15.9.
∫x
4
∫
17.15.11.
∫x
∫
17.15.13.
∫x
(16)
2
dx
1
1
−1 x
4
4 = −
4 2 −
6 tan
2
(x + a )
2a x
2a
a
dx
1
1
x
⎛ x − a⎞
=
ln
tan −1
−
a
− a 4 4 a 3 ⎜⎝ x + a⎟⎠ 2a 3
x dx
1
⎛ x 2 − a2 ⎞
4 =
2 ln ⎜ 2
−a
4a
⎝ x + a 2 ⎟⎠
x 3dx
1
= ln ( x 4 − a 4 )
4
− a4 4
4
3
dx
1
1
1
x
⎛ x − a⎞
=
+
ln
tan −1
+
a
( x 4 − a 4 ) a 4 x 4 a 5 ⎜⎝ x + a⎟⎠ 2a 5
dx
1
1
⎛ x 2 − a2 ⎞
4
4 =
4 2 +
6 ln ⎜ 2
x ( x − a ) 2a x
4a
⎝ x + a 2 ⎟⎠
3
Integrals Involving xn ⴞ an
∫
⎛ xn ⎞
dx
1
=
ln
⎜
⎟
x ( x n + a n ) na n ⎝ x n + a n ⎠
17.16.2.
∫
x n−1dx 1
= ln ( x n + a n )
x n + an n
17.16.3.
x m dx
∫ ( x + a n )r =
17.16.4.
∫x
17.16.5.
∫x
17.16.1.
17.16.6.
2a
⎡ −1 ⎛ x 2 ⎞
⎛ x 2⎞⎤
⎢tan ⎜1 − a ⎟ − tan −1 ⎜1 + a ⎟ ⎥
2 ⎢⎣
⎝
⎠
⎝
⎠ ⎥⎦
dx
1
⎛ x 4 − a4 ⎞
4 =
4 ln ⎜
x ( x − a ) 4a
⎝ x 4 ⎟⎠
17.15.12.
∫
1
5
x
x 2 dx
1 ⎛ x − a⎞ 1
ln
tan −1
=
+
a
x 4 − a 4 4 a ⎜⎝ x + a⎟⎠ 2a
17.15.10.
17.15.14.
87
n
∫
m
x m − n dx
∫ ( x + a n )r −1 − a n
n
dx
1
=
( x n + a n )r a n
∫x
m
x m − n dx
∫ ( x n + a n )r
dx
1
−
( x n + a n )r −1 a n
⎛ x n + an − an ⎞
dx
1
ln
=
x n + a n n a n ⎜⎝ x n + a n + a n ⎟⎠
dx
1
⎛ x n − an ⎞
n =
n ln ⎜
x ( x − a ) na
⎝ x n ⎟⎠
n
∫x
m−n
dx
( x n + a n )r
TABLES OF SPECIAL INDEFINITE INTEGRALS
88
x n −1dx 1
= ln ( x n − a n )
n
− an n
17.16.7.
∫x
17.16.8.
x m dx
x m − n dx
x m − n dx
n
a
=
+
n
r
n
n
r
n
∫ (x − a )
∫ ( x − a ) ∫ ( x − a n )r −1
17.16.9.
∫x
17.16.10.
∫x
17.16.11.
∫x
n
m
dx
1
= n
n r
(x − a )
a
n
dx
x n − an
2
=
n an
∫x
m−n
cos −1
dx
1
− n
n
n r
(x − a )
a
∫x
m
dx
( x − a n )r −
n
an
xn
m
⎛ x + a cos[(2k − 1)π /2m]⎞
x p−1dx
1
(2k − 1) pπ
sin
tan −1 ⎜
=
2m
2m− p ∑
ma
2
m
+a
⎝ a sin[(2k − 1)π /2m] ⎟⎠
k =1
2m
−
m
⎞
1
(2k − 1) pπ ⎛ 2
(2k − 1)π
cos
ln ⎜ x + 2ax cos
+ a 2⎟
∑
2ma 2 m − p k =1
2m
2
m
⎝
⎠
where 0 < p 2m.
17.16.12.
∫x
m −1
⎞
x p−1dx
1
kpπ ⎛ 2
kπ
cos
ln ⎜ x − 2ax cos
+ a 2⎟
=
2m
2m− p ∑
2ma
m
m
−a
⎝
⎠
k =1
2m
−
⎛ x − a cos (kπ /m)⎞
1 m −1 kpπ
sin
tan −1 ⎜
ma 2 m − p ∑
m
⎝ a sin (kπ /m) ⎟⎠
k =1
1
{ln ( x − a) + (−1) p ln ( x + a)}
2ma 2 m − p
+
where 0 < p 2m.
17.16.13.
∫
m
⎛ x + a cos[2kπ /(2m + 1)]⎞
x p−1dx
2(−1) p−1
2kpπ
sin
tan −1 ⎜
=
2 m +1
2 m +1
2 m − p+1 ∑
x
(2m + 1)a
+a
2m + 1
⎝ a sin[2kπ /(2m + 1)] ⎟⎠
k =1
−
m
⎞
pπ ⎛ 2
(−1) p−1
2kp
2kπ
ln ⎜ x + 2ax cos
cos
+ a 2⎟
2 m − p+1 ∑
(2m + 1)a
2m + 1 ⎝
2m + 1
⎠
k =1
+
(−1) p−1 ln( x + a)
(2m + 1)a 2 m − p+1
where 0 < p 2m + 1.
17.16.14.
∫
m
⎛ x − a cos[2kπ /(2m + 1)]⎞
x p−1dx
2kpπ
−2
sin
tan −1 ⎜
=
∑
2 m +1
2 m +1
2 m − p+1
x
(2m + 1)a
2m + 1
−a
⎝ a sin[2kπ /(2m + 1)] ⎟⎠
k =1
+
m
⎞
1
2kpπ ⎛ 2
2kπ
cos
ln ⎜ x − 2ax cos
+ a 2⎟
∑
2 m − p+1
(2m + 1)a
2m + 1 ⎝
2m + 1
⎠
k =1
+
where 0 < p 2m + 1.
ln ( x − a)
(2m + 1)a 2 m − p+1
TABLES OF SPECIAL INDEFINITE INTEGRALS
(17)
89
Integrals Involving sin ax
17.17.1.
∫ sin ax dx = −
cos ax
a
17.17.2.
∫ x sin ax dx =
sin ax x cos ax
−
a
a2
17.17.3.
∫x
17.17.4.
∫x
17.17.5.
sin ax
(ax )3 (ax )5
dx
=
ax
−
+
− ⋅⋅⋅
∫ x
3 ⋅ 3! 5 ⋅ 5!
17.17.6.
∫
17.17.7.
∫ sin ax = α ln(csc ax − cot ax ) = α ln tan
17.17.8.
∫ sin ax = a
17.17.9.
2x
⎛ 2 x2 ⎞
cos ax
2 sin ax + ⎜ 3 −
a ⎟⎠
a
⎝a
2
sin ax dx =
3
⎛ 3x 2 6 ⎞
⎛ 6x x 3 ⎞
sin ax dx = ⎜ 2 − 4 ⎟ sin ax + ⎜ 3 − ⎟ cos ax
a⎠
a ⎠
⎝a
⎝a
sin ax
sin ax
cos ax
+ a∫
dx = −
dx (See 17.18.5.)
x2
x
x
dx
1
1
ax
2
⎫⎪
2(22 n −1 − 1) Bn (ax )2 n+1
(ax )3 7(ax )5
1 ⎧⎪
+ ⎬
+
++
ax +
2 ⎨
(2n + 1)!
18
1800
⎪⎭
⎩⎪
x sin 2ax
∫ sin 2 ax dx = 2 − 4a
x dx
x 2 x sin 2ax cos 2ax
−
−
4
4a
8a 2
17.17.10.
∫ x sin
17.17.11.
∫ sin
17.17.12.
∫ sin ax dx =
17.17.13.
∫ sin
17.17.14.
∫ sin
17.17.15.
∫ sin px sin qx dx =
17.17.16.
∫ 1 − sin ax = a tan ⎜⎝ 4 +
17.17.17.
∫ 1 − sin ax = a tan ⎜⎝ 4 +
17.17.18.
∫ 1 + sin ax = − α tan ⎜⎝ 4 −
17.17.19.
∫ 1 + sin ax = − a tan ⎜⎝ 4 −
3
2
ax dx =
ax dx = −
4
cos ax cos3 ax
+
3a
a
3x sin 2ax sin 4 ax
−
+
8
4a
32a
dx
1
= − cot ax
2
a
ax
cos ax
ax
dx
1
ln tan
=−
+
3
2
ax
2a sin 2 ax 2a
sin ( p − q) x sin ( p + q) x
−
2( p − q)
2( p + q)
(If p = ± q, see 17.17.9.)
dx
1
⎛π
ax ⎞
2 ⎟⎠
x dx
x
⎛π
⎛ π ax ⎞
ax ⎞ 2
+ ln sin ⎜ − ⎟
2 ⎟⎠ a 2
⎝4 2 ⎠
dx
1
⎛π
ax ⎞
2 ⎟⎠
x dx
x
⎛π
ax ⎞ 2
⎛ π ax ⎞
+ ln sin ⎜ + ⎟
2 ⎟⎠ a 2
⎝4 2 ⎠
TABLES OF SPECIAL INDEFINITE INTEGRALS
90
dx
17.17.20.
∫ (1 − sin ax)
17.17.21.
∫ (1 + sin ax )
17.17.22.
=
2
=−
dx
∫
1
⎛ π ax ⎞ 1
⎛ π ax ⎞
tan ⎜ + ⎟ +
tan 3 ⎜ + ⎟
a
2a
4
2
6
⎝
⎝4 2 ⎠
⎠
2
⎛ π ax ⎞ 1
⎛ π ax ⎞
1
tan ⎜ − ⎟ −
tan 3 ⎜ − ⎟
2a
⎝ 4 2 ⎠ 6a
⎝4 2 ⎠
1
2
⎧
−1 p tan 2 ax + q
tan
⎪a p2 − q 2
p2 − q 2
⎪
dx
=
p + q sin ax ⎨
⎛ p tan 12 ax + q − q 2 − p2 ⎞
⎪
1
ln
⎜
⎟
⎪
2
2
⎝ p tan 12 ax + q + q 2 − p2 ⎠
⎩a q − p
(If p = ± q, see 17.17.16 and 17.17.18.)
17.17.23.
dx
∫ ( p + q sin ax)
2
=
q cos ax
p
+
a( p2 − q 2 )( p + q sin ax ) p2 − q 2
dx
∫ p + q sin ax
(If p = ± q, see 17.17.20 and 17.17.21.)
17.17.24.
∫p
⎧
p2 − q 2 tan ax
1
−1
tan
⎪
2
2
p
dx
⎪ ap p − q
=⎨
2
2
2
p − q sin ax
⎛ q 2 − p2 tan ax + p⎞
1
⎪
ln
⎜ 2
⎟
2
2
⎪
⎝ q − p2 tan ax − p⎠
⎩2ap q − p
17.17.25.
∫
17.17.26.
∫x
17.17.27.
∫
17.17.28.
∫ sin
17.17.29.
∫ sin
17.17.30.
∫ sin
(18)
p2 + q 2 tan ax
p
dx
1
=
tan −1
+ q 2 sin 2 ax ap p2 + q 2
2
sin ax dx = −
m
x m cos ax mx m −1 sin ax m(m − 1)
+
−
a
a2
a2
sin ax
sin ax
a
+
dx = −
xn
(n − 1) x n −1 n − 1
n
ax dx = −
∫
m−2
sin ax dx
cos ax
dx (See 17.18.30.)
x n −1
sin n−1 ax cos ax n − 1
+
an
n
dx
n−2
− cos ax
=
+
n
n −1
ax a(n − 1) sin ax n − 1
∫x
∫ sin
∫ sin
n− 2
ax dx
dx
n− 2
ax
x dx
− x cos ax
1
n−2
+
=
−
n
ax a(n − 1) sin n−1 ax a 2 (n − 1)(n − 2) sin n− 2 ax n − 1
Integrals Involving cos ax
sin ax
a
17.18.1.
∫ cos ax dx =
17.18.2.
∫ x cos ax dx =
17.18.3.
∫ x 2 cos ax dx =
cos ax x sin ax
+
a
a2
2x
⎛ x2 2 ⎞
cos
+
ax
⎜⎝ a − a 3 ⎟⎠ sin ax
a2
x dx
n− 2
ax
∫ sin
TABLES OF SPECIAL INDEFINITE INTEGRALS
91
17.18.4.
⎛ 3x 2 6 ⎞
⎛ x 3 6x⎞
3
cos
=
−
cos
x
ax
dx
ax
+
⎜
⎟
⎜⎝ a − a 3 ⎟⎠ sin ax
2
4
∫
a ⎠
⎝a
17.18.5.
cos ax
(ax )2 (ax )4 (ax )6
dx
=
ln
x
−
+
−
+
∫ x
2 ⋅ 2! 4 ⋅ 4 ! 6 ⋅ 6!
17.18.6.
∫
17.18.7.
∫ cos ax = a ln (sec ax + tan ax ) = a ln tan ⎜⎝ 4 +
17.18.8.
En (ax )2 n+ 2
x dx
1 ⎧(ax )2 (ax )4 5(ax )6
⎫
=
+
+
+
+
+ ⎬
⎨
∫ cos ax a 2 ⎩ 2
(2n + 2)(2n)!
8
144
⎭
17.18.9.
∫ cos
cos ax
cos ax
−a
dx = −
x2
x
dx
2
∫
sin ax
dx (See 17.17.5.)
x
1
1
ax dx =
x 2 x sin 2ax cos 2ax
+
+
4
4a
8a 2
∫ x cos2 ax dx =
17.18.11.
∫ cos3 ax dx =
sin ax sin 3 ax
−
3a
a
17.18.12.
∫ cos
3x sin 2ax sin 4 ax
+
+
8
4a
32a
17.18.13.
∫ cos
17.18.14.
∫ cos
17.18.15.
∫ cos ax cos px dx =
17.18.16.
∫ 1 − cos ax = − a cot
17.18.17.
∫ 1 − cos ax = − a cot
17.18.18.
∫ 1 + cos ax = a tan
17.18.19.
∫ 1 + cos ax = a tan
17.18.20.
∫ (1 − cos ax)
17.18.21.
∫ (1 + cos ax)
17.18.22.
ax dx =
dx
2
ax
=
ax ⎞
2 ⎟⎠
x sin 2ax
+
2
4a
17.18.10.
4
⎛π
tan ax
a
sin ax
dx
1
⎛ π ax ⎞
ln tan ⎜ + ⎟
=
+
3
ax 2a cos 2 ax 2a
⎝4 2 ⎠
dx
1
ax
2
x dx
x
ax 2
ax
+ ln sin
2 a2
2
dx
1
ax
2
x dx
x
ax 2
ax
+ ln cos
2 a2
2
dx
2
=−
2
=
dx
∫
sin(a − p) x sin(a + p) x
+
2(a − p)
2(a + p)
(If a = ± p, see 17.18.9.)
1
ax 1
ax
cot
cot 3
−
2a
2 6a
2
1
ax 1
ax
tan
tan 3
+
2a
2 6a
2
⎧
2
1
tan −1 ( p − q) / ( p + q) tan ax
⎪
2
2
2
dx
⎪ a p −q
=⎨
⎛ tan 1 ax + (q + p) / (q − p) ⎞
p + q cos ax ⎪
1
2
ln ⎜
⎪ a q 2 − p2 ⎝ tan 1 ax − (q + p) / (q − p) ⎟⎠
2
⎩
(If p = ± q, see 17.18.16
and 17.18.18.)
TABLES OF SPECIAL INDEFINITE INTEGRALS
92
dx
17.18.23.
∫ ( p + q cos ax )
17.18.24.
∫p
2
q sin ax
p
−
a(q 2 − p2 )( p + q cos ax ) q 2 − p2
(If p = ± q see 17.18.19
and 17.18.20.)
dx
∫ p + q cos ax
dx
1
p tan ax
=
tan −1
+ q 2 cos 2 ax ap p2 + q 2
p2 + q 2
2
1
⎧
−1 p tan ax
⎪ap p2 − q 2 tan
p2 − q 2
dx
⎪
=
⎛ p tan ax − q 2 − p2 ⎞
p2 − q 2 cos 2 ax ⎨
1
⎪
ln
⎪2ap q 2 − p2 ⎜⎝ p tan ax + q 2 − p2 ⎟⎠
⎩
17.18.25.
∫
17.18.26.
∫ x m cos ax dx =
17.18.27.
∫
17.18.28.
∫ cosn ax dx =
17.18.29.
∫ cos
17.18.30.
∫ cos
(19)
=
x m sin ax mx m −1
m(m − 1)
+
cos ax −
a
a2
a2
cos ax
cos ax
a
−
dx = −
n
n −1
x
(n − 1) x
n −1
dx
n
ax
=
∫
∫x
m−2
cos ax dx
sin ax
dx (Seee 17.17.27.)
x n −1
sin ax cos n−1 ax n − 1
+
an
n
∫ cos
n−2
sin ax
+
a(n − 1) cos n−1 ax b − 1
∫ cos
n− 2
ax dx
dx
n− 2
ax
x dx
x sin ax
1
n−2
=
−
+
n
ax a(n − 1) cos n−1 ax a 2 (n − 1)(n − 2) cos n− 2 ax n − 1
Integrals Involving sin ax and cos ax
17.19.1.
∫ sin ax cos ax dx =
sin 2 ax
2a
17.19.2.
sin px cos qx dx = −
cos( p − q) x cos( p + q) x
−
2( p − q)
2( p + q)
17.19.3.
∫ sin n ax cos ax dx =
sin n+1 ax
(n + 1)a
17.19.4.
∫ cos ax sin ax dx = −
cos n+1 ax
(n + 1)a
17.19.5.
∫ sin
x sin 4 ax
−
8
32a
17.19.6.
∫ sin ax cos ax = a ln tan ax
17.19.7.
∫ sin
17.19.8.
∫ sin ax cos
17.19.9.
∫ sin
n
2
ax cos 2 ax dx =
dx
2
(If n = −1, see 17.20.1.)
1
dx
1
1
⎛ π ax ⎞
= ln tan ⎜ + ⎟ −
ax cos ax a
⎝ 4 2 ⎠ a sin ax
dx
2
(If n = −1, see 17.21.1.)
2
ax
=
ax
1
1
ln tan
+
a
2 a cos ax
dx
2 cot 2ax
=−
a
ax cos 2 ax
x dx
∫ cos
n− 2
ax
TABLES OF SPECIAL INDEFINITE INTEGRALS
93
17.19.10.
sin 2 ax
sin ax 1
⎛ ax π ⎞
∫ cos ax dx = − a + a ln tan ⎜⎝ 2 + 4 ⎟⎠
17.19.11.
∫
17.19.12.
∫ cos ax (1 ± sin ax ) = ∓ 2a(1 ± sin ax ) + 2a ln tan ⎜⎝ 2
17.19.13.
∫ sin ax (1 ± cos ax ) = ± 2a(1 ± cos ax ) + 2a ln tan
17.19.14.
∫ sin ax ± cos ax = a
17.19.15.
∫ sin ax ± cos ax = 2 ∓ 2a ln (sin ax ± cos ax )
17.19.16.
∫ sin ax ± cos ax = ± 2 + 2a ln (sin ax ± cos ax )
17.19.17.
∫ p + q cos ax = − aq ln ( p + q cos ax)
17.19.18.
∫ p + q sin ax = aq ln ( p + q sin ax)
17.19.19.
∫ ( p + q cos ax )
17.19.20.
∫ ( p + q sin ax)
17.19.21.
17.19.22.
cos 2 ax
cos ax 1
ax
dx =
+ ln tan
sin ax
2
a
a
dx
1
1
⎛ ax
dx
1
1
ax
2
dx
sin ax dx
1
x
sin ax dx
⎛ ax π ⎞
ln tan ⎜ ± ⎟
⎝ 2 8⎠
2
1
x
cos ax dx
π⎞
+ ⎟
4⎠
1
1
cos ax dx
1
sin ax dx
n
cos ax dx
n
=
1
aq(n − 1)( p + q cos ax )n−1
=
−1
aq(n − 1)( p + q sin ax )n−1
∫
⎛ ax + tan −1 (q / p) ⎞
dx
1
ln tan ⎜
=
⎟⎠
p sin ax + q cos ax a p2 + q 2
2
⎝
∫
⎧
⎛ p + (r − q) tan(ax / 2 ) ⎞
2
⎪
tan −1 ⎜
⎟
⎪ a r 2 − p2 − q 2
r 2 − p2 − q 2
⎝
⎠
dx
=⎨
p sin ax + q cos ax + r ⎪
⎛ p − p2 + q 2 − r 2 + (r − q) tan (ax / 2 ) ⎞
1
ln ⎜
⎟
⎪
2
2
2
⎝ p + p2 + q 2 − r 2 + (r − q) tan (ax / 2 ) ⎠
⎩a p + q − r
(If r = q see 17.19.23. If r2 = p2 + q2 see 17.19.24.)
17.19.23.
dx
1
⎛
∫ p sin ax + q(1 + cos ax ) = ap ln ⎜⎝q + p tan
ax ⎞
2 ⎟⎠
dx
−1
⎛ π ax + tan −1 (q / p)⎞
=
tan
⎜⎝ 4 ∓
⎟⎠
2
p sin ax + q cos ax ± p2 + q 2 a p2 + q 2
17.19.24.
∫
17.19.25.
∫p
2
17.19.26.
∫p
2
dx
1
⎛ p tan ax ⎞
=
tan −1 ⎜
sin 2 ax + q 2 cos 2 ax apq
⎝ q ⎟⎠
dx
1
⎛ p tan ax − q⎞
ln ⎜
=
2
2
apq
2
sin ax − q cos ax
⎝ p tan ax + q⎟⎠
2
TABLES OF SPECIAL INDEFINITE INTEGRALS
94
17.19.27.
⎧ sin m −1 ax cos n+1 ax m − 1
sin m − 2 ax cos n ax dx
+
⎪−
a(m + n)
m+n ∫
m
n
sin
ax
cos
ax
dx
=
⎨
m +1
n −1
∫
⎪ sin ax cos ax + n − 1 sin m ax cos n− 2 ax dx
m+n ∫
a (m + n )
⎩
17.19.28.
sin m −1 ax
m − 1 sin m − 2 ax
⎧
−
−
1
n
⎪ a(n − 1) cos ax n − 1 ∫ cos n− 2 ax dx
⎪
m
sin m +1 ax
sin ax
m−n+2
sin m ax
⎪
dx
=
−
⎨
∫ cosn−2 ax dx
∫ cosn ax
n −1
a(n − 1) cos n−1 ax
⎪
m − 1 sin m − 2 ax
− sin m −1 ax
⎪
dx
+
n −1
⎪⎩ a(m − n) co
os ax m − n ∫ cos n ax
17.19.29.
− cos m −1 ax
m − 1 cos m − 2 ax
⎧
⎪ a(n − 1) sin n−1 ax − n − 1 ∫ sin n− 2 ax dx
⎪ − cos m +1 ax
cos m ax
cos m ax
m−n+2
⎪
dx
=
−
⎨
n
n
−
1
∫ sin n−2 ax dx
∫ sin ax
n −1
a(n − 1) sin ax
⎪
cos m −1 ax
m − 1 cos m − 2 ax
⎪
+
n −1
∫ sin n ax dx
⎩⎪ a(m − n) sin ax m − n
17.19.30.
1
m+n−2
⎧
⎪ a(n − 1) sin m −1 ax cos n−1 ax + n − 1
dx
∫ sin m ax cosn ax = ⎨
−1
m+n−2
⎪
+
m −1
⎪⎩a(m − 1) sin m −1 ax cos n−1 ax
(20)
dx
ax cos n− 2 ax
dx
∫ sin m−2 ax cosn ax
∫ sin
m
Integrals Involving tan ax
1
1
17.20.1.
∫ tan ax dx = − a ln cos ax = a ln sec ax
17.20.2.
∫ tan
17.20.3.
∫ tan3 ax dx =
17.20.4.
∫ tan
17.20.5.
∫
17.20.6.
∫ tan ax = a ln sin ax
17.20.7.
∫ x tan ax dx =
2
n
ax dx =
tan ax
−x
a
tan 2 ax 1
+ ln cos ax
2a
a
ax sec 2 ax dx =
tan n+1 ax
(n + 1)a
sec 2 ax
1
dx = ln tan ax
tan ax
a
dx
1
22 n (22 n − 1) Bn (ax )2 n+1
1 ⎧(ax )3 (ax )5 2(ax )7
⎫
+ ⎬
+
+
+
+
⎨
2
(
2
n
+
1
)!
3
15
105
a ⎩
⎭
22 n (22 n − 1) Bn (ax )2 n−1
tan ax
(ax )3 2(ax )5
+
++
+
dx = ax +
9
75
(2n − 1)(2n)!
x
17.20.8.
∫
17.20.9.
∫ x tan
2
ax dx =
x tan ax 1
x2
+ 2 ln cos ax −
2
a
a
TABLES OF SPECIAL INDEFINITE INTEGRALS
dx
17.20.10.
∫ p + q tan ax =
17.20.11.
∫ tan
(21)
n
ax dx =
95
px
q
+
ln (q sin ax + p cos ax )
p 2 + q 2 a( p 2 + q 2 )
tan n −1 ax
− tan n − 2 ax dx
(n − 1)a ∫
Integrals Involving cot ax
1
17.21.1.
∫ cot ax dx = a ln sin ax
17.21.2.
∫ cot
17.21.3.
∫ cot ax dx = −
17.21.4.
∫ cot n ax csc2 ax dx = −
17.21.5.
∫
17.21.6.
∫ cot ax = − a ln cos ax
17.21.7.
∫ x cot ax dx =
17.21.8.
∫
17.21.9.
∫ x cot
2
ax dx = −
3
cot ax
−x
a
cot 2 ax 1
− ln sin ax
2a
a
cot n+1 ax
(n + 1)a
csc 2 ax
1
dx = − ln cot ax
cot ax
a
dx
1
1
a2
22 n Bn (ax )2 n+1
(ax )3 (ax )5
⎪⎧
⎪⎫
−
− −
− ⎬
⎨ax −
(2n + 1)!
9
225
⎪⎩
⎪⎭
22 n Bn (ax )2 n −1
cot ax
1 ax (ax )3
−
−
−
− −
dx = −
x
ax 3
135
(2n − 1)(2n)!
2
ax dx = −
17.21.10.
∫ p + q cot ax =
px
q
−
ln (q sin ax + q cos ax )
p 2 + q 2 a( p 2 + q 2 )
17.21.11.
∫ cot n ax dx = −
cot n −1 ax
− cot n − 2 ax dx
(n − 1)a ∫
(22)
dx
x cot ax 1
x2
+ 2 ln sin ax −
2
a
a
Integrals Involving sec ax
1
1
⎛ ax
π⎞
+ ⎟
4⎠
17.22.1.
∫ sec ax dx = a ln (sec ax + tan ax) = a ln tan ⎜⎝ 2
17.22.2.
∫ sec ax dx =
tan ax
a
17.22.3.
∫ sec ax dx =
sec ax tan ax 1
+
ln (sec ax + tan ax )
2a
2a
2
3
TABLES OF SPECIAL INDEFINITE INTEGRALS
96
sec n ax
na
17.22.4.
∫ secn ax tan ax dx =
17.22.5.
∫ sec ax =
17.22.6.
∫ x sec ax dx =
17.22.7.
En (ax )2 n
sec ax
(ax )2 5(ax )4 61(ax )6
+
dx
=
ln
x
+
+
+
+
+
∫ x
4
96
4320
2n(2n)!
17.22.8.
∫ x sec ax dx = a tan ax + a
17.22.9.
∫ q + p sec ax = q − q ∫ p + q cos ax
dx
sin ax
a
1
x
2
dx
x
2
ln cos ax
p
dx
sec n − 2 ax tan ax n − 2
+
a(n − 1)
n −1
∫ sec
n−2
17.23.1.
∫ csc ax dx = a ln (csc ax − cot ax) = a ln tan
ax
2
17.23.2.
∫ csc ax dx = −
cot ax
a
17.23.3.
∫ csc ax dx = −
csc ax cot ax 1
ax
+
ln tan
2a
2a
2
17.23.4.
∫ cscn ax cot ax dx = −
17.23.5.
∫ csc ax = −
17.23.6.
∫ x csc ax dx =
17.23.7.
∫
17.23.8.
∫ x csc ax dx = −
17.23.9.
∫ q + p csc ax = q − q ∫ p + q sin ax
17.22.10.
(23)
∫ sec
E (ax )2 n+ 2
⎪⎧(ax )2 (ax )4 5(ax )6
⎪⎫
+ ⎬
+
+
++ n
⎨
(2n + 2)(2n)!
8
144
⎪⎩ 2
⎪⎭
1
a2
n
ax dx =
ax dx
Integrals Involving csc ax
17.23.10.
1
1
2
3
dx
csc n ax
na
cos ax
a
2(22 n −1 − 1) Bn (ax )2 n+1
(ax )3 7(ax )5
⎪⎧
⎪⎫
+ ⎬
+
++
⎨ax +
(2n + 1)!
18
1800
⎪⎩
⎪⎭
1
a2
2(22 n −1 − 1) Bn (ax )2 n −1
csc ax
1 ax 7(ax )3
+
dx = −
+
+
++
(2n − 1)(2n)!
x
ax 6
1080
2
dx
∫ cscn ax dx = −
x cot ax 1
+ 2 ln sin ax
a
a
x
p
dx
csc n − 2 ax cot ax n − 2
+
a(n − 1)
n −1
(See 17.17.22.)
∫ csc
n−2
ax dx
TABLES OF SPECIAL INDEFINITE INTEGRALS
(24)
Integrals Involving Inverse Trigonometric Functions
x
x
dx = x sin −1 + a 2 − x 2
a
a
17.24.1.
∫ sin
17.24.2.
∫ x sin −1
x
x x a2 − x 2
⎛ x 2 a2 ⎞
dx = ⎜ − ⎟ sin −1 +
4⎠
4
a
a
⎝2
17.24.3.
∫ x 2 sin −1
x
x3
x ( x 2 + 2a 2 ) a 2 − x 2
sin −1 +
dx =
3
9
a
a
17.24.4.
∫
sin −1 ( x /a)
x ( x/a)3 1 i 3( x/a)5 1 i 3 i 5( x /a)7
+
+
+
dx = +
x
a 2i 3i 3 2i 4 i 5i 5 2i 4 i 6i 7i 7
17.24.5.
∫
sin −1 ( x /a)
sin −1 ( x/a) 1 ⎛ a + a 2 − x 2 ⎞
dx = −
− ln ⎜
2
⎟
x
a ⎝
x
x
⎠
17.24.6.
⎛
∫ ⎜⎝sin −1
x
x⎞
x⎞
⎛
dx = x ⎜sin −1 ⎟ − 2 x + 2 a 2 − x 2 sin −1
a
a⎟⎠
a
⎝
⎠
17.24.7.
∫ cos
x
x
dx = x cos −1 − a 2 − x 2
a
a
17.24.8.
∫ x cos−1
x
x x a2 − x 2
⎛ x 2 a2 ⎞
dx = ⎜ − ⎟ cos −1 −
a
a
4⎠
4
⎝2
17.24.9.
∫x
x
x3
x ( x 2 + 2a 2 ) a 2 − x 2
dx =
cos −1 −
a
a
3
9
2
−1
−1
2
cos −1
2
17.24.10.
cos −1 ( x/a)
π
sin −1 ( x/a)
dx
x
ln
=
−
∫ x
∫ x dx (See 17.224.4.)
2
17.24.11.
∫
17.24.12.
⎛
∫ ⎜⎝cos−1
17.24.13.
∫ tan
17.24.14.
∫ x tan
17.24.15.
∫x
17.24.16.
∫
tan −1 ( x/a)
x ( x/a)3 ( x/a)5 ( x/a)7
+
−
dx = −
+
x
a
32
52
72
17.24.17.
∫
tan −1 ( x/a)
1
x 1 ⎛ x 2 + a2 ⎞
dx = − tan −1 −
ln
2
x
x
a 2a ⎜⎝ x 2 ⎠⎟
cos −1 ( x /a)
cos −1 ( x/a) 1 ⎛ a + a 2 − x 2 ⎞
dx = −
+ ln ⎜
2
⎟
x
a ⎝
x
x
⎠
2
−1
2
2
x
x⎞
x⎞
⎛
dx = x ⎜ cos −1 ⎟ − 2 x − 2 a 2 − x 2 cos −1
a
a⎟⎠
a
⎝
⎠
x
x a
dx = x tan −1 − ln ( x 2 + a 2 )
a
a 2
−1
tan −1
1
x
x ax
dx = ( x 2 + a 2 ) tan −1 −
2
a
a 2
x
x3
x ax 2 a 3
dx =
+ ln ( x 2 + a 2 )
tan −1 −
a
a
3
6
6
97
TABLES OF SPECIAL INDEFINITE INTEGRALS
98
x
x a
+ ln (x 2 + a 2 )
dx = x cot −1
a
a 2
17.24.18.
∫ cot
17.24.19.
∫ x cot
17.24.20.
∫ x 2 cot −1
17.24.21.
cot −1 ( x/a)
π
tan −1 ( x /a)
=
ln
−
dx
x
∫ x
∫ x dx
2
17.24.22.
cot −1 ( x/a)
cot −1 ( x /a) 1 ⎛ x 2 + a 2 ⎞
=
+
dx
ln
∫ x2
2a ⎜⎝ x 2 ⎟⎠
x
17.24.23.
⎧
−1
⎪x sec
x
−1
sec
=
dx ⎨
∫
a
⎪x sec −1
⎩
17.24.24.
⎧ x2
a x 2 − a2
−1 x
−
sec
⎪
x
⎪
2
a
∫ x sec −1 a dx = ⎨ x22
2
x
a
x
− a2
⎪ sec −1 +
⎪⎩ 2
2
a
17.24.25.
⎧ x3
ax x 2 − a 2 a 3
−1 x
−
− ln(x + x 2 − a 2 )
sec
⎪
⎪3
6
6
a
2
−1 x
∫ x sec a dx = ⎨ x 3
2
2
3
−
x
ax
x
a
a
⎪ sec −1 +
+ ln(x + x 2 − a 2 )
⎪⎩ 3
6
6
a
17.24.26.
sec −1 ( x/a)
π
a (a/x )3 1i3(a/x )5 1 i 3 i 5(a/x )7
=
ln
+
+
+
dx
x
+
+ ⋅⋅⋅
∫ x
2
x 2 i 3 i 3 2 i 4 i 5i 5 2 i 4 i 6 i 7i 7
17.24.27.
⎧ sec −1 ( x /a)
+
⎪−
x
⎪
−1
sec ( x/a)
⎪
∫ x 2 dx = ⎨ sec −1 ( x/a)
⎪−
−
x
⎪
⎩⎪
17.24.28.
⎧
x
x csc −1 + a ln ( x + x 2 − a 2 )
⎪
x
∫ csc −1 a dx = ⎨ −1 xa
⎪x csc
− a ln ( x + x 2 − a 2 )
a
⎩
17.24.29.
⎧ x2
⎪⎪ csc −1
−1 x
∫ x csc a dx = ⎨ x22
⎪ csc −1
⎪⎩ 2
17.24.30.
⎧x3
x ax x 2 − a 2 a 3
+ ln ( x + x 2 − a 2 )
⎪ csc −1 +
3
6
6
a
x
⎪
∫ x 2 csc −1 a dx = ⎨ 3
⎪x
ax x 2 − a 2 a 3
−1 x
−
− ln ( x + x 2 − a 2 )
⎪ csc
6
6
a
⎩3
−1
−1
1
x
x ax
+
dx = ( x 2 + a 2 ) cot −1
2
2
a
a
x
x3
x
ax 2
a3
+
−
ln ( x 2 + a 2 )
cot −1
dx =
3
6
6
a
a
(See 17.24.16.)
x π
<
a 2
π
x
< sec −1 < π
2
a
x
− a ln ( x + x 2 − a 2 )
a
x
+ a ln (x + x 2 − a 2 )
a
0 < sec −1
x π
<
a 2
π
x
< sec −1 < π
2
a
0 < sec −1
x π
<
a 2
π
x
< sec −1 < π
2
a
0 < sec −1
x π
<
a
2
x 2 − a2
ax
0 < sec −1
x 2 − a2
ax
π
x
< sec −1 < π
2
a
x π
<
a 2
π
x
− < csc −1 < 0
2
a
0 < csc −1
x
a x 2 − a2
x
π
+
<
0 < csc −1
a
a
2
2
x a x 2 − a2
π
x
−
− < csc −1
<0
2
2
a
a
0 < csc −1
−
x π
<
a 2
π
x
< csc −1 < 0
2
a
TABLES OF SPECIAL INDEFINITE INTEGRALS
99
csc −1 ( x /a)
⎛ a (a /x )3 1 i 3(a /x )5 1 i 3 i 5(a /x )7
⎞
+
+
+ ⋅⋅⋅ ⎟
dx = − ⎜ +
2i 4 i 6 i 7i 7
x
⎝ x 2i 3 i 3 2i 4 i 5 i 5
⎠
17.24.31.
∫
17.24.32.
⎧ csc −1 ( x /a)
−
⎪−
x
csc −1 ( x /a)
⎪
=
dx
⎨
∫ x2
⎪ csc −1 ( x /a)
+
⎪−
x
⎩
x 2 − a2
ax
0 < csc −1
x 2 − a2
ax
−
π
x
< csc −1 < 0
2
a
17.24.33.
∫ x m sin −1
1
x
x m +1
x
dx =
sin −1 −
a
m +1
a m +1
∫
17.24.34.
∫x
cos −1
1
x
x m +1
x
dx =
cos −1 +
a
m +1
a m +1
∫
a2 − x 2
17.24.35.
∫x
tan −1
x
x m +1
x
a
tan −1 −
dx =
a
m +1
a m +1
∫
x m +1
dx
x + a2
17.24.36.
∫x
cot −1
x
x m +1
x
a
cot −1 +
dx =
a
m +1
a m +1
∫
x m +1
dx
x + a2
x m dx
17.24.37.
⎧ x m +1 sec −1 ( x/a)
a
−
⎪
m +1
m +1
x
⎪
∫ x m sec −1 a dx = ⎨ x m+1 sec −1 ( x/a) a
⎪
+
⎪
m +1
m +1
⎩
17.24.38.
(25)
m
m
m
⎧ x m +1 sec −1 ( x /a)
a
+
⎪
m +1
m +1
x
⎪
∫ x m csc −1 a dx = ⎨ x m+1 csc −1 ( x /a) a
⎪
−
⎪
m +1
m +1
⎩
x π
<
a 2
x m +1
dx
a2 − x 2
x m +1
dx
2
2
∫
∫
∫
∫
x −a
2
2
x m dx
x −a
2
2
x m dx
x −a
2
2
x m dx
x −a
2
2
0 < sec −1
x π
<
a 2
π
x
< sec −1 < π
2
a
0 < csc −1
−
x π
<
a
2
π
x
< csc −1 < 0
2
a
Integrals Involving eax
e ax
a
17.25.1.
∫e
17.25.2.
∫ xeax dx =
17.25.3.
∫x e
17.25.4.
∫ x neax dx =
ax
dx =
2 ax
e ax
a
⎛
1⎞
⎜⎝ x − a ⎟⎠
e ax ⎛ 2 2 x 2 ⎞
+
x −
a ⎜⎝
a a 2 ⎟⎠
dx =
=
x n e ax n
−
a
a
e ax
a
∫x
n −1 ax
e dx
n(n − 1) x n− 2
(−1)n n !⎞
⎛ n nx n−1
−
+
−
⋅⋅⋅
x
⎜⎝
a
a2
a n ⎟⎠
17.25.5.
e ax
ax (ax )2 (ax )3
=
ln
+
+
+
+ ⋅⋅⋅
dx
x
∫ x
1 i 1! 2 i 2! 3 i3!
17.25.6.
∫x
e ax
n
dx =
a
−e ax
+
n −1
(n − 1) x n−1
e ax
∫x
n −1
dx
if n = positivee integer
TABLES OF SPECIAL INDEFINITE INTEGRALS
100
17.25.7.
∫
dx
x
1
=
−
ln (p + qe ax )
p + qe ax
p ap
17.25.8.
∫
dx
x
1
1
= 2 +
−
ln (p + qe ax )
( p + qe ax )2
p
ap( p + qe ax ) ap2
dx
+ qe − ax
⎧ 1
⎛ p ax ⎞
tan −1 ⎜
e
⎪
⎝ q ⎟⎠
⎪⎪a pq
=⎨
⎛ e ax − − q /p ⎞
⎪
1
ln
⎪
⎜ ax
⎟
⎪⎩2a − pq ⎝ e + − q /p ⎠
∫ pe
ax
17.25.10.
∫e
ax
sin bx dx =
e ax (a sin bx − b cos bx )
a2 + b2
17.25.11.
∫e
ax
cos bx dx =
e ax (a cos bx + b sin bx )
a2 + b2
17.25.12.
∫ xe
17.25.13.
∫ xe
17.25.14.
∫ eax ln x dx =
17.25.15.
∫ eax sin n bx dx =
e ax sin n−1 bx
n(n − 1)b 2 ax n− 2
(
a
sin
bx
−
nb
co
s
bx
)
+
e sin bx dx
a2 + n2b2
a2 + n2b2 ∫
17.25.16.
∫ eax cosn bx dx =
e ax cos n−1 bx
n(n − 1)b 2 ax
(a cos bx + nb sin bx ) + 2
e cos n− 2 bx dx
2
2 2
a +n b
a + n2b2 ∫
17.25.9.
(26)
ax
sin bx dx =
xe ax (a sin bx − b cos bx ) e ax {(a 2 − b 2 ) sin bx − 2ab cos bx}
−
(a 2 + b 2 ) 2
a2 + b2
ax
cos bx dx =
xe ax (a cos bx + b sin bx ) e ax {(a 2 − b 2 ) cos bx + 2ab sin bx}
−
(a 2 + b 2 ) 2
a2 + b2
e ax ln x 1
−
a
a
e ax
∫ x dx
Integrals Involving ln x
17.26.1.
∫ ln x dx = x ln x − x
17.26.2.
∫ x ln x dx =
17.26.3.
∫x
17.26.4.
∫
ln x
1
dx = ln 2 x
x
2
17.26.5.
∫
ln x
ln x 1
−
2 dx = −
x
x
x
17.26.6.
∫ ln
17.26.7.
ln n x dx ln n+1 x
∫ x = n +1
17.26.8.
∫
m
2
x2
2
ln x dx =
⎛
1⎞
⎜⎝ ln x − ⎟⎠
2
1 ⎞
x m +1 ⎛
ln x −
m + 1 ⎜⎝
m + 1⎟⎠
( If m = −1, see 17.26.4.)
x dx = x ln 2 x − 2 x ln x + 2 x
dx
= ln (ln x )
x ln x
(If n = −1, see 17.26.8.)
TABLES OF SPECIAL INDEFINITE INTEGRALS
dx
ln 2 x ln 3 x
= ln (ln x ) + ln x +
+
+ ⋅⋅⋅
2 i 2 ! 3 i 3!
ln x
∫
17.26.9.
(m + 1)2 ln 2 x
(m + 1)3 ln 3 x
x m dx
= ln (ln x ) + (m + 1) ln x +
+
+ ⋅⋅⋅
ln x
2 i 2!
3 i 3!
17.26.10.
∫
17.26.11.
∫ ln x dx = x ln
17.26.12.
n
n
∫ ln
x−n
n −1
x dx
x m +1 ln n x
n
−
m +1
m +1
If m = –1, see 17.26.7.
∫x
m
ln n x dx =
17.26.13.
∫ ln (x
17.26.14.
∫ ln ( x
17.26.15.
∫
(27)
∫x
m
ln n −1 xdx
x
a
2
+ a 2 ) dx = x ln (x 2 + a 2 ) − 2 x + 2a tan −1
2
⎛ x + a⎞
− a 2 ) dx = x ln ( x 2 − a 2 ) − 2 x + a ln ⎜
⎝ x − a ⎟⎠
x m ln (x 2 ± a 2 ) dx =
2
x m +1 ln ( x 2 ± a 2 )
−
m +1
m +1
∫
x m+2
dx
x ± a2
2
Integrals Involving sinh ax
cosh ax
a
17.27.1.
∫ sinh ax dx =
17.27.2.
∫ x sinh ax dx =
17.27.3.
⎛ x2
2⎞
2x
2
sinh
x
ax
dx
=
∫
⎜⎝ a + a 3 ⎟⎠ cosh ax − a 2 sinh ax
17.27.4.
∫
sinh ax
(ax )3 (ax )5
dx = ax +
+
+ ⋅⋅⋅
3 i3!
5 i5!
x
17.27.5.
∫
sinh ax
sinh ax
dx = −
+a
x
x2
17.27.6.
∫ sinh ax
17.27.7.
101
dx
=
x cosh ax sinh ax
−
a
a2
∫
cosh ax
dx
x
(See 17.28.4.)
1
ax
ln tanh
a
2
2(−1)n (22 n − 1) Bn (ax )2 n+1
x dx
1 ⎧
(ax )3 7(ax )5
⎫
+ ⋅⋅⋅⎬
=
ax
−
+
−
⋅⋅⋅
+
⎨
∫ sinh ax a 2 ⎩
(2n + 1)!
18
1800
⎭
17.27.8.
∫ sinh
17.27.9.
∫ x sinh
2
ax dx =
2
sinh ax cosh ax x
−
2a
2
ax dx =
x sinh 2ax
cosh 2ax x 2
−
−
4a
4
8a 2
TABLES OF SPECIAL INDEFINITE INTEGRALS
102
dx
coth ax
a
17.27.10.
∫ sinh
17.27.11.
∫ sinh ax sinh px dx =
2
ax
=−
sinh (a + p) x sinh (a − p) x
−
2(a + p)
2(a − p)
For a = ± p see 17.27.8.
x m cosh ax
a
m m −1
x cosh ax dx
a ∫
17.27.12.
∫x
17.27.13.
∫ sinh n ax dx =
17.27.14.
∫
sinh ax
a
− sinh ax
dx =
+
(n − 1) x n −1
xn
n −1
17.27.15.
∫
dx
n−2
− cosh ax
=
−
n
n −1
sinh ax
a(n − 1) sinh ax
n −1
17.27.16.
∫ sinh
(28)
m
sinh ax dx =
−
sinh n−1 ax cosh ax
an
−
∫
n −1
n
∫ sinh
n− 2
cosh ax
dx
x n −1
(See 17.28.12.)
ax dx
(See 17.28.14.)
dx
∫ sinh
n−2
ax
x dx
− x cosh ax
1
n−2
=
− 2
−
n
n −1
n−2
ax a(n − 1)sinh ax a (n − 1)(n − 2)sinh ax n − 1
Integrals Involving cosh ax
sinh ax
a
17.28.1.
∫ cosh ax dx =
17.28.2.
∫ x cosh ax dx =
17.28.3.
∫ x 2 cosh ax dx = −
17.28.4.
cosh ax
(ax )2 (ax )4 (ax )6
ln
+ ⋅⋅⋅
dx
=
x
+
+
+
∫ x
2 i 2! 4 i 4 ! 6 i 6!
17.28.5.
∫
17.28.6.
∫ cosh ax
x sinh ax cosh ax
−
a
a2
2 x cosh ax ⎛ x 2
2⎞
+⎜
+ 3 ⎟ sinh ax
2
a
⎝a a ⎠
cosh ax
cosh ax
dx = −
+a
x
x2
dx
=
∫
sinh ax
dx
x
(See 17.27.4.)
2
tan −1 e ax
a
(−1)n En (ax )2 n+ 2
x dx
1 ⎧(ax )2 (ax )4 5(ax )6
⎫
+ ⋅⋅⋅⎬
= 2 ⎨
−
+
+ ⋅⋅⋅ +
(2n + 2)(2n)!
8
144
cosh ax a ⎩ 2
⎭
17.28.7.
∫
17.28.8.
∫ cosh
17.28.9.
∫ x cosh 2 ax dx =
2
ax dx =
x sinh ax cosh ax
+
2
2a
x 2 x sinh 2ax
cosh 2ax
+
−
4
4a
8a 2
x dx
∫ sinh
n−2
ax
TABLES OF SPECIAL INDEFINITE INTEGRALS
dx
103
tanh ax
a
17.28.10.
∫ cosh
17.28.11.
∫ cosh ax cosh px dx =
17.28.12.
∫x
17.28.13.
∫ cosh
17.28.14.
∫
cosh ax
a
− cosh ax
dx =
+
xn
(n − 1) x n −1
n −1
17.28.15.
∫
sinh ax
dx
n−2
=
+
n
n −1
cosh ax
a(n − 1) cosh ax
n −1
17.28.16.
∫
1
n−2
x dx
x sinh ax
+
=
+
n
n −1
2
n−2
cosh ax
a(n − 1) cosh ax (n − 1) (n − 2)a cosh ax n − 1
(29)
2
ax
=
cosh ax dx =
m
n
ax dx =
sinh (a + p) x
sinh (a − p) x
+
2(a − p)
2(a + p)
x m sinh ax
a
−
m m −1
x sinh ax dx
a ∫
cosh n −1 ax sinh ax
an
n −1
n
+
∫
∫ cosh
dx
∫ cosh
∫ sinh ax cosh ax dx =
sinh 2 ax
2a
17.29.2.
∫ sinh px cosh qx dx =
cosh ( p + q) x cosh ( p − q) x
+
2( p + q)
2( p − q)
17.29.3.
∫ sinh
17.29.4.
∫ sinh ax cosh ax
17.29.5.
∫ sinh
17.29.6.
∫ cosh ax dx =
17.29.7.
∫
dx
2
=
sinh 4 ax x
−
32a
8
1
ln tanh ax
a
dx
2 coth 2ax
=−
ax cosh 2 ax
a
sinh 2 ax
sinh ax 1 −1
tan sinh ax
a
a
cosh 2 ax
cosh ax 1
ax
+ ln tanh
dx =
sinh ax
2
a
a
ax dx
(See 17.27.14.)
17.29.1.
ax cosh 2 ax dx =
n−2
sinh ax
dx
x n −1
Integrals Involving sinh ax and cosh ax
2
(See 17.27.12.)
n−2
ax
x dx
∫ cosh
n−2
ax
TABLES OF SPECIAL INDEFINITE INTEGRALS
104
(30)
Integrals Involving tanh ax
1
ln cosh ax
a
17.30.1.
∫ tanh ax dx
17.30.2.
∫ tanh
17.30.3.
∫ tanh3 ax dx =
17.30.4.
∫ x tanh ax dx = a
17.30.5.
2
=
ax dx = x −
1
tanh 2 ax
ln cosh ax −
a
2a
⎫⎪
(−1)n −1 22 n (22 n − 1) Bn (ax )2 n+1
1 ⎧⎪ (ax )3 (ax )5 2(ax )7
−
+
− ⋅⋅⋅
+ ⋅⋅⋅⎬
2 ⎨
15
105
(2n + 1)!
⎪⎭
⎩⎪ 3
2
1
x
x tanh ax
∫ x tanh 2 ax dx = 2 − a + a 2 ln cosh ax
(−1)n −1 22 n (22 n − 1) Bn (ax )2 n −1
tanh ax
(ax )3 2(ax )5
+
− ⋅⋅⋅
+ ⋅⋅⋅
dx = ax −
9
75
(2n − 1)(2n)!
x
17.30.6.
∫
17.30.7.
∫ p + q tanh ax
17.30.8.
∫ tanh n ax dx =
(31)
tanh ax
a
dx
=
px
q
−
ln (q sinh ax + p cosh ax )
p 2 − q 2 a( p 2 − q 2 )
− tanh n−1 ax
+ ∫ tanh n− 2 ax dx
a(a − 1)
Integrals Involving coth ax
1
ln sinh ax
a
17.31.1.
∫ coth ax dx =
17.31.2.
∫ coth
17.31.3.
∫ coth3 ax dx =
1
coth 2 ax
ln sinh ax −
a
2a
17.31.4.
∫ x coth ax dx =
(−1)n−1 22 n Bn (ax )2 n+1
1 ⎧
(ax )3 (ax )5
⎫
−
+ ⋅⋅⋅
+ ⋅⋅⋅⎬
2 ⎨ax +
(2n + 1)!
9
225
a ⎩
⎭
17.31.5.
∫ x coth
17.31.6.
∫
17.31.7.
∫ p + q coth ax
17.31.8.
∫ coth n ax dx = −
2
ax dx = x −
2
ax dx =
coth ax
a
x 2 x coth ax
1
−
+ 2 ln sinh ax
2
a
a
(−1)n 22 n Bn (ax )2 n−1
coth ax
1 ax (ax )3
+
−
+ ⋅⋅⋅
+ ⋅⋅⋅
dx = −
(2n − 1)(2n)!
x
ax 3
135
dx
=
px
q
−
ln ( p sinh ax + q cosh ax )
p 2 − q 2 a( p 2 − q 2 )
coth n−1 ax
+
a(n − 1)
∫ coth
n− 2
ax dx
TABLES OF SPECIAL INDEFINITE INTEGRALS
(32)
Integrals Involving sech ax
2
17.32.1.
∫ sech ax dx = a tan
17.32.2.
∫ sech
17.32.3.
∫ sech
17.32.4.
∫ x sech ax dx =
17.32.5.
∫ x sech
17.32.6.
∫
17.32.7.
∫ sech n ax dx =
(33)
105
−1
e ax
2
ax dx =
tanh ax
a
3
ax dx =
1
sech ax tanh ax
tan −1 sinh ax
+
2a
2a
2
(−1)n En (ax )2 n+ 2
1 ⎧ (ax )2 (ax )4
5(ax )6
⎫
−
+
+ ⋅⋅⋅
+ ⋅⋅⋅⎬
2 ⎨
(2n + 2)(2n)!
8
144
a ⎩ 2
⎭
ax dx =
x tanh ax 1
− 2 ln cosh ax
a
a
(−1)n En (ax )2 n
sech ax
(ax )2 5(ax )4 61(ax )6
+ ⋅⋅⋅
dx = ln x −
+
−
+ ⋅⋅⋅
4
96
4320
2n(2n)!
x
sech n− 2 ax tanh ax n − 2
+
a(n − 1)
n −1
∫ sech
n− 2
ax dx
Integrals Involving csch ax
1
17.33.1.
∫ csch ax dx = a ln tanh
17.33.2.
∫ csch
17.33.3.
∫ csch
17.33.4.
∫ x csch ax dx = a
17.33.5.
∫ x csch
17.33.6.
∫
17.33.7.
∫ csch
ax
2
2
ax dx = −
coth ax
a
3
ax dx = −
csch ax coth ax
1
ax
ln tanh
−
2a
2a
2
2(−1)n (22 n−1 − 1) Bn (ax )2 n+1
1 ⎧
(ax )3 7(ax )5
⎫
+
+ ⋅⋅⋅ +
+ ⋅⋅⋅⎬
2 ⎨ ax −
(
2
n
+
1
)!
18
1800
⎭
⎩
2
ax dx = −
x coth ax 1
+ 2 ln sinh ax
a
a
(−1)n 2(22 n−1 − 1) Bn (ax )2 n−1
1
csch ax
ax 7(ax )3
−
+
+ ⋅⋅⋅
+ ⋅⋅⋅
dx = −
6
1080
(2n − 1) (2n)!
x
ax
n
ax dx =
− csch n− 2 ax coth ax n − 2
−
a(n − 1)
n −1
∫ csch
n− 2
ax dx
TABLES OF SPECIAL INDEFINITE INTEGRALS
106
(34)
Integrals Involving Inverse Hyperbolic Functions
x
x
dx = x sinh −1 − x 2 + a 2
a
a
17.34.1.
∫ sinh
17.34.2.
∫ x sinh
17.34.3.
⎧ x ( x /a)3 1 i 3( x /a)5 1 i 3 i 5( x /a)7
|x| < a
⎪ a − 2 i3 i3 + 2 i 4 i 5 i 5 − 2 i 4 i 6 i 7i7 + ⋅⋅⋅
⎪
⎪⎪ ln 2 (2 x /a) (a /x )2 1 i 3(a /x )4 1 i3 i5(a /x )6
sinh −1 ( x /a)
=
−
+
−
dx
+ ⋅⋅⋅
x>a
⎨
∫
x
2
2i2i2 2 i 4 i 4 i 4 2i 4 i 6 i 6 i 6
⎪
⎪
4
6
2
2
⎪− ln (−2 x /a) + (a /x ) − 1 i 3(a /x ) + 1 i3 i5(a /x ) − ⋅⋅⋅
x < −a
2
2i2i2
⎪⎩
2 i 4 i 4 i 4 2i 4 i 6 i 6 i 6
17.34.4.
⎧x cosh −1 ( x /a) − x 2 − a 2 , cosh −1 ( x /a) > 0
x
⎪
∫ cosh a dx = ⎨
−1
2
2
−1
⎩⎪x cosh ( x /a) + x − a , cosh ( x /a) < 0
17.34.5.
1
⎧1
(2 x 2 − a 2 ) cosh −1 ( x /a) − x x 2 − a 2 , cosh −1 ( x /a) > 0
⎪
4
4
x
⎪
∫ x cosh −1 a dx = ⎨1
⎪ (2 x 2 − a 2 ) cosh −1 ( x /a) + 1 x x 2 − a 2 , cosh −1 ( x /a) < 0
⎪⎩4
4
17.34.6.
−1
−1
x
x
x 2 + a2
⎛ x 2 a2 ⎞
dx = ⎜ + ⎟ sinh −1 − x
a
a
4⎠
4
⎝2
−1
cosh −1 ( x /a)
(a /x )2 1 i 3(a /x )4 1 i 3 i 5(a /x )6
⎡1 2
⎤
ln
(
)
=
±
+
+
2
+
+ ⋅⋅⋅⎥
dx
x
/
a
∫
⎢⎣ 2
x
2i2i2 2i4i4i4 2i4i6i6i6
⎦
+ if cosh −1 ( x /a) > 0, − if cosh −1 ( x /a) < 0
17.34.7.
∫ tanh
17.34.8.
∫ x tanh
17.34.9.
∫
x
x a
dx = x tanh −1 + ln (a 2 − x 2 )
a
a 2
−1
−1
x
ax 1 2
x
+ ( x − a 2 ) tanh −1
dx =
2 2
a
a
tanh −1 ( x /a)
x ( x /a)3 ( x /a)5
+
+ ⋅⋅⋅
dx = +
x
a
32
52
x
a
dx = x coth −1 x + ln ( x 2 − a 2 )
a
2
17.34.10.
∫ coth
17.34.11.
∫ x coth
17.34.12.
coth −1 ( x /a)
⎛ a (a /x )3 (a /x )5
⎞
=
−
dx
⎜⎝ x + 32 + 52 + ⋅⋅⋅⎟⎠
∫
x
17.34.13.
−1
−1
−1
x
⎪⎧x sech ( x /a) + a sin ( x /a), sech ( x /a) > 0
∫ sech a dx = ⎨⎪x sech −1 ( x /a) − a sin −1 ( x /a), sech −1 ( x /a) < 0
⎩
17.34.14.
∫ csch
−1
−1
x
ax 1 2
x
dx =
+ ( x − a 2 ) coth −1
a
a
2 2
−1
−1
x
x
x
dx = x csch −1 ± a sinh −1
a
a
a
(+ if x > 0, − if x < 0)
TABLES OF SPECIAL INDEFINITE INTEGRALS
1
x
x m +1
x
sinh −1 −
dx =
a
m +1
a m +1
17.34.15.
∫ x m sinh −1
17.34.16.
⎧ x m +1
−1
⎪ m + 1 cosh
x
⎪
∫ x m cosh −1 a dx = ⎨ x m+1
⎪
−1
⎪ m + 1 cosh
⎩
107
x m +1
∫
dx
x 2 + a2
1
x
−
a m +1
∫
x
1
+
a m +1
∫
x m +1
x 2 − a2
x m +1
x −a
2
2
17.34.17.
∫ x m tanh −1
x
x m +1
x
a
dx =
tanh −1 −
a
m +1
a m +1
x m +1
∫ a 2 − x 2 dx
17.34.18.
∫x
1
x
x m +1
x
coth −1 −
dx =
a
m +1
a m +1
∫a
17.34.19.
⎧ x m +1
−1
⎪ m + 1 sech
x
⎪
∫ x m sech −1 a dx = ⎨ x m+1
⎪
−1
⎪ m + 1 sech
⎩
17.34.20.
m
coth −1
∫ x m csch −1
cosh −1 ( x /a) > 0
dx
cosh −1 ( x /a) < 0
x m +1
dx
− x2
2
x m dx
x
a
+
a m +1
∫
x
1
−
a m +1
∫
a2 − x 2
∫
x m dx
x 2 + a2
x
x m +1
x
a
csch −1 ±
dx =
a
m +1
a m +1
dx
a −x
2
2
x m dx
sech −1 ( x /a) > 0
sech −1 ( x /a) < 0
(+ if x > 0, − if x < 0)
18
DEFINITE INTEGRALS
Definition of a Definite Integral
Let f(x) be defined in an interval a x b. Divide the interval into n equal parts of length x = (b − a)/n. Then
the definite integral of f(x) between x = a and x = b is defined as
18.1.
∫
b
f ( x ) dx = lim{ f (a) Δx + f (a + Δx ) Δx + f (a + 2Δx ) Δx + + f (a + (n − 1) Δx ) Δx}
n→∞
a
The limit will certainly exist if f(x) is piecewise continuous.
d
If f ( x ) =
g( x ), then by the fundamental theorem of the integral calculus the above definite integral can
dx
be evaluated by using the result
18.2.
∫
b
f ( x ) dx = ∫
a
b
d
g( x ) dx = g( x ) = g(b) − g(a)
a
dx
b
a
If the interval is infinite or if f(x) has a singularity at some point in the interval, the definite integral is called
an improper integral and can be defined by using appropriate limiting procedures. For example,
∞
b
18.3.
∫
a
18.4.
∫
−∞
∫
a
18.5.
18.6.
∫
f ( x ) dx = lim ∫ f ( x ) dx
b→∞ a
∞
b
b
a
b
f ( x ) dx = lim ∫ f ( x ) dx
a→−∞ a
b→∞
f ( x ) dx = lim ∫
b −∈
f ( x ) dx = lim ∫
b
α
∈→ 0
∈→ 0
a +∈
f ( x ) dx
if b is a singular point..
f ( x ) dx if a is a singulaar point.
General Formulas Involving Definite Integrals
b
18.7.
∫
a
18.8.
∫
a
18.9.
∫
18.10.
∫
18.11.
∫
108
b
a
a
b
a
b
a
{ f ( x ) ± g( x ) ± h( x ) ± } dx =
∫
b
b
a
f ( x ) dx ±
∫
b
a
cf ( x ) dx = c ∫ f ( x ) dx where c is any constant.
a
f ( x ) dx = 0
a
f ( x ) dx = − ∫ f ( x ) dx
b
c
b
a
c
f ( x ) dx = ∫ f ( x ) dx + ∫ f ( x ) dx
b
g( x ) dx ± ∫ h( x ) dx ± a
DEFINITE INTEGRALS
18.12.
∫
b
109
f ( x ) dx = (b − a) f (c)
a
where c is between a and b.
This is called the mean value theorem for definite integrals and is valid if f(x) is continuous in
a x b.
18.13.
∫
b
b
f ( x ) g( x ) dx = f (c) ∫ g( x ) dx
a
a
where c is between a and b
This is a generalization of 18.12 and is valid if f(x) and g(x) are continuous in a x b and g(x) 0.
Leibnitz’s Rules for Differentiation of Integrals
18.14.
d
dα
∫
φ 2 (α )
φ 1 (α )
F ( x , α ) dx = ∫
φ 2 (α )
φ 1(α )
dφ
dφ
∂F
dx + F (φ2 , α ) 2 − F (φ1 , α ) 1
dα
dα
dα
Approximate Formulas for Definite Integrals
In the following the interval from x = a to x = b is subdivided into n equal parts by the points
a = x0, x1, x2, …, xn–1, xn = b and we let y0 = f(x0), y1 = f(x1), y2 = f(x2), …, yn = f(xn), h = (b – a)/n.
Rectangular formula:
18.15.
∫
b
a
f ( x ) dx ≈ h( y0 + y1 + y2 + + yn−1 )
Trapezoidal formula:
18.16.
∫
b
a
f ( x ) dx ≈
h
( y + 2 y1 + 2 y2 + + 2 yn−1 + yn )
2 0
Simpson’s formula (or parabolic formula) for n even:
18.17.
∫
b
a
f ( x ) dx ≈
h
( y + 4 y1 + 2 y2 + 4 y3 + + 2 yn− 2 + 4 yn−1 + yn )
3 0
Definite Integrals Involving Rational or Irrational Expressions
18.18.
∫
18.19.
∫
∞
0
∞
0
∞
dx
π
=
x 2 + a 2 2a
x p−1dx
π
=
,
1+ x
sin pπ
0 < p<1
x m dx
π a m+1− n
=
,
x n + a n n sin[(m + 1)π /n]
∫
0
18.21.
∫
x m dx
π sin mβ
=
0 1 + 2 x cos β + x 2
sin mπ sin β
18.22.
∫
0
18.20.
18.23.
∫
∞
a
a
0
dx
π
=
2
2
a −x
2
a 2 − x 2 dx =
π a2
4
0 < m +1< n
DEFINITE INTEGRALS
110
18.24.
18.25.
∫
∫
a
0
∞
0
x m (a n − x n ) p dx =
a m +1+ np Γ [(m + 1)/n] Γ ( p + 1)
nΓ [(m + 1)/n + p + 1]
x m dx
(−1)r −1 π a m +1− nr Γ [(m + 1)/n]
,
=
( x n + a n )r n sin[(m + 1)π /n](r − 1)! Γ[(m + 1) /n − r + 1]
0 < m + 1 < nr
Definite Integrals Involving Trigonometric Functions
All letters are considered positive unless otherwise indicated.
18.26.
18.27.
∫
∫
π
0
π
0
π
18.28.
∫
0
18.29.
∫
0
18.30.
∫
0
18.31.
∫
0
18.32.
∫
0
18.33.
18.34.
18.35.
∫
∫
∫
0
m, n integers and m + n even
⎧⎪
sin mx cos nx dx = ⎨
⎪⎩2m/ (m 2 − n 2 ) m, n integers and m + n odd
π /2
π /2
π /2
0
∞
0
∞
0
∞
18.36.
∫
0
18.37.
∫
0
∞
m, n integers and m = n
⎧⎪ 0 m, n integers and m ≠ n
cos mx cos nx dx = ⎨
ntegers and m = n
⎩⎪π /2 m, n in
π /2
∞
m, n integers and m ≠ n
⎪⎧ 0
sin mx sin nx dx = ⎨
⎪⎩π /2
sin 2 x dx =
∫
sin 2 m x dx =
π /2
0
∫
sin 2 m +1 x dx =
cos 2 x dx =
π /2
0
∫
π
4
cos 2 m x dx =
π /2
0
1 i 3 i 5 2 m − 1 π
,
2 i 4 i 6 2m 2
cos 2 m +1 x dx =
sin 2p−1 x cos 2 q−1 x dx =
2 i 4 i 6 2m
,
1 i 3 i 5 2 m + 1
Γ ( p) Γ (q)
2Γ ( p + q)
⎧ π /2 p > 0
sin px
⎪⎪
dx = ⎨ 0 p = 0
x
⎪
⎪⎩−π /2 p < 0
⎧ 0
⎪⎪
sin px cos qx
dx = ⎨π /2
x
⎪
⎩⎪π /4
⎧⎪π p/2
sin px sin qx
dx = ⎨
2
x
⎪⎩π q/2
sin 2 px
πp
dx =
2
x2
1 − cos px
πp
dx =
2
x2
p>q>0
0< p<q
p=q>0
0 < pq
pq > 0
m = 1, 2,…
m = 1, 2,…
DEFINITE INTEGRALS
∞
18.38.
∫
0
18.39.
∫
0
18.40.
∫
0
18.41.
∫
0
18.42.
∫
0
18.43.
∫
18.44.
∫
18.45.
∫
18.46.
∫
18.47.
∫
18.48.
∫
∞
∞
∞
∞
x sin mx
π − ma
e
2
2 dx =
2
x +a
sin mx
π
dx = 2 (1 − e − ma )
x(x 2 + a2 )
2a
0
π /2
0
2π
0
2π
0
∫
0
∫
0
18.52.
∫
0
18.53.
∫
0
18.54.
∫
18.55.
∫
2π
a2 − b2
dx
=
a + b cos x
2π
a2 − b2
cos −1 (b / a)
dx
=
a + b cos x
a2 − b2
dx
=
(a + b sin x )2
2π
0
dx
2π a
=
(a + b cos x )2 (a 2 − b 2 )3/2
dx
2π
=
,
1 − 2a cos x + a 2 1 − a 2
∞
∫
sin ax 2 dx =
0 < a <1
sin ax n dx =
∞
∞
∞
0
∞
0
∞
∫
cos ax 2 dx =
1
2
∞
0
a <1
a >1
m = 0, 1, 2,…
π
2a
n >1
1
π
Γ (1/n) cos ,
2n
na1/n
cos ax n dx =
sin x
dx =
x
∞
0
a 2 < 1,
1
π
Γ (1/n) sin ,
2n
na1/n
∞
0
∫
cos mx dx
π am
,
2 =
1 − 2a cos x + a
1 − a2
π
18.50.
dx
=
a + b sin x
⎧⎪(π /a) ln (1 + a),
x sin x dx
=
1 − 2a cos x + a 2 ⎨π ln (1 + 1/a),
⎪⎩
π
0
∫
cos mx
π − ma
e
2
2 dx =
a
2
x +a
2π
∫
18.56.
cos px − cos qx
π (q − p)
dx =
2
2
x
0
18.49.
18.51.
cos px − cos qx
q
dx = ln
x
p
2π
0
111
n >1
cos x
π
dx =
2
x
sin x
π
dx =
,
2Γ ( p) sin ( pπ /2)
xp
cos x
π
dx =
,
2Γ ( p) cos ( pπ /2)
xp
sin ax 2 cos 2bx dx =
1
2
π
2a
0 < p<1
0 < p<1
b2
b2 ⎞
⎛
cos
−
sin
⎜⎝
a
a ⎟⎠
DEFINITE INTEGRALS
112
18.57.
18.58.
∫
∫
18.59.
∫
18.60.
∫
18.61.
∫
18.62.
∫
∞
0
∞
0
∞
0
∞
0
cos ax 2 cos 2bx dx =
π
2a
b2
b2 ⎞
⎛
cos
sin
+
⎜⎝
a
a ⎟⎠
sin 3 x
3π
dx =
8
x3
sin 4 x
π
dx =
3
x4
tan x
π
dx =
x
2
π /2
0
π /2
0
dx
π
=
1 + tan m x 4
{
}
x
1 1
1
1
dx = 2 2 − 2 + 2 − 2 + sin x
1 3
5
7
tan −1 x
1 1
1
1
dx = 2 − 2 + 2 − 2 + x
1 3
5
7
1
∫
0
18.64.
∫
sin −1 x
π
dx = ln 2
0
2
x
18.65.
∫
0
18.66.
∫
18.67.
∫
18.63.
1
2
1
1
∞ cos x
1 − cos x
dx − ∫
dx = γ
1
x
x
∞
0
∞
0
⎛ 1
⎞ dx
⎜⎝ 1 + x 2 − cos x⎟⎠ x = γ
tan −1 px − tan −1 qx
π p
dx = ln
x
2 q
Definite Integrals Involving Exponential Functions
Some integrals contain Euler’s constant g = 0.5772156 . . . (see 1.3, page 3).
∞
a
18.68. ∫ e − ax cos bx dx = 2
0
a + b2
18.69.
∫
18.70.
∫
18.71.
18.72.
18.73.
∫
∞
0
e − ax sin bx dx =
∞ − ax
e
0
∞ − ax
e
0
∞
∫
0
∫
0
b
sin bx
dx = tan −1
x
a
− e − bx
b
dx = ln
x
a
e − ax dx =
∞
2
b
a2 + b2
1 π
2 a
e − ax cos bx dx =
2
1 π − b /4 a
e
2 a
2
DEFINITE INTEGRALS
18.74.
∫
∞
0
2
∞
18.75.
∫
−∞
18.76.
∫
0
18.77.
∫
0
18.78.
∫
18.79.
∫
18.80.
∫
∞
∞
2
π
∫
∞
e − x dx
p
2
2
x n e − ax dx =
x m e − ax dx =
2
0
∞
0
2
2
π ( b − 4 ac )/4 a
e
a
2
Γ(n + 1)
a n+1
Γ[(m + 1)/2]
2a ( m +1)/2
e − ( ax + b/x ) dx =
∞
2
e − ( ax + bx +c ) dx =
∞
0
1 π ( b − 4 ac )/4 a
b
e
erfc
2 a
2 a
e − ( ax + bx +c ) dx =
where erfc (p) =
113
1 π −2
e
2 a
ab
x dx
π2
1 1
1
1
=
+
+
+
+
=
6
e x − 1 12 22 32 4 2
x n−1
1
1
⎛1
⎞
dx = Γ(n) ⎜ n + n + n + ⎟
ex − 1
2
3
⎝1
⎠
For even n this can be summed in terms of Bernoulli numbers (see pages 142–143).
18.81.
18.82.
∫
∫
∞
0
∞
0
x dx
π2
1 1
1
1
= 2 − 2 + 2 − 2 + =
x
12
e +1 1
2
3
4
x n−1
1
1
⎛1
⎞
dx = Γ(n) ⎜ n − n + n − ⎟
x
e +1
2
3
⎝1
⎠
For some positive integer values of n the series can be summed (see 23.10).
18.83.
∫
18.84.
∫
∞
0
sin mx
m
1
1
dx = coth −
4
2 2m
e 2π x − 1
⎛ 1
− x ⎞ dx
⎜⎝ 1 + x − e ⎟⎠ x = γ
∞
0
2
∞ −x
e
− e− x
dx = 12 γ
x
18.85.
∫
18.86.
e− x ⎞
⎛ 1
−
∫0 ⎜⎝ e x − 1 x ⎟⎠ dx = γ
18.87.
∫
18.88.
∫
18.89.
∫
0
∞
− e − bx
1 ⎛ b 2 + p2 ⎞
dx = ln ⎜ 2
x sec px
2 ⎝ a + p2 ⎟⎠
∞ − ax
e
0
− e − bx
b
a
dx = tan −1 − tan −1
x csc px
p
p
∞ − ax
e
0
∞ − ax
e
0
(1 − cos x )
a
dx = cot −1 a − ln (a 2 + 1)
2
x2
DEFINITE INTEGRALS
114
Definite Integrals Involving Logarithmic Functions
18.90.
∫
1
0
x m (ln x )n dx =
(−1)n n !
(m + 1)n+1
m > −1,
n = 0, 1, 2,…
If n ≠ 0, 1, 2, … replace n! by Γ(n + 1).
18.91.
ln x
π2
dx
=
−
∫0 1 + x
12
18.92.
ln x
π2
dx
=
−
∫0 1 − x
6
18.93.
18.94.
18.95.
18.96.
18.97.
1
1
∫
∫
∫
∫
∫
ln (1 + x )
π2
dx =
x
12
1
0
ln (1 − x )
π2
dx = −
x
6
1
0
1
0
1
0
ln x ln (1 + x ) dx = 2 − 2 ln 2 −
ln x ln (1 − x ) dx = 2 −
∞
0
0 < p<1
xm − xn
m +1
dx = ln
ln x
n +1
1
∫
0
18.99.
∫
0
18.100.
∫
18.101.
π2
⎛ e x + 1⎞
ln
dx
=
∫0 ⎜⎝ e x − 1⎟⎠
4
18.102.
∫
∞
e − x ln x dx = −γ
∞
0
π
(γ + 2 ln 2)
4
e − x ln x dx = −
2
∞
∫
π /2
0
π /2
0
π
18.104.
∫
0
18.105.
∫
0
18.106.
∫
0
18.107.
π2
6
x p−1 ln x
dx = −π 2 csc pπ cot pπ
1+ x
18.98.
18.103.
π2
12
∫
2π
0
∫
π /2
0
(ln sin x )2 dx =
x ln sin x dx = −
π /2
π
ln sin x dx =
∫
ln cos x dx = −
π /2
0
π
ln 2
2
(ln cos x )2 dx =
π
π3
(ln 2)2 +
2
24
π2
ln 2
2
sin x ln sin x dx = ln 2 − 1
ln (a + b sin x ) dx =
∫
2π
0
ln (a + b cos x ) dx = 2π ln (a + a 2 − b 2 )
⎛ a + a2 − b2 ⎞
ln(a + b cos x ) dx = π ln ⎜
⎟
2
⎝
⎠
DEFINITE INTEGRALS
π
18.108.
∫
18.109.
∫
0
18.110.
∫
0
18.111.
∫
0
⎧⎪2π ln a, a b > 0
ln (a 2 − 2ab cos x + b 2 ) dx = ⎨
⎪⎩2π ln b, b a > 0
π /4
π /2
a
0
115
ln (1 + tan x ) dx =
π
ln 2
8
⎛ 1 + b cos x ⎞
1
sec x ln ⎜
dx = {(cos −1 a)2 − (cos −1 b)2}
⎟
2
⎝ 1 + a cos x ⎠
x⎞
⎛
⎛ sin a sin 2a sin 3a
⎞
ln ⎜ 2 sin ⎟ dx = − ⎜ 2 +
+ 2 + ⎟
2⎠
22
3
⎝
⎝ 1
⎠
See also 18.102.
Definite Integrals Involving Hyperbolic Functions
18.112.
∫
18.113.
∫
18.114.
∫
18.115.
∫
∞
0
∞
0
∞
0
∞
0
sin ax
π
aπ
dx =
tanh
2b
2b
sinh bx
cos ax
π
aπ
sech
dx =
2b
2b
cosh bx
x dx
π2
= 2
sinh ax 4 a
{
}
x n dx
2n+1 − 1
1
1
1
= n n+1 Γ (n + 1) n+1 + n+1 + n+1 + sinh ax 2 a
1
2
3
If n is an odd positive integer, the series can be summed.
18.116.
∫
18.117.
∫
∞
0
∞
0
1
sinh ax
π
aπ
dx =
csc
−
b 2a
2b
e bx + 1
aπ
sinh ax
π
1
dx =
cot
−
b
2a 2b
e bx − 1
Miscellaneous Definite Integrals
18.118.
∫
f (ax ) − f (bx )
b
dx = { f (0) − f (∞)}ln
x
a
∞
0
This is called Frullani’s integral. It holds if f ′(x) is continuous and
18.119.
∫
18.120.
∫
1
0
dx 1 1
1
x = 1 + 2 + 3 +
x
1 2
3
a
−a
(a + x )m −1 (a − x )n−1 dx = (2a)m + n−1
Γ (m ) Γ (n)
Γ (m + n)
∫
∞
0
f ( x ) − f (∞)
dx converges.
x
Section V: Differential Equations and Vector Analysis
19
BASIC DIFFERENTIAL EQUATIONS
and SOLUTIONS
DIFFERENTIAL EQUATION
SOLUTION
19.1. Separation of variables
f1 ( x )
g ( y)
dx + ∫ 2
dy = c
f2 ( x )
g1 ( y)
∫
f1(x) g1(y) dx + f2(x) g2(y) dy = 0
19.2. Linear first order equation
ye ∫
dy
+ p( x ) y = Q( x )
dx
Pdx
= ∫ Qe ∫
Pdx
dx + c
19.3. Bernoulli’s equation
dy
+ P( x ) y = Q( x ) y n
dx
e
∫
(1− n ) Pdx
= (1 − n) ∫ Qe
∫
(1− n ) Pdx
dx + c
where y = y1−n. If n = 1, the solution is
ln y = ∫ (Q − P) dx + c
19.4. Exact equation
M(x, y)dx + N(x, y)dy = 0
where ∂M/∂y = ∂N/∂x.
⎛
∂
⎞
∫ M ∂x + ∫ ⎜⎝ N − ∂y ∫ M ∂x⎟⎠ dy = c
where ∂x indicates that the integration is to be performed with respect to x keeping y constant.
19.5 Homogeneous equation
⎛ y⎞
dy
= F⎜ ⎟
dx
⎝ x⎠
ln x =
d
∫ F ( ) − + c
where y = y/x. If F(y) = y, the solution is y = cx.
116
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS
117
19.6.
y F(xy) dx + x G(xy) dy = 0
ln x =
G ( ) d
∫ {G( ) − F ( )} + c
where y = xy. If G(y) = F(y), the solution is xy = c.
19.7. Linear, homogeneous second order
equation
d2y
dy
+a
+ by = 0
2
dx
dx
Let m1, m2 be the roots of m2 + am + b = 0. Then there
are 3 cases.
Case 1. m1, m2 real and distinct:
y = c1e m x + c2 e m x
1
Case 2.
2
m1, m2 real and equal:
y = c1 e m x + c2 x e m x
1
a, b are real constants.
Case 3.
m1 = p + qi,
1
m2 = p − qi:
y = e px (c1 cos qx + c2 sin qx )
where p = −a/2, q = b − a 2 /4 .
19.8. Linear, nonhomogeneous second
order equation
There are 3 cases corresponding to those of entry 19.7
above.
Case 1.
y = c1e m x + c2 e m x
2
d y
dy
+a
+ by = R( x )
2
dx
dx
1
+
a, b are real constants.
2
em x
e − m x R( x ) dx
m1 − m2 ∫
1
1
+
em x
e − m x R( x ) dx
m2 − m1 ∫
2
2
Case 2.
y = c1 e m x + c2 x e m x
1
1
+ xe m x ∫ e − m x R( x ) dx
1
1
− e m x ∫ xe − m x R( x ) dx
1
1
Case 3.
y = e px (c1 cos qx + c2 sin qx )
+
e px sin qx − px
∫ e R( x ) cos qx dx
q
−
e px cos qx − px
∫ e R( x ) sin qx dx
q
BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS
118
19.9. Euler or Cauchy equation
x2
Putting x = et, the equation becomes
d2y
dy
+ (a − 1) + by = S (e t )
dt 2
dt
d2y
dy
+ ax
+ by = S ( x )
2
dx
dx
and can then be solved as in entries 19.7 and 19.8
above.
19.10. Bessel’s equation
x2
d2y
dy
+x
+ ( 2 x 2 − n 2 ) y = 0
2
dx
dx
y = c1 J n ( x ) + c2 Yn (x )
See 27.1 to 27.15.
19.11. Transformed Bessel’s equation
x2
d2y
dy
+ (2 p + 1) x
+ (a 2 x 2 r + β 2 ) y = 0
dx 2
dx
⎛α ⎞
⎛ α ⎞ ⎪⎫
⎪⎧
y = x − p ⎨c1 J q/r ⎜ x r ⎟ + c2 Yq/r ⎜ x r ⎟ ⎬
⎝r ⎠
⎝ r ⎠ ⎪⎭
⎪⎩
where q =
p2 − β 2 .
19.12. Legendre’s equation
(1 − x 2 )
d2y
dy
− 2x
+ n(n + 1) y = 0
dx 2
dx
y = c1 Pn ( x ) + c2 Qn ( x )
See 28.1 to 28.48.
20
FORMULAS from VECTOR ANALYSIS
Vectors and Scalars
Various quantities in physics such as temperature, volume, and speed can be specified by a real number. Such
quantities are called scalars.
Other quantities such as force, velocity, and momentum require for their specification a direction as well
as magnitude. Such quantities are called vectors. A vector is represented by an arrow or directed line
segment indicating direction. The magnitude of the vector is determined by the length of the arrow,
using an appropriate unit.
Notation for Vectors
A vector is denoted by a bold faced letter such as A (Fig. 20.1). The magnitude is denoted by |A| or A. The
tail end of the arrow is called the initial point, while the head is called the terminal point.
Fundamental Definitions
1.
Equality of vectors. Two vectors are equal if they have the same
magnitude and direction. Thus, A = B in (Fig. 20-1).
2.
Multiplication of a vector by a scalar. If m is any real number
(scalar), then mA is a vector whose magnitude is |m| times the
magnitude of A and whose direction is the same as or opposite
to A according as m > 0 or m < 0. If m = 0, then mA = 0 is called
the zero or null vector.
Fig. 20-1
3. Sums of vectors. The sum or resultant of A and B is a vector C = A + B formed by placing the
initial point B on the terminal point A and joining the initial point of A to the terminal point of B as
in Fig. 20-2b. This definition is equivalent to the parallelogram law for vector addition as indicated in
Fig. 20-2c. The vector A − B is defined as A + (−B).
Fig. 20-2
Extension to sums of more than two vectors are immediate. Thus, Fig. 20-3 shows how to obtain the sum E
of the vectors A, B, C, and D.
119
FORMULAS FROM VECTOR ANALYSIS
120
Fig. 20-3
4.
Unit vectors. A unit vector is a vector with unit magnitude. If A is a vector, then a unit vector in the
direction of A is a = A/A where A > 0.
Laws of Vector Algebra
If A, B, C are vectors and m, n are scalars, then:
20.1. A + B = B + A
Commutative law for addition
20.2. A + (B + C) = (A + B) + C
Associative law for addition
20.3. m(nA) = (mn)A = n(mA)
Associative law for scalar multiplication
20.4. (m + n)A = mA + nA
Distributive law
20.5. m(A + B) = mA + mB
Distributive law
Components of a Vector
A vector A can be represented with initial point at the
origin of a rectangular coordinate system. If i, j, k are unit
vectors in the directions of the positive x, y, z axes, then
z
A
k
20.6. A = A1i + A2j + A3k
i
where A1i, A2j, A3k are called component vectors of A in
the i, j, k directions and A1, A2, A3 are called the components
of A.
Dot or Scalar Product
20.7. A • B = AB cos q
0qp
where q is the angle between A and B.
Fundamental results follow:
j
A3k
A2j
x
Fig. 20-4
y
A2i
FORMULAS FROM VECTOR ANALYSIS
20.8. A • B = B • A
Commutative law
20.9. A • (B + C) = A • B + A • C
Distributive law
121
20.10. A • B = A1B1 + A2B2 + A3B3
where A = A1i + A2j + A3k, B = B1i + B2j + B3k.
Cross or Vector Product
20.11. A × B = AB sin q u
0qp
where q is the angle between A and B and u is a unit vector
perpendicular to the plane of A and B such that A, B, u form
a right-handed system (i.e., a right-threaded screw rotated
through an angle less than 180° from A to B will advance
in the direction of u as in Fig. 20-5).
Fundamental results follow:
i
A
×
=
A
B
20.12.
1
B1
j
A2
B2
k
A3
B3
u
B
A
= ( A2 B3 − A3 B2 )i + ( A3 B1 − A1 B3 ) j + ( A1 B2 − A2 B1 )k
Fig. 20-5
20.13. A × B = − (B × A)
20.14. A × (B + C) = A × B + A × C
20.15. | A × B | = area of parallelogram having sides A and B
Miscellaneous Formulas Involving Dot and Cross Products
20.16.
A1 A2
A i (B × C) = B1 B2
C1 C2
A3
B3 = A1 B2C3 + A2 B3C1 + A3 B1C2 − A3 B2C1 − A2 B1C3 − A1 B3C2
C3
20.17. | A • (B × C) | = volume of parallelepiped with sides A, B, C
20.18. A × (B × C) = B(A • C) − C(A • B)
20.19. (A × B) × C = B(A • C) − A(B • C)
20.20.
(A × B) • (C × D) = (A • C)(B • D) − (A • D)(B • C)
20.21. (A × B) × (C × D) = C{A • (B × D)}− D{A • (B × C)}
= B{A • (C × D)} − A{B • (C × D)}
FORMULAS FROM VECTOR ANALYSIS
122
Derivatives of Vectors
The derivative of a vector function A(u) = A1(u)i + A2(u)j + A3(u)k of the scalar variable u is given by
20.22.
dA
dA
dA
A(u + Δu) − A(u) dA1
= lim
=
i+ 2 j+ 3 k
Δ
u
→
0
du
du
Δu
du
du
Partial derivatives of a vector function A(x, y, z) are similarly defined. We assume that all derivatives exist
unless otherwise specified.
Formulas Involving Derivatives
20.23.
d
dB dA
(A i B) = A i
+
iB
du
du du
20.24.
d
dB dA
(A × B) = A ×
+
×B
du
du du
20.25.
⎛ dB
⎛
⎞
d
dA
dC ⎞
{A i (B × C )} =
i (B × C ) + A i ⎜
× C⎟ + A i ⎜ B ×
du
du
du ⎟⎠
⎝ du
⎝
⎠
20.26.
Ai
dA
dA
=A
du
du
20.27.
Ai
dA
=0
du
if | A | is a constant
The Del Operator
The operator del is defined by
20.28.
∇=i
∂
∂
∂
+ j +k
∂x
∂y
∂z
In the following results we assume that U = U(x, y, z), V = V(x, y, z), A = A(x, y, z) and B = B(x, y, z) have
partial derivatives.
The Gradient
⎛ ∂
∂
∂⎞
∂U
∂U
∂U
20.29. Gradient of U = grad U = ∇U = ⎜ i + j + k ⎟ U =
i+
j+
k
∂y
∂z ⎠
∂x
∂y
∂z
⎝ ∂x
The Divergence
⎛ ∂
∂
∂⎞
20.30. Divergence of A = div A = ∇ • A = ⎜ i + j + k ⎟ i ( A1 i + A2 j + A3 k)
∂y
∂z ⎠
⎝ ∂x
=
∂A1 ∂A2 ∂A3
+
+
∂y
∂z
∂x
FORMULAS FROM VECTOR ANALYSIS
The Curl
20.31. Curl of A = curl A = ∇ × A
⎛ ∂
∂
∂⎞
= ⎜ i + j + k ⎟ × ( A1 i + A2 j + A3 k)
∂
∂
∂
x
y
z⎠
⎝
i
∂
=
∂x
A1
j
∂
∂y
A2
k
∂
∂z
A3
⎛ ∂A ∂A ⎞
⎛ ∂A ∂A ⎞
⎛ ∂A
∂A ⎞
= ⎜ 3 − 2 ⎟ i + ⎜ 1 − 3⎟ j + ⎜ 2 − 1⎟ k
∂x ⎠
∂y ⎠
∂z ⎠
⎝ ∂x
⎝ ∂z
⎝ ∂y
The Laplacian
20.32. Laplacian of U = ∇ 2U = ∇ i (∇U ) =
20.33.
Laplacian of A = ∇ 2 A =
∂ 2U ∂ 2U ∂ 2U
+
+ 2
∂x 2 ∂y 2
∂z
∂2 A ∂2 A ∂2 A
+ 2 + 2
∂x 2
∂y
∂z
The Biharmonic Operator
20.34. Biharmonic operator on U = ∇ 4U = ∇ 2 (∇ 2U )
=
∂ 4U
∂ 4U ∂ 4U ∂ 4U
∂ 4U
∂ 4U
2
+
2
2
+
+
+
+
∂y 2 ∂z 2
∂x 4
∂y 4
∂z 4
∂x 2 ∂y 2
∂x 2 ∂z 2
Miscellaneous Formulas Involving ∇
20.35.
∇(U + V ) = ∇U + ∇V
20.36.
∇ i (A + B) = ∇ i A + ∇ i B
20.37.
∇ × (A + B) = ∇ × A + ∇ × B
20.38.
∇ i (UA) = (∇U ) i A + U (∇ i A)
20.39.
∇ × (UA) = (∇U ) × A + U (∇ × A)
20.40.
∇ i (A × B) = B i (∇ × A) − A i (∇ × B)
20.41.
∇ × (A × B) = (B i ∇) A − B(∇ i A) − (A i ∇)B + A(∇ i B)
20.42.
∇(A i B) = (B i ∇) A + ( A i ∇)B + B × (∇ × A) + A × (∇ × B)
20.43.
∇ × (∇U ) = 0,
20.44.
∇ i (∇ × A) = 0, that is, the divergence of the curl of A is zero.
20.45.
∇ × (∇ × A) = ∇(∇ i A) − ∇ 2 A
that is, the curl of the gradient of U is zero.
123
FORMULAS FROM VECTOR ANALYSIS
124
Integrals Involving Vectors
d
B(u). then the indefinite integral of A(u) is as follows:
du
If A(u) =
20.46.
∫ A(u)du = B(u) + c,
c = constant vector
The definite integral of A(u) from u = a to u = b in this case is given by
20.47.
∫
b
A(u) du = B(b) − B(a)
a
The definite integral can be defined as in 18.1.
Line Integrals
z
Consider a space curve C joining two points P1(a1, a2, a3) and
P2(b1, b2, b3) as in Fig. 20-6. Divide the curve into n parts by
points of subdivision (x1, y1, z1), . . . , (xn−1, yn−1, zn−1). Then the
line integral of a vector A(x, y, z) along C is defined as
P2
C
(xp , yp , zp)
P1
20.48.
∫
c
A i dr =
∫
P2
n→∞
P1
y
n
A i dr = lim ∑ A( x p , y p , z p ) i Δrp
p =1
x
Fig. 20-6
where Δrp = Δx p i + Δy p j + Δz pk, Δx p = x p+1 − x p , Δy p = y p+1 − y p ,
Δz p = z p+1 − z p and where it is assumed that as n → ∞ the largest
of the magnitudes |rp | approaches zero. The result 20.48 is a
generalization of the ordinary definite integral (see 18.1).
The line integral 20.48 can also be written as
20.49.
∫
C
A i dr =
∫
( A1dx + A2 dy + A3 dz )
C
using A = A1i + A2j + A3k
dr = dxi + dyj + dzk.
and
Properties of Line Integrals
20.50.
20.51.
p2
∫
p1
∫
P1
P1
A i dr = − ∫ A i dr
P2
P2
A i dr =
∫
P3
P1
P2
A i dr + ∫ A i dr
P3
Independence of the Path
In general, a line integral has a value that depends on the particular path C joining points P1 and P2 in a region
. However, in the case of A = ∇f or ∇ × A = 0 where f and its partial derivatives are continuous in , the
line integral ∫ A i dr is independent of the path. In such a case,
C
20.52.
∫
C
A i dr =
∫
P2
P1
A i dr = φ (P2 ) − φ (P1 )
where f(P1) and f(P2) denote the values of f at P1 and P2, respectively. In particular if C is a closed curve,
FORMULAS FROM VECTOR ANALYSIS
20.53.
∫
C
125
A i dr = ∫ A i dr = 0
C
where the circle on the integral sign is used to emphasize that C is closed.
Multiple Integrals
Let F(x, y) be a function defined in a region of the
xy plane as in Fig. 20-7. Subdivide the region into n
parts by lines parallel to the x and y axes as indicated.
Let Ap = xp yp denote an area of one of these parts.
Then the integral of F(x, y) over is defined as
20.54.
∫
n
F( x , y) dA = lim ∑ F( x p , y p )ΔAp
n→∞
p =1
provided this limit exists.
In such a case, the integral can also be written as
20.55.
b
∫ ∫
x =a
f2 (x)
y = f1 (x)
Fig. 20-7
F( x , y) dy dx
{∫
⎫
F( x , y) dy⎬ dx
⎭
where y = f1(x) and y = f2(x) are the equations of curves PHQ and PGQ, respectively, and a and b are the x
coordinates of points P and Q. The result can also be written as
=
20.56.
∫
b
x =a
d
g2( y )
y =c
x = g1( y )
∫ ∫
f2( x )
y = f1( x )
F ( x , y) dx dy =
∫
d
y =c
{∫
g2( y )
x = g1( y )
}
F ( x , y) dx dy
where x = g1(y), x = g2(y) are the equations of curves HPG and HQG, respectively, and c and d are the y
coordinates of H and G.
These are called double integrals or area integrals. The ideas can be similarly extended to triple or volume
integrals or to higher multiple integrals.
Surface Integrals
z
Subdivide the surface S (see Fig. 20-8) into n elements of area
ΔS p , p = 1, 2, … , n, Let A( x p , y p , z p ) = A p where ( x p , y p , z p )
is a point P in Sp. Let Np be a unit normal to Sp at P. Then
the surface integral of the normal component of A over S is
defined as
20.57.
∫
Np
γ
ΔSp
S
n
S
A i N dS = lim ∑ A p i N p ΔS p
n→∞
y
p =1
Δ xp Δyp
x
Fig. 20-8
FORMULAS FROM VECTOR ANALYSIS
126
Relation Between Surface and Double Integrals
If is the projection of S on the xy plane, then (see Fig. 20-8)
20.58.
∫
S
A i N dS = ∫
∫ AiN
dx dy
Nik
The Divergence Theorem
Let S be a closed surface bounding a region of volume V; and suppose N is the positive (outward drawn)
normal and dS = N dS. Then (see Fig. 20-9)
20.59.
∫
V
∇ i A dV =
∫
S
A i dS
The result is also called Gauss’ theorem or Green’s theorem.
z
z
N
N
S
dS
S
dS
C
y
y
x
x
Fig. 20-10
Fig. 20-9
Stokes’ Theorem
Let S be an open two-sided surface bounded by a closed non-intersecting curve C(simple closed curve) as
in Fig. 20-10. Then
20.60.
∫
C
A i dr =
∫
S
(∇ × A) i dS
where the circle on the integral is used to emphasize that C is closed.
Green’s Theorem in the Plane
20.61.
∫
C
(P dx + Q dy) =
∫
⎛ ∂Q ∂P ⎞
−
dx dy
R⎜
⎝ ∂x ∂y ⎟⎠
where R is the area bounded by the closed curve C. This result is a special case of the divergence theorem
or Stokes’ theorem.
FORMULAS FROM VECTOR ANALYSIS
127
Green’s First Identity
20.62.
∫
V
{(φ ∇ 2ψ + ( ∇φ ) i (∇ψ )}dV = ∫ (φ ∇ψ ) i dS
where f and y are scalar functions.
Green’s Second Identity
20.63.
∫
V
(φ ∇ 2ψ − ψ ∇ 2φ ) dV =
∫
S
(φ ∇ψ − ψ ∇φ ) i dS
Miscellaneous Integral Theorems
20.64.
∫
∇ × A dV =
∫
φ dr =
V
20.65.
C
∫
S
∫
S
dS × A
dS × ∇φ
Curvilinear Coordinates
A point P in space (see Fig. 20-11) can be located by
rectangular coordinates (x, y, z,) or curvilinear coordinates
(u1, u2, u3) where the transformation equations from one
set of coordinates to the other are given by
20.66.
z
u3 curve
e3
x = x (u1, u2 , u3)
c2
u2 =
y = y(u1, u2 , u3)
z = z (u1, u2 , u3)
u1 curve
e1
P
u3 =
c3
u1 =
c1
e2
u2 curve
y
If u2 and u3 are constant, then as u1 varies, the position
vector r = xi + yj + zk of P describes a curve called
the u1 coordinate curve. Similarly, we define the u2 and
u3 coordinate curves through P. The vectors ∂r/∂u1,
∂r/∂u2 , ∂r/∂u3 represent tangent vectors to the u1, u2,
u3 coordinate curves. Letting e1, e2, e3 be unit tangent
vectors to these curves, we have
20.67.
∂r
= h1e1 ,
∂u1
∂r
= h2 e 2 ,
∂u2
x
Fig. 20-11
∂r
= h3 e3
∂u3
where
20.68.
h1 =
∂r
,
∂u1
h2 =
∂r
,
∂u2
h3 =
∂r
∂u3
are called scale factors. If e1, e2, e3 are mutually perpendicular, the curvilinear coordinate system is called
orthogonal.
FORMULAS FROM VECTOR ANALYSIS
128
Formulas Involving Orthogonal Curvilinear Coordinates
∂r
∂r
∂r
du +
du +
du = h1 du1e1 + h2 du2 e 2 + h3 du3 e 3
∂u1 1 ∂u2 2 ∂u3 3
20.69.
dr =
20.70.
ds 2 = dr i dr = h12 du12 + h22 du22 + h32 du32
where ds is the element of are length.
If dV is the element of volume, then
20.71.
dV = | (h1e1du1 ) i (h2 e 2 du2 ) × (h3 e 3 du3 ) | = h1h2 h3 du1du2 du3
=
∂r ∂r
∂r
∂( x , y, z )
du1 du2 du3 =
du du du
×
i
∂(u1, u2 , u3 ) 1 2 3
∂u1 ∂u2 ∂u3
where
20.72.
∂x /∂u1
∂( x , y, z )
= ∂y/∂u1
∂(u1, u2 , u3 )
∂z/∂u1
∂x /∂u2
∂y/∂u2
∂z/∂u2
∂x /∂u3
∂y/∂u3
∂z/du3
sometimes written J(x, y, z; u1, u2, u3), is called the Jacobian of the transformation.
Transformation of Multiple Integrals
Result 20.72 can be used to transform multiple integrals from rectangular to curvilinear coordinates. For
example, we have
20.73.
∫ ∫ ∫ F ( x , y, z) dx dy dz = ∫ ∫ ∫ G(u , u , u )
1
2
3
′
∂( x , y, z )
du du du
∂(u1, u2 , u3) 1 2 3
where ′ is the region into which is mapped by the transformation and G(u1, u2, u3) is the value of F(x, y, z)
corresponding to the transformation.
Gradient, Divergence, Curl, and Laplacian
In the following, Φ is a scalar function and A = A1e1 + A2e2 + A3e3 is a vector function of orthogonal curvilinear coordinates u1, u2, u3.
20.74. Gradient of Φ = grad Φ = ∇Φ =
e1 ∂ Φ e 2 ∂ Φ e3 ∂ Φ
+
+
h1 ∂u1 h2 du2 h3 ∂u3
1
h1h2 h3
20.75.
Divergence of A = div A = ∇ i A =
20.76.
h1e1
1
∂
Curl of A = curl A = ∇ × A =
h1h2 h3 ∂u1
h1 A1
=
1
h2 h3
⎤
⎡ ∂
∂
∂
(h1h2 A3 )⎥
(h2 h3 A1 ) +
(h3 h1 A2 ) +
⎢
∂u3
∂u2
⎦
⎣ ∂u1
h2 e 2
∂
∂u2
h2 A2
h3 e 3
∂
∂u3
h3 A3
⎤
⎤
⎡ ∂
1 ⎡ ∂
∂
∂
(h3 A3 ) −
(h2 A2 )⎥ e1 +
(h1 A1 ) −
(h3 A3 )⎥ e 2
⎢
⎢
∂
∂
h
h
u
u
u
u
∂
∂
3
1 3 ⎣
3
1
⎦
⎦
⎣ 2
+
1
h1h2
⎤
⎡ ∂
∂
(h2 A2 ) −
(h1 A1 )⎥ e 3
⎢
∂u2
⎦
⎣ ∂u1
FORMULAS FROM VECTOR ANALYSIS
20.77. Laplacian of Φ = ∇ 2 Φ =
1
h1h2 h3
129
⎡ ∂ ⎛ h2 h3 ∂ Φ⎞
∂ ⎛ h1h2 ∂ Φ⎞ ⎤
∂ ⎛ h3 h1 ∂ Φ⎞
+
+
⎢
⎟⎥
⎟
⎜
⎟
⎜
⎜
⎢⎣ ∂u1 ⎝ h1 ∂u1 ⎠ ∂u2 ⎝ h2 ∂u2 ⎠ ∂u3 ⎝ h3 ∂u3 ⎠ ⎥⎦
Note that the biharmonic operator ∇ 4 Φ = ∇ 2 (∇ 2 Φ) can be obtained from 20.77.
Special Orthogonal Coordinate Systems
Cylindrical Coordinates (r, q, z) (See Fig. 20-12)
20.78. x = r cos q,
20.79.
h12 = 1,
20.80.
∇2Φ =
y = r sin q,
h22 = r 2 ,
z=z
h32 = 1
∂2 Φ 1 ∂ Φ 1 ∂2 Φ ∂2 Φ
+
+
+ 2
∂ r 2 r ∂r r 2 ∂θ 2
∂z
ez
z
z
er
eq
ef
(r, q, z)
P
P(r, q, f,)
er
z
z
x
x
q
y
y
r
f
y
eq
x
x
Fig. 20-12.
Fig. 20-13.
Cylindrical coordinates.
Spherical coordinates.
Spherical Coordinates (r, q, f) (See Fig. 20-13)
20.81. x = r sin q cos f,
y = r sin q sin f,
20.82.
h12 = 1,
h32 = r 2 sin 2 θ
20.83.
∇2Φ =
h22 = r 2 ,
z = r cos q
1
∂2 Φ
1 ∂ ⎛ 2 ∂Φ ⎞
1
∂ ⎛
∂Φ ⎞
+ 2 2
+ 2
r
sin θ
⎜
⎟
⎜
⎟
2
r ∂r ⎝ dr ⎠ r sin θ ∂θ ⎝
∂θ ⎠ r sin θ ∂φ 2
Parabolic Cylindrical Coordinates (u, y, z)
20.84.
x = 12 (u 2 − 2 ),
20.85.
h12 = h22 = u 2 + 2 ,
20.86.
∇2Φ =
y = u ,
z=z
h32 = 1
1 ⎛ ∂2 Φ ∂2 Φ ⎞ ∂2 Φ
+
+
u 2 + 2 ⎜⎝ ∂u 2 ∂ 2 ⎟⎠ ∂z 2
The traces of the coordinate surfaces on the xy plane
are shown in Fig. 20-14. They are confocal parabolas
with a common axis.
Fig. 20-14
y
FORMULAS FROM VECTOR ANALYSIS
130
Paraboloidal Coordinates (u, y, f)
20.87.
x = u cos φ ,
where
y = u sin φ ,
u 0,
20.88.
h12 = h22 = u 2 + 2 ,
20.89.
∇2Φ =
0,
z = 12 (u 2 − 2 )
0 φ < 2π
h32 = u 2 2
∂ ⎛ ∂Φ ⎞
1
1
1 ∂2 Φ
∂ ⎛ ∂Φ ⎞
u
+
+
u(u 2 + 2 ) ∂u ⎜⎝ ∂u ⎟⎠ (u 2 + 2 ) ∂ ⎜⎝ ∂ ⎟⎠ u 2 2 ∂φ 2
Two sets of coordinate surfaces are obtained by revolving the parabolas of Fig. 20-14 about the x axis
which is then relabeled the z axis.
Elliptic Cylindrical Coordinates (u, y, z)
20.90.
x = a cosh u cos ,
where
u 0,
y = a sinh u sin ,
0 < 2π ,
20.91.
h12 = h22 = a 2 (sinh 2 u + sin 2 ),
20.92.
∇2Φ =
z=z
− ∞< z < ∞
h32 = 1
⎛ ∂2 Φ ∂2 Φ ⎞ ∂2 Φ
1
+
+
2
a (sinh u + sin ) ⎜⎝ ∂u 2 ∂ 2 ⎟⎠ ∂z 2
2
2
The traces of the coordinate surfaces on the xy plane are shown in Fig. 20-15. They are confocal ellipses
and hyperbolas.
Fig. 20-15.
Elliptic cylindrical coordinates.
FORMULAS FROM VECTOR ANALYSIS
131
Prolate Spheroidal Coordinates (x, h, f)
20.93.
x = a sinh ξ sin η cos φ ,
y = a sinh ξ sin η sin φ ,
ξ 0,
where
20.94.
h12 = h22 = a 2 (sinh 2 ξ sin 2 η),
20.95.
∇2Φ =
z = a cosh ξ cos η
0 η π,
0 φ < 2π
h32 = a 2 sinh 2 ξ sin 2 η
∂ ⎛
∂Φ ⎞
1
sinh ξ
∂ξ ⎟⎠
a 2 (sinh 2 ξ + sin 2 η) sinh ξ ∂ξ ⎜⎝
+
∂ ⎛
∂Φ ⎞
∂2 Φ
1
1
sin
η
+
⎜
⎟
a 2 (sinh 2 ξ + sin 2 η) sin η ∂η ⎝
∂η ⎠ a 2 sinh 2 ξ sin 2 η ∂φ 2
Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 20-15 about the x axis which
is relabeled the z axis. The third set of coordinate surfaces consists of planes passing through this axis.
Oblate Spheroidal Coordinates (x, h, f)
20.96.
x = a cosh ξ cos η cos φ ,
y = a cosh ξ cos η sin φ ,
where
ξ 0,
20.97.
h12 = h22 = a 2 (sinh 2 ξ + sin 2 η),
20.98.
∇2Φ =
z = a sinh ξ sin η
− π /2 η π /2,
0 φ < 2π
h32 = a 2 cosh 2 ξ cos 2 η
∂ ⎛
∂Φ ⎞
1
cosh ξ
⎜
2
∂ξ ⎠⎟
a (sinh ξ + sin η) cosh ξ ∂ξ ⎝
2
2
+
∂ ⎛
∂Φ ⎞
∂2 Φ
1
1
cos η
+ 2
⎜
⎟
2
2
2
a (sinh ξ + sin η) cos η ∂η ⎝
∂η ⎠ a cosh ξ cos η ∂φ 2
2
2
Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 20-15 about the y axis which
is relabeled the z axis. The third set of coordinate surfaces are planes passing through this axis.
Bipolar Coordinates (u, y, z)
20.99.
x=
a sinh ,
cosh − cos u
where
y=
a sin u
,
cosh − cos u
0 u < 2π ,
z=z
−∞< < ∞,
−∞< z < ∞
or
20.100.
x 2 + (y − a cot u)2 = a 2 csc 2 u,
( x − a coth )2 + y 2 = a 2 csch 2 ,
z=z
FORMULAS FROM VECTOR ANALYSIS
132
20.101.
h12 = h22 =
20.102.
∇2Φ =
a2
,
(cosh − cos u)2
(cosh − cos u)2
a2
h32 = 1
⎛ ∂2 Φ ∂2 Φ ⎞ ∂2 Φ
⎜⎝ ∂u 2 + ∂ 2 ⎟⎠ + ∂z 2
The traces of the coordinate surfaces on the xy plane are shown in Fig. 20-16.
Fig. 20-16.
Bipolar coordinates.
Toroidal Coordinates (u, y, f)
a sinh cos φ
,
cosh − cos u
20.103.
x=
20.104.
h12 = h22 =
20.105.
∇ 2Φ =
y=
a sinh sin φ
,
cosh − cos u
a2
,
(cosh − cos u)2
h32 =
z=
a sin u
cosh − cos u
a 2 sinh 2 (cosh − cos u)2
(cosh − cos u)3 ∂ ⎛
1
∂Φ⎞
2
⎜
a
∂u ⎝ cosh − cos u ∂u ⎟⎠
+
(cosh − cos u)3 ∂ ⎛
sinh ∂Φ⎞ (cosh − cos u)2 ∂ 2Φ
+
a 2 sinh ∂ ⎜⎝ cosh − cos u ∂ ⎟⎠
a 2 sinh 2 ∂f 2
The coordinate surfaces are obtained by revolving the curves of Fig. 20.16 about the y axis which is
relabeled the z axis.
Conical Coordinates (l, m, v)
μ v
,
ab
20.106.
x=
20.107.
h12 = 1,
y=
h22 =
(μ 2 − a 2 )(v 2 − a 2 )
,
a
a2 − b2
2 (μ 2 − v 2 )
,
(μ 2 − a 2 )(b 2 − μ 2 )
h32 =
z=
(μ 2 − b 2 )(v 2 − b 2 )
b
b2 − a2
2 (μ 2 − v 2 )
(v 2 − a 2 )(v 2 − b 2 )
FORMULAS FROM VECTOR ANALYSIS
133
Confocal Ellipsoidal Coordinates (l, m, y)
⎧ x2
y2
z2
⎪ a 2 − + b 2 − + c 2 − = 1,
⎪
z2
y2
⎪ x2
+ 2
+ 2
= 1,
⎨ 2
⎪a − μ b − μ c − μ
⎪ x2
y2
z2
+ 2
+ 2
= 1,
⎪ 2
⎩a − v b − v c − v
20.108.
< c2 < b2 < a2
c2 < μ < b2 < a2
c2 < b2 < v < a2
or
20.109.
⎧ 2 (a 2 − )(a 2 − μ )(a 2 − v)
⎪x =
(a 2 − b 2 )(a 2 − c 2 )
⎪
⎪ 2 (b 2 − )(b 2 − μ )(b 2 − v)
⎨y =
(b 2 − a 2 )(a 2 − c 2 )
⎪
⎪
(c 2 − )(c 2 − μ )(c 2 − v)
⎪z 2 =
(c 2 − a 2 )(c 2 − b 2 )
⎩
20.110.
⎧ 2
⎪h1 = 4(a 2
⎪
⎪ 2
⎨h2 =
4(a 2
⎪
⎪
⎪h32 =
4(a 2
⎩
(μ − )(v − )
− )(b 2 − )(c 2 − )
(v − μ )( − μ )
− μ )(b 2 − μ )(c 2 − μ )
( − v)(μ − v)
− v)(b 2 − v)(c 2 − v)
Confocal Paraboloidal Coordinates (l, m, v)
20.111.
20.112.
.
20.113.
⎧ x2
h2
⎪ a 2 − + b 2 − = z − ,
⎪
y2
⎪ x2
+ 2
= z − μ,
⎨ 2
⎪a − μ b − μ
⎪ x2
y2
+ 2
= z − v,
⎪ 2
⎩a − v b − v
or
− ∞< < b 2
b2 < μ < a2
a2 < v < ∞
⎧ 2 (a 2 − )(a 2 − μ )(a 2 − v)
⎪x =
b2 − a2
⎪⎪
⎨ 2 (b 2 − )(b 2 − μ )(b 2 − v)
⎪y =
a2 − b2
⎪
⎪⎩z = + μ + v − a 2 − b 2
⎧ 2
(μ − )(v − )
⎪h1 = 4(a 2 − )(b 2 − )
⎪
⎪ 2
(v − μ )( − μ )
⎨h2 =
a 2 − μ )(b 2 − μ )
4
(
⎪
⎪
( − v)(μ − v)
⎪h32 =
16(a 2 − v)(b 2 − v)
⎩
Section VI: Series
21
SERIES of CONSTANTS
Arithmetic Series
21.1.
a + (a + d ) + (a + 2 d ) + ⋅⋅⋅ + {a + (n − 1)d } = 12 n{2 a + (n − 1)d } = 12 n(a + l )
where l a (n 1)d is the last term.
Some special cases are
21.2.
1 + 2 + 3 + ⋅⋅⋅ + n = 12 n(n + 1)
21.3.
1 + 3 + 5 + ⋅⋅⋅ + (2 n − 1) = n 2
Geometric Series
21.4.
a + ar + ar 2 + ar 3 + ⋅⋅⋅ + ar n−1 =
a(1 − r n ) a − rl
=
1− r
1− r
where l ar n1 is the last term and r ≠ 1.
If 1 < r < 1, then
21.5.
a + ar + ar 2 + ar 3 + ⋅⋅⋅ =
a
1− r
Arithmetic-Geometric Series
21.6.
a + (a + d )r + (a + 2 d )r 2 + ⋅⋅⋅ + {a + (n − 1)d }r n−1 =
a(1 − r n ) rd {1 − nr n−1 + (n − 1)r n }
+
1− r
(1 − r )2
where r ≠ 1.
If 1 < r < 1, then
21.7.
a + (a + d )r + (d + 2 d )r 2 + ⋅⋅⋅ =
a
rd
+
1 − r (1 − r )2
Sums of Powers of Positive Integers
21.8.
1p + 2 p + 3 p + ⋅⋅⋅ + n p =
n p+1 1 p B1 pn p−1 B2 p( p − 1)( p − 2 )n p− 3
+ ⋅⋅⋅
+ n +
−
4!
p +1 2
2!
where the series terminates at n2 or n according as p is odd or even, and Bk are the Bernoulli numbers (see
page 142).
134
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SERIES OF CONSTANTS
135
Some special cases are
21.9.
1 + 2 + 3 + ⋅⋅⋅ + n =
n(n + 1)
2
21.10.
12 + 2 2 + 32 + ⋅⋅⋅ + n 2 =
n(n + 1)(2 n + 1)
6
21.11.
13 + 2 3 + 33 + ⋅⋅⋅ + n 3 =
n 2 (n + 1)2
= (1 + 2 + 3 + ⋅⋅⋅ + n )2
4
21.12.
14 + 2 4 + 34 + ⋅⋅⋅ + n 4 =
n(n + 1)(2 n + 1)(3n 2 + 3n − 1)
30
If Sk = 1k + 2 k + 3k + ⋅⋅⋅ + n k where k and n are positive integers, then
21.13.
⎛ k + 1⎞
⎛ k + 1⎞
⎛ k + 1⎞
k +1
⎜⎝ 1 ⎟⎠ S1 + ⎜⎝ 2 ⎟⎠ S2 + ⋅⋅⋅ + ⎜⎝ k ⎟⎠ Sk = (n + 1) − (n + 1)
Series Involving Reciprocals of Powers of Positive Integers
21.14.
1−
1 1 1 1
+ − + − ⋅⋅⋅ = ln 2
2 3 4 5
21.15.
1−
π
1 1 1 1
+ − + − ⋅⋅⋅ =
3 5 7 9
4
21.16.
1−
π 3 1
1 1
1
1
+ −
+
− ⋅⋅⋅ =
+ ln 2
4 7 10 13
9
3
21.17.
1−
π 2
1 1
1
1
+ −
+
− ⋅⋅⋅ =
+
5 9 13 17
8
21.18.
1 1 1
1
1
π 3
1
− + −
+ − ⋅⋅⋅ =
+ ln 2
2 5 8 11 14
9
3
21.19.
1
1
1
1
π2
2 +
2 +
2 +
2 + ⋅⋅⋅ =
6
1
2
3
4
21.20.
1
1
1
1
π4
4 +
4 +
4 +
4 + ⋅⋅⋅ =
90
1
2
3
4
21.21.
1
1
1
1
π6
6 +
6 +
6 +
6 + ⋅⋅⋅ =
945
1
2
3
4
21.22.
1
1
1
1
π2
2 −
2 +
2 −
2 + ⋅⋅⋅ =
12
1
2
3
4
21.23.
1
1
1
1
7π 4
4 −
4 +
4 −
4 + ⋅⋅⋅ =
720
1
2
3
4
21.24.
1
1
1
1
31π 6
6 −
6 +
6 −
6 + ⋅⋅⋅ =
30, 240
1
2
3
4
21.25.
1
1
1
1
π2
2 +
2 +
2 +
2 + ⋅⋅⋅ =
8
1
3
5
7
21.26.
1
1
1
1
π4
4 +
4 +
4 +
4 + ⋅⋅⋅ =
96
1
3
5
7
2 ln (1 + 2 )
4
SERIES OF CONSTANTS
136
21.27.
1
1
1
1
π6
6 +
6 +
6 +
6 + ⋅⋅⋅ =
960
1
3
5
7
21.28.
1
1
1
1
π3
3 −
3 +
3 −
3 + ⋅⋅⋅ =
32
1
3
5
7
21.29.
1
1
1
1
3π 3 2
3 +
3 −
3 −
3 + ⋅⋅⋅ =
128
1
3
5
7
21.30.
1
1
1
1
1
+
+
+
+ ⋅⋅⋅ =
1i 3 3 i5 5 i 7 7 i9
2
21.31.
1
1
1
1
3
+
+
+
+ ⋅⋅⋅ =
1i 3 2 i 4 3 i5 4 i6
4
21.32.
1
1
1
1
π2 −8
2 +
2
2 +
2
2 +
2
2 + ⋅⋅⋅ =
16
1 i3
3 i5
5 i7
7 i9
21.33.
1
1
1
4π 2 − 39
2
2 +
2
2
2 +
2
2
2 + ⋅⋅⋅ =
16
1 i 2 i3
2 i3 i 4
3 i4 i5
21.34.
1 u a −1du
1
1
1
1
−
+
−
+ ⋅⋅⋅ = ∫
0 1 + ud
a a + d a + 2 d a + 3d
2
2
21.35.
2 2 p−1π 2 p Bp
1
1
1
1
2p +
2p +
2p +
2 p + ⋅⋅⋅ =
(2 p)!
1
2
3
4
21.36.
(2 2 p − 1) π 2 p Bp
1
1
1
1
+
+
+
+
⋅⋅⋅
=
2(2 p)!
12 p 32 p 5 2 p 7 2 p
21.37.
(2 2 p−1 − 1) π 2 p Bp
1
1
1
1
2p −
2p +
2p −
2 p + ⋅⋅⋅ =
(2 p)!
1
2
3
4
21.38.
1
12 p+1
−
1
32 p+1
+
1
5 2 p+1
−
1
7 2 p+1
+ ⋅⋅⋅ =
π 2 p+1E p
2 2 p+ 2 (2 p)!
Miscellaneous Series
21.39.
1
sin(n + 1/2)α
+ cos α + cos 2α + ⋅⋅⋅ + cos nα =
2
2 sin(α /2)
21.40.
sin α + sin 2α + sin 3α + ⋅⋅⋅ + sin nα =
21.41.
1 + r cos α + r 2 cos 2α + r 3 cos 3α + ⋅⋅⋅ =
21.42.
r sin α + r 2 sin 2α + r 3 sin 3α + ⋅⋅⋅ =
21.43.
1 + r cos α + r 2 cos 2α + ⋅⋅⋅ + r n cos nα =
21.44.
r sin α + r 2 sin 2α + ⋅⋅⋅ + r n sin nα =
sin[1/2(n + 1)]α sin 1/2nα
sin (α /2)
1 − r cos α
, |r |< 1
1 − 2r cos α + r 2
r sin α
, |r |< 1
1 − 2r cos α + r 2
r n+ 2 cos nα − r n+1 cos(n + 1)α − r cos α + 1
1 − 2 r cos α + r 2
r sin α − r n+1 sin(n + 1))α + r n+ 2 sin nα
1 − 2 r cos α + r 2
SERIES OF CONSTANTS
137
The Euler-Maclaurin Summation Formula
n −1
21.45.
∑ F (k ) = ∫
k =1
n
0
1
F (k )dk − {F (0) + F (n)}
2
+
1
1
{F ′(n) − F (0)} −
{F ′′′(n) − F ′′′(0)}
720
12
+
1
1
{F ( vii ) (n) − F ( vii ) (0)}
{F ( v)) (n) − F ( v ) (0)} −
30, 240
1, 209, 600
+ ⋅⋅⋅ (−1) p−1
Bp
{F ( 2 p−1) (n) − F ( 2 p−1) (0)} + ⋅⋅⋅
(2 p)!
The Poisson Summation Formula
∞
21.46.
∞
{
∑ F (k ) = ∑ ∫
k =−∞
m =−∞
∞
}
e 2π imx F ( x )dx
−∞
22
TAYLOR SERIES
Taylor Series for Functions of One Variable
22.1.
f ( x ) = f (a ) + f ′(a )( x − a ) +
f ′′(a )( x − a )2
f ( n−1) (a )( x − a )
+ ⋅⋅⋅ +
(n − 1)!
2!
n −1
+ Rn
where Rn, the remainder after n terms, is given by either of the following forms:
22.2. Lagrange’s form: Rn =
22.3. Cauchy’s form: Rn =
f ( n ) (ξ )( x − a )n
n!
f ( n ) (ξ )( x − ξ )n−1 ( x − a )
(n − 1)!
The value x, which may be different in the two forms, lies between a and x. The result holds if f(x) has
continuous derivatives of order n at least.
If lim Rn = 0, the infinite series obtained is called the Taylor series for f(x) about x a. If a 0, the series
n→∞
is often called a Maclaurin series. These series, often called power series, generally converge for all values of x
in some interval called the interval of convergence and diverge for all x outside this interval.
Some series contain the Bernoulli numbers Bn and the Euler numbers En defined in Chapter 23, pages
142143.
Binomial Series
22.4.
(a + x )n = a n + na n−1 x +
n(n − 1) n− 2 2 n(n − 1)(n − 2 ) n− 3 3
a x + ⋅⋅⋅
a x +
2!
3!
⎛ n⎞
⎛ n⎞
⎛ n⎞
= a n + ⎜ ⎟ a n−1 x + ⎜ ⎟ a n− 2 x 2 + ⎜ ⎟ a n− 3 x 3 + ⋅⋅⋅
⎝ 3⎠
⎝1⎠
⎝ 2⎠
Special cases are
22.5.
(a + x )2 = a 2 + 2 ax + x 2
22.6.
(a + x )3 = a 3 + 3a 2 x + 3ax 2 + x 3
22.7.
(a + x )4 = a 4 + 4 a 3 x + 6 a 2 x 2 + 4 ax 3 + x 4
22.8.
(1 + x )−1 = 1 − x + x 2 − x 3 + x 4 − ⋅⋅⋅
1 < x < 1
22.9.
(1 + x )−2 = 1 − 2 x + 3x 2 − 4 x 3 + 5 x 4 − ⋅⋅⋅
1 < x < 1
(1 + x )−3 = 1 − 3x + 6 x 2 − 10 x 3 + 15 x 4 − ⋅⋅⋅
1 < x < 1
22.10.
138
TAYLOR SERIES
139
22.11.
(1 + x )−1/ 2 = 1 −
1
1i 3 2 1i 3i 5 3
x+
x −
x + ⋅⋅⋅
2
2i4
2i4i6
1 < x 1
22.12.
(1 + x )1/ 2 = 1 +
1
1 2
1i 3 3
x−
x +
x − ⋅⋅⋅
2
2i4
2i4i6
1 < x 1
22.13.
(1 + x )−1/ 3 = 1 −
1
1i 4 2 1i 4 i 7 3
x+
x −
x + ⋅⋅⋅
3
3i6
3i6i9
1 < x 1
22.14.
(1 + x )1/ 3 = 1 +
1
2 2
2i5 3
x−
x +
x − ⋅⋅⋅
3
3i6
3i6i9
1 < x 1
Series for Exponential and Logarithmic Functions
x2 x3
+ + ⋅⋅⋅
2 ! 3!
∞ < x < ∞
22.15.
ex = 1 + x +
22.16.
a x = e x ln a = 1 + x ln a +
22.17.
ln (1 + x ) = x −
22.18.
1 ⎛1 + x⎞
x3 x5 x7
ln ⎜
=x+
+
+
+ ⋅⋅⋅
⎟
2 ⎝1 − x⎠
3
5
7
1 < x < 1
22.19.
3
5
1 ⎛ x − 1⎞
⎪⎧⎛ x − 1⎞ 1 ⎛ x − 1⎞
⎪⎫
ln x = 2 ⎨⎜
+ ⋅⋅⋅⎬
+ ⎜
+ ⎜
⎟
⎟
⎟
x
+
x
x
+
1
3
1
5
+
1
⎝
⎝
⎠
⎠
⎝
⎠
⎪⎩
⎪⎭
x>0
22.20.
⎛ x − 1⎞ 1 ⎛ x − 1⎞ 1 ⎛ x − 1⎞
+
+
+ ⋅ ⋅⋅
ln x = ⎜
⎝ x ⎟⎠ 2 ⎜⎝ x ⎟⎠ 3 ⎜⎝ x ⎟⎠
( x ln a )2 ( x ln a )3
+
+ ⋅⋅⋅
2!
3!
x2 x3 x4
+
−
+ ⋅⋅⋅
2
3
4
2
∞ < x < ∞
1 < x 1
3
x
1
2
Series for Trigonometric Functions
22.21.
sin x = x −
x3 x5 x7
+ − +
3! 5! 7!
− ∞< x < ∞
22.22.
cos x = 1 −
x2 x4 x6
+
− +
2! 4 ! 6 !
− ∞< x < ∞
22.23.
tan x = x +
2 2 n (2 2 n − 1)Bn x 2 n−1
x 3 2 x 5 17 x 7
+
+
+
++
3
15
315
(2 n )!
| x |<
22.24.
cot x =
22.25.
sec x = 1 +
22.26.
csc x =
22.27.
sin −1 x = x +
22.28.
cos −1 x =
2 2 n Bn x 2 n−1
1 x x 3 2x5
− −
−
−−
−
x 3 45 945
(2 n )!
E x 2n
x 2 5 x 4 61x 6
+
+
++ n
+
2
24
720
(2 n )!
2(2 2 n−1 − 1)Bn x 2 n−1
1 x 7x 3
31x 5
+ +
+
++
+
x 6 360 15, 120
(2 n )!
1 x3 1 i 3 x5 1 i 3 i 5 x7
+
+
+
2 3 2i4 5 2i4i6 7
1 x3 1 i 3 x5
π
π ⎛
⎞
− sin −1 x = − ⎜ x +
+
+ ⎟
2
2 ⎝
2 3 2i4 5
⎠
π
2
0< |x| <π
|x|<
π
2
0< |x| <π
|x|<1
|x|<1
TAYLOR SERIES
140
22.29.
22.30.
22.31.
22.32.
⎧
x3 x5 x7
x−
+
−
+
⎪
3
5
7
tan −1 x = ⎨
⎪± π − 1 + 1 − 1 + ⎩ 2 x 3x 3 5x 5
⎧π ⎛
⎞
x3 x5
− ⎜ x − + − ⎟
⎪
π
⎪
3
5
⎠
cot −1 x = − tan −1 x = ⎨ 2 ⎝
2
⎪ pπ + 1 − 1 + 1 − ⎪⎩
x 3x 3 5x 5
| x | <1
(+ if x 1, − if x − 1)
| x | <1
( p = 0 if x > 1, p = 1 if x < − 1)
⎞
π ⎛1
1
1⋅ 3
−⎜ +
+
+ ⎟
3
5
2 ⎝ x 2 i 3x
2 i 4 i 5x
⎠
1
1
1
⋅
3
csc −1 x = sin −1 (1/x ) = +
+
+
x 2 i 3x 3 2 i 4 i 5x 5
sec −1 x = cos −1 (1/x ) =
|x|>1
|x|>1
Series for Hyperbolic Functions
22.33.
sinh x = x +
x3 x5 x7
+ + +
3! 5! 7!
− ∞< x < ∞
22.34.
cosh x = 1 +
x2 x4 x6
+ + +
2! 4 ! 6!
− ∞< x < ∞
22.35.
tanh x = x −
(−1)n−1 2 2 n (2 2 n − 1)Bn x 2 n−1
x 3 2 x 5 17 x 7
+
−
+
+
3
15
315
(2 n )!
|x|<
22.36.
coth x =
22.37.
sech x = 1 −
22.38.
csch x =
22.39.
⎧
x3
1 i 3x 5
1 i 3 i 5x 7
⎪⎪x − 2 i 3 + 2 i 4 i 5 − 2 i 4 i 6 i 7 + sinh −1 x = ⎨
⎛
⎞
⎪± ln | 2 x | + 1 − 1 i 3 + 1 i 3 i 5 − 2
4
6
⎜
⎟⎠
2
2
x
2
4
4
x
2
4
6
i
i
i
i
i
i
6
x
⎩⎪ ⎝
22.40.
22.41.
22.42.
(−1)n−1 2 2 n Bn x 2 n−1
1 x x 3 2x5
+
+ −
+
+
x 3 45 945
(2 n)!
(−1)n En x 2 n
x 2 5 x 4 61x 6
+
−
+
+
2
24
720
(2 n )!
(−1)n 2(2 2 n−1 − 1)Bn x 2 n−1
1 x 7x 3
31x 5
− +
−
+
+
x 6 360 15, 120
(2 n )!
⎛ 1
⎞ ⎪⎫
1i 3
1i 3i 5
⎪⎧
+ ⎟ ⎬
cosh −1 x = ± ⎨ln(2 x ) − ⎜
+
+
6
2
4
2 i 4 i 4x
2 i 4 i 6 i 6x
⎝ 2 i 2x
⎠ ⎪⎭
⎪⎩
3
5
7
x
x
x
tanh −1 x = x +
+
+
+
3
5
7
coth −1 x =
1
1
1
1
+
+
+
+
x 3x 3 5 x 5 7 x 7
π
2
0< |x| <π
|x|<
π
2
0< |x| <π
| x | <1
⎡ + if x 1 ⎤
⎢− if x − 1⎥
⎣
⎦
⎡+ if cosh −1 x > 0, x 1⎤
⎢− if cosh −1 x < 0, x 1⎥
⎣
⎦
|x|<1
|x|>1
Miscellaneous Series
x2 x4 x5
−
−
+
2
8 15
22.43.
e sin x = 1 + x +
22.44.
⎛ x 2 x 4 31x 6
⎞
e cos x = e ⎜1 −
+
−
+ ⎟
2
6
720
⎝
⎠
−∞< x < ∞
−∞< x < ∞
TAYLOR SERIES
141
x 2 x 3 3x 4
+
+
+
2
2
8
|x|<
π
2
22.45.
e tan x = 1 + x +
22.46.
e x sin x = x + x 2 +
x3 x5 x6
2n / 2 sin (nπ /4) x n
−
−
+ +
+
3 30 90
n!
−∞< x < ∞
22.47.
e x cos x = 1 + x −
x3 x4
2 n /2 cos(nπ / 4 )x n
−
++
+
3
6
n!
−∞< x < ∞
22.48.
ln | sin x | = ln | x | −
22.49.
ln | cos x | = −
22.50.
ln | tan x | = ln | x | +
22.51.
ln(1 + x )
= x − (1 + 12 ) x 2 + (1 + 12 + 13 ) x 3 − 1+ x
2 2 n−1 Bn x 2 n
x2
x4
x6
+
−
−
−−
6 180 2835
n(2 n )!
2 2 n−1 (2 2 n − 1)Bn x 2 n
x 2 x 4 x 6 17 x 8
−
−
−
−−
+
2 12 45 2520
n(2 n )!
2 2 n (2 2 n−1 − 1)Bn x 2 n
x 2 7 x 4 62 x 6
+
+
++
+
3
90 2835
n(2 n )!
0< |x| <π
|x|<
π
2
0< |x|<
π
2
| x | <1
Reversion of Power Series
Suppose
22.52.
y = C1 x + C2 x 2 + C3 x 3 + C4 x 4 + C5 x 5 + C6 x 6 + then
22.53.
x = C1 y + C2 y 2 + C3 y 3 + C4 y 4 + C5 y 5 + C6 y 6 + where
22.54.
c1C1 = 1
22.55.
c13C2 = −c2
22.56.
c15C3 = 2c22 − c1c3
22.57.
c17C4 = 5c1c2 c3 − 5c23 − c12 c4
22.58.
c19C5 = 6c12 c2 c4 + 3c12 c32 − c13c5 + 14 c24 − 21c1c22 c3
22.59.
c111C6 = 7c13c2 c5 + 84 c1c23c3 + 7c13c3c4 − 28c12 c2 c32 − c14 c6 − 28c12 c22 c4 − 42c25
Taylor Series for Functions of Two Variables
22.60.
f ( x, y) = f (a, b ) + ( x − a ) f x (a, b ) + ( y − b ) f y (a, b )
+
1
{( x − a )2 f xx (a, b ) + 2( x − a )( y − b ) f xy (a, b ) + ( y − b )2 f yy (a, b )} + 2!
where f x (a, b ), f y (a, b ),… denote partial derivatives with respect to x, y, … evaluated at x a, y b.
23
BERNOULLI and EULER NUMBERS
Definition of Bernoulli Numbers
The Bernoulli numbers B1 , B2 , B3 ,… are defined by the series
23.1.
x
x B x2 B x4 B x6
= 1− + 1 − 2 + 3 −
2
2!
4!
6!
e −1
23.2.
1−
x
x
x B x2 B x4 B x6
cot = 1 + 3 + 3 + 2
2
2!
4!
6!
| x | < 2π
|x| <π
Definition of Euler Numbers
The Euler numbers E1, E2, E3, … are defined by the series
23.3.
sech x = 1 −
23.4.
sec x = 1 +
E1 x 2 E2 x 4 E3 x 6
+
−
+
2!
4!
6!
E1 x 2 E2 x 4 E3 x 6
+
−
+
2!
4!
6!
|x|<
π
2
|x|<
π
2
Table of First Few Bernoulli and Euler Numbers
Bernoulli Numbers
142
Euler Numbers
B1 = 1/6
E1 = 1
B2 = 1/30
E2 = 5
B3 = 1/42
E3 = 61
B4 = 1/30
E4 = 1385
B5 = 5/66
E5 = 50, 521
B6 = 691/2730
E6 = 2, 702, 765
B7 = 7/6
E7 = 199, 360, 981
B8 = 3617/510
E8 = 19, 391, 512, 145
B9 = 43, 867/798
E9 = 2, 404, 879, 675, 441
B10 = 174, 611/330
E10 = 370, 371, 188, 237, 525
B11 = 854, 513138
/
E11 = 69, 348, 874, 393, 137, 901
B12 = 236, 364, 091/2730
E12 = 15, 514, 534, 163, 557, 086, 905
BERNOULLI AND EULER NUMBERS
Relationships of Bernoulli and Euler Numbers
23.5.
⎛ 2 n + 1⎞ 2
⎛ 2 n + 1⎞ 4
⎛ 2 n + 1⎞ 6
n −1
2n
⎜⎝ 2 ⎟⎠ 2 B1 − ⎜⎝ 4 ⎟⎠ 2 B2 + ⎜⎝ 6 ⎟⎠ 2 B3 − (−1) (2 n + 1)2 Bn = 2n
23.6.
⎛ 2 n⎞
⎛ 2 n⎞
⎛ 2 n⎞
En = ⎜ ⎟ En−1 − ⎜ ⎟ En− 2 + ⎜ ⎟ En− 3 − (−1)n
⎝ 2⎠
⎝ 4⎠
⎝ 6⎠
23.7.
Bn =
⎫
⎧⎛ 2 n − 1⎞
2n
⎛ 2 n − 1⎞
⎛ 2 n − 1⎞
E − (−1)n−1 ⎬
E −
E +
2 2 n (2 2 n − 1) ⎨⎩⎜⎝ 1 ⎟⎠ n−1 ⎜⎝ 3 ⎟⎠ n− 2 ⎜⎝ 5 ⎟⎠ n− 3
⎭
Series Involving Bernoulli and Euler Numbers
23.8.
23.9.
23.10.
23.11.
1
1
⎫
⎧
⎨1 + 2 2 n + 32 n + ⎬
⎭
⎩
2(2 n )! ⎧
1
1
⎫
1+
Bn = 2 n
+
+ ⎬
(2 − 1)π 2 n ⎨⎩ 32 n 5 2 n
⎭
Bn =
(2 n )!
2 2 n−1 π 2 n
2(2 n )!
1
1
⎫
⎧
+
− ⎬
1−
(2 2 n−1 − 1)π 2 n ⎨⎩ 2 2 n 32 n
⎭
2 n+ 2
2 (2 n )! ⎧
1
1
⎫
1−
En =
+
− ⎬
π 2 n+1 ⎨⎩ 32 n+1 5 2 n+1
⎭
Bn =
Asymptotic Formula for Bernoulli Numbers
23.12.
Bn ~ 4 n 2 n (π e)−2 n π n
143
24
FOURIER SERIES
Definition of a Fourier Series
The Fourier series corresponding to a function f (x) defined in the interval c x c + 2 L where c and
L > 0 are constants, is defined as
24.1.
∞
a0
nπx
n π x⎞
⎛
+ ∑ ⎜ an cos
+ bn sin
2 n=1 ⎝
L
L ⎟⎠
where
24.2.
⎧
⎪an =
⎨
⎪bn =
⎩
1
L
1
L
∫
∫
c+ 2 L
c
c+ 2 L
c
nπx
dx
L
nπx
f ( x )sin
dx
L
f ( x ) cos
If f (x) and f ′(x) are piecewise continuous and f (x) is defined by periodic extension of period 2L, i.e.,
f (x 2L) f (x), then the series converges to f (x) if x is a point of continuity and to 12 { f ( x + 0) + f ( x − 0)} if
x is a point of discontinuity.
Complex Form of Fourier Series
Assuming that the series 24.1 converges to f (x), we have
24.3.
f (x ) =
∞
∑c e
in π x/L
n
n =−∞
where
24.4.
1
cn =
2L
∫
c+ 2 L
c
f ( x )e
− in π x/L
⎧12 (an − ibn )
⎪
dx = ⎨12 (a− n + ib− n )
⎪⎩12 a0
n>0
n<0
n=0
Parseval’s Identity
24.5.
1
L
∫
c+ 2 L
c
{ f ( x )}2 dx =
∞
a02
+ ∑ (an2 + bn2 )
2 n=1
Generalized Parseval Identity
24.6.
1
L
∫
c+ 2 L
c
f ( x )g( x ) dx =
∞
a0 c0
+ ∑ (an cn + bn d n )
2
n =1
where an, bn and cn, dn are the Fourier coefficients corresponding to f (x) and g(x), respectively.
144
FOURIER SERIES
145
Special Fourier Series and Their Graphs
24.7.
⎧ 1 0< x<π
f (x ) = ⎨
⎩−1 −π < x < 0
4 ⎛ sin x sin 3x sin 5 x
⎞
+
+
+ ⎟
3
5
π ⎜⎝ 1
⎠
Fig. 24-1
24.8.
0< x<π
⎧ x
f (x ) = | x | = ⎨
π
−
x
−
<x<0
⎩
π 4 ⎛ cos x cos 3x cos 5 x
⎞
−
+
+
+ ⎟
2 π ⎜⎝ 12
32
52
⎠
Fig. 24-2
24.9.
f ( x ) = x, − π < x < π
⎛ sin x sin 2 x sin 3x
⎞
2⎜
−
+
− ⎟
2
3
⎝ 1
⎠
Fig. 24-3
24.10.
f ( x ) = x, 0 < x < 2π
⎛ sin x sin 2 x sin 3x
⎞
π − 2⎜
+
+
+ ⎟
2
3
⎝ 1
⎠
Fig. 24-4
24.11.
f ( x ) = | sin x |, − π < x < π
2 4 ⎛ cos 2 x cos 4 x cos 6 x
⎞
−
+
+
+ ⎟
3i5
5i7
π π ⎜⎝ 1 i 3
⎠
Fig. 24-5
FOURIER SERIES
146
24.12.
⎧sin x 0 < x < π
f (x ) = ⎨
π < x < 2π
⎩ 0
1 1
2 ⎛ cos 2 x cos 4 x cos 6 x
⎞
+ sin x − ⎜
+
+
+ ⎟
π 2
π ⎝ 1i 3
3i5
5i7
⎠
Fig. 24-6
24.13.
0< x<π
⎧ cos x
f (x ) = ⎨
π
−
cos
x
−
<x<0
⎩
8 ⎛ sin 2 x 2 sin 4 x 3 sin 6 x
⎞
+
+
+ ⎟
π ⎜⎝ 1 i 3
3i5
5i7
⎠
Fig. 24-7
24.14.
f (x ) = x 2 , − π < x < π
π2
⎛ cos x cos 2 x cos 3x
⎞
− 4⎜ 2 −
+
− ⎟
3
22
32
⎝ 1
⎠
Fig. 24-8
24.15.
f ( x ) = x (π − x ), 0 < x < π
π 2 ⎛ cos 2 x cos 4 x cos 6 x
⎞
−
+
+
+ ⎟
6 ⎜⎝ 12
22
32
⎠
Fig. 24-9
24.16.
f ( x ) = x (π − x )(π + x ), − π < x < π
⎛ sin x sin 2 x sin 3x
⎞
+
− ⎟
12 ⎜ 3 −
23
33
⎝ 1
⎠
Fig. 24-10
FOURIER SERIES
24.17.
147
0 < x < π −α
⎧0
⎪
f ( x ) = ⎨1 π − α < x < π + α
⎪⎩0 π + α < x < 2π
α 2 ⎛ sin α cos x sin 2α cos 2 x
−
−
π π ⎜⎝
1
2
+
24.18.
sin 3α cos 3x
⎞
− ⎟
3
⎠
Fig. 24-11
0< x<π
⎧ x (π − x )
f (x ) = ⎨
π
π
−
x
(
−
x
)
−
<x<0
⎩
8 ⎛ sin x sin 3x sin 5 x
⎞
+
+
+ ⎟
π ⎜⎝ 13
33
53
⎠
Fig. 24-12
Miscellaneous Fourier Series
24.19.
f ( x ) = sin μ x, − π < x < π , μ ≠ integer
2 sin μπ ⎛ sin x
2 sin 2 x 3 sin 3x
⎞
+
− ⎟
⎜
2
2 − 2
π
2 − μ 2 32 − μ 2
⎝1 − μ
⎠
24.20.
f ( x ) = cos μ x, − π < x < π , μ ≠ integer
cos 2 x
cos 3x
2 μ sin μπ ⎛ 1
cos x
⎞
⎜⎝ 2 μ 2 + 12 − μ 2 − 2 2 − μ 2 + 32 − μ 2 − ⎟⎠
π
24.21.
f ( x ) = tan −1[(a sin x ) / (1 − a cos x )], − π < x < π , | a | < 1
a sin x +
24.22.
a2
a3
sin 2 x + sin 3x + 2
3
f ( x ) = ln (1 − 2a cos x + a 2 ), − π < x < π , | a | < 1
⎛
⎞
a3
a2
− 2 ⎜ a cos x + cos 2 x + cos 3x + ⎟
2
3
⎝
⎠
24.23.
f (x) =
1 −1
tan [(2a sin x ) / (1 − a 2 )], − π < x < π , | a | < 1
2
a sin x +
24.24.
f (x) =
a3
a5
sin 3x + sin 5x + 3
5
1 −1
tan [(2a cos x ) / (1 − a 2 )], − π < x < π , | a | < 1
2
a cos x −
a3
a5
cos 3x + cos 5x − 3
5
FOURIER SERIES
148
24.25.
f (x ) = e μx , − π < x < π
∞
(−1)n (μ cos nx − n sin nx )⎞
2 sinh μπ ⎛ 1
+∑
⎜
⎟⎠
π
μ 2 + n2
⎝ 2 μ n=1
24.26.
f ( x ) = sinh μ x, − π < x < π
2 sinh μπ ⎛ sin x
2 sin 2 x 3 sin 3x
⎞
+
− ⎟
⎜
2
2 − 2
π
2 + μ 2 32 + μ 2
⎝1 + μ
⎠
24.27.
f ( x ) = cosh μ x, − π < x < π
2 μ sinh μπ ⎛ 1
cos 3x
cos x
cos 2 x
⎞
⎜⎝ 2 μ 2 − 12 + μ 2 + 2 2 + μ 2 − 32 + μ 2 + ⎟⎠
π
24.28.
f ( x ) = ln | sin 12 x |, 0 < x < π
cos x cos 2 x cos 3x
⎛
⎞
+ ⎟
− ⎜ ln 2 +
+
+
3
1
2
⎝
⎠
24.29.
f ( x ) = ln | cos 12 x |, − π < x < π
cos x cos 2 x cos 3x
⎛
⎞
+ ⎟
− ⎜ ln 2 −
+
−
3
1
2
⎝
⎠
24.30.
f ( x ) = 16 π 2 − 12 π x + 14 x 2 , 0 x 2π
cos x cos 2 x cos 3x
+
+
+
32
12
22
24.31.
f (x ) =
1
12
x ( x − π )( x − 2π ), 0 x 2π
sin x sin 2 x sin 3x
+
+
+
33
13
23
24.32.
f (x ) =
1
90
π 4 − 121 π 2 x 2 + 121 π x 3 −
1
48
x 4 , 0 x 2π
cos x cos 2 x cos 3x
+
+
+
24
34
14
Section VII: Special Functions and Polynomials
25
THE GAMMA FUNCTION
Definition of the Gamma Function ⌫(n) for n > 0
25.1.
∞
Γ(n) = ∫ t n−1e − t dt
0
n>0
Recursion Formula
25.2.
Γ (n + 1) = nΓ (n)
If n = 0, 1, 2, …, a nonnegative integer, we have the following (where 0! = 1):
25.3.
Γ(n + 1) = n!
The Gamma Function for n < 0
For n < 0 the gamma function can be defined by using 25.2, that is,
25.4.
Γ (n) =
Γ (n + 1)
n
Graph of the Gamma Function
5
Γ(n)
4
3
2
1
−5
−4
−3
−2
−1
1
−1
2
3
4
5
n
−2
−3
−4
−5
Fig. 25-1
149
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
THE GAMMA FUNCTION
150
Special Values for the Gamma Function
25.5.
Γ( 12 ) = π
25.6.
Γ(m + 12 ) =
25.7.
Γ(− m + 12 ) =
1i 3 i 5 ⋅⋅⋅ (2m − 1)
2m
π
(−1)m 2m π
1i 3 i 5 ⋅⋅⋅ (2m − 1)
m = 1, 2, 3, ...
m = 1, 2, 3, ...
Relationships Among Gamma Functions
π
sin pπ
25.8.
Γ ( p)Γ (1− p) =
25.9.
22 x −1 Γ ( x )Γ ( x + 12 ) = π Γ (2 x )
This is called the duplication formula.
25.10.
⎛
⎛
m − 1⎞
1⎞ ⎛
2⎞
Γ ( x )Γ ⎜ x + ⎟ Γ ⎜ x + ⎟ ⋅⋅⋅ Γ ⎜ x +
= m1/ 2 − mx (2π )( m −1)/ 2 Γ (mx )
m
m
m ⎟⎠
⎝
⎠ ⎝
⎠
⎝
For m = 2 this reduces to 25.9.
Other Definitions of the Gamma Function
1 i 2 i 3 ⋅⋅⋅ k
kx
( x + 1)( x + 2) ⋅⋅⋅ ( x + k )
25.11.
Γ( x + 1) = lim
25.12.
∞ ⎧
⎫⎪
1
x⎞
⎪⎛
= xeγ x ∏ ⎨⎜1 + ⎟ e − x / m ⎬
Γ( x )
⎪⎭
m =1 ⎪
⎩⎝ m ⎠
k →∞
This is an infinite product representation for the gamma function where g is Euler’s constant defined
in 1.3, page 3.
Derivatives of the Gamma Function
∞
25.13.
Γ ′(1) = ∫ e − x ln x dx = −γ
25.14.
⎛1
Γ ′( x )
= −γ + ⎜ −
Γ(x)
⎝1
0
⎛1
1⎞ ⎛ 1
1 ⎞
1 ⎞
+⎜ −
+ ⋅⋅⋅
+ ⋅⋅⋅ + ⎜ −
⎟
⎟
x ⎠ ⎝ 2 x + 1⎠
⎝ n x + n − 1⎟⎠
Here again is Euler’s constant g.
THE GAMMA FUNCTION
151
Asymptotic Expansions for the Gamma Function
25.15.
⎧
⎫
1
1
139
Γ( x + 1) = 2π x x x e − x ⎨1 +
+
−
+ ⋅⋅ ⋅⎬
2
3
x
x
,
x
12
288
51
840
⎩
⎭
This is called Stirling’s asymptotic series.
If we let x = n a positive integer in 25.15, then a useful approximation for n! where n is large (e.g.,
n > 10) is given by Stirling’s formula
25.16.
n! ~ 2π n n ne − n
where ~ is used to indicate that the ratio of the terms on each side approaches 1 as n → ∞.
Miscellaneous Results
25.17.
| Γ (ix ) |2 =
π
x sinh π x
26
THE BETA FUNCTION
Definition of the Beta Function B(m, n)
26.1.
1
B(m, n) = ∫ t m −1 (1 − t )n−1 dt
0
m > 0, n > 0
Relationship of Beta Function to Gamma Function
26.2.
B( m , n ) =
Γ (m )Γ (n)
Γ (m + n)
Extensions of B(m, n) to m < 0, n < 0 are provided by using 25.4.
Some Important Results
26.3.
B( m , n ) = B( n , m )
26.4.
B( m , n ) = 2 ∫
26.5.
B( m , n ) = ∫
26.6.
B(m, n) = r n (r + 1)m
152
π /2
0
∞
0
sin 2 m −1 θ cos 2 n−1 θ dθ
t m −1
dt
(1 + t )m + n
∫
(1 − t )n−1
dt
(r + t )m + n
1 m −1
0
t
27
BESSEL FUNCTIONS
Bessel’s Differential Equation
27.1.
x 2 y n + xy′ + ( x 2 − n 2 ) y = 0
n0
Solutions of this equation are called Bessel functions of order n.
Bessel Functions of the First Kind of Order n
27.2.
⎧
⎫
xn
x2
x4
+
− ⋅⋅⋅⎬
⎨1 −
2 Γ (n + 1) ⎩ 2(2n + 2) 2 i 4(2n + 2)(2n + 4)
⎭
Jn ( x ) =
n
∞
(−1) k ( x /2)n+ 2 k
k ! Γ (n + k + 1)
=∑
k =0
27.3.
J− n ( x ) =
x −n
x2
x4
⎫
⎧
1−
+
− ⋅⋅⋅⎬
⎨
2 Γ (1 − n) ⎩ 2(2 − 2n) 2 i 4(2 − 2n)(4 − 2n)
⎭
−n
∞
=∑
k =0
27.4.
(−1) k ( x / 2)2 k − n
k ! Γ ( k + 1 − n)
J − n ( x ) = (−1)n J n ( x )
n = 0, 1, 2, ...
If n ≠ 0, 1, 2, ..., J n ( x ) and J–n (x) are linearly independent.
If n ≠ 0, 1, 2, ..., J n ( x ) is bounded at x = 0 while J–n (x) is unbounded.
For n = 0, 1 we have
x2
x4
x6
+
−
+ ⋅⋅⋅
22 22 i 4 2 22 i 4 2 i62
27.5.
J0 ( x ) = 1 −
27.6.
J1 ( x ) =
27.7.
J 0(x) = − J1 (x)
x
x3
x5
x7
− 2 + 2 2 2 − 2 2 2 + ⋅⋅⋅
2 2 i4 2 i4 i6
2 i 4 i6 i8
Bessel Functions of the Second Kind of Order n
27.8.
⎧ J n ( x ) cos nπ − J − n ( x )
⎪
sin n π
⎪
Yn ( x ) = ⎨
J p ( x ) cos pπ − J − p ( x )
⎪
⎪lim
p→ n
sin pπ
⎩
n ≠ 0, 1, 2, ...
n = 0, 1, 2, ...
This is also called Weber’s function or Neumann’s function [also denoted by Nn(x)].
153
BESSEL FUNCTIONS
154
For n = 0, 1, 2, …, L’ Hospital’s rule yields
27.9.
Yn ( x ) =
2
1 n −1 (n − k − 1)!
( x / 2)2 k − n
{ln ( x / 2) + γ }J n ( x ) − ∑
π
π k =0
k!
−
( x / 2)2 k + n
1 ∞
(−1) k {Φ(k ) + Φ(n + k )}
∑
π k =0
k !(n + k )!
where g = .5772156 … is Euler’s constant (see 1.20) and
27.10.
Φ( p) = 1 +
1 1
1
+ + ⋅⋅⋅ + ,
P
2 3
Φ (0) = 0
For n = 0,
2
2
{ln ( x / 2) + γ }J 0 ( x ) +
π
π
27.11.
Y0 ( x ) =
27.12.
Y− n ( x ) = (−1)n Yn ( x )
( )
(
n = 0, 1, 2, ...
For any value n 0, J n ( x ) is bounded at x = 0 while Yn(x) is unbounded.
General Solution of Bessel’s Differential Equation
27.13.
y = AJ n ( x ) + BJ − n ( x )
n ≠ 0, 1, 2, .. .
27.14.
y = AJ n ( x ) + BYn ( x )
all n
27.15.
y = AJ n ( x ) + BJ n ( x ) ∫
dx
xJ n2 ( x )
all n
where A and B are arbitrary constants.
Generating Function for Jn(x)
27.16.
e x ( t −1/t ) / 2 =
∞
∑ J ( x )t
n
n
n =−∞
Recurrence Formulas for Bessel Functions
2n
J ( x ) − J n−1 ( x )
x n
27.17.
J n+1 ( x ) =
27.18.
J n′ ( x ) = 12 {J n −1 ( x ) − J n+1 ( x )}
27.19.
xJ n′ ( x ) = xJ n−1 ( x ) − nJ n ( x )
27.20.
xJ n′ ( x ) = nJ n ( x ) − xJ n+1 ( x )
)
1
1 1
x4
x6
⎫
⎧x 2
⎨ 22 − 22 4 2 1 + 2 + 22 4 2 62 1 + 2 + 3 − ⋅⋅⋅⎬
⎭
⎩
BESSEL FUNCTIONS
27.21.
d n
{x J n ( x )} = x n J n−1 ( x )
dx
27.22.
d −n
{x J n ( x )} = − x − n J n+1 ( x )
dx
155
The functions Yn(x) satisfy identical relations.
Bessel Functions of Order Equal to Half an Odd Integer
In this case the functions are expressible in terms of sines and cosines.
2
sin x
πx
27.23.
J1/ 2 ( x ) =
27.24.
J −1/ 2 ( x ) =
27.25.
J3 / 2 ( x ) =
2
cos x
πx
2
πx
⎛ sin x
⎞
⎜⎝ x − cos x⎟⎠
⎞
2 ⎛ cos x
+ sin x⎟
π x ⎜⎝ x
⎠
27.26.
J −3 / 2 ( x ) =
27.27.
J5 / 2 ( x ) =
⎫⎪
⎞
2 ⎧⎪ ⎛ 3
3
⎨ ⎜ 2 − 1⎟ sin x − cos x⎬
π x ⎩⎪ ⎝ x
x
⎠
⎪⎭
27.28.
J −5 / 2 ( x ) =
⎛3 ⎞
2 ⎧⎪ 3
⎪⎫
⎨ sin x + ⎜ − 1⎟ cos x⎬
π x ⎩⎪ x
⎝x ⎠
⎭⎪
For further results use the recurrence formula. Results for Y1/ 2 ( x ), Y3 / 2 ( x ), ... are obtained from 27.8.
Hankel Functions of First and Second Kinds of Order n
27.29.
H n(1) ( x ) = J n ( x ) + iYn ( x )
27.30.
H n( 2) ( x ) = J n ( x ) − iYn ( x )
Bessel’s Modified Differential Equation
27.31.
x 2 y′′ + xy′ − ( x 2 + n 2 ) y = 0
n0
Solutions of this equation are called modified Bessel functions of order n.
Modified Bessel Functions of the First Kind of Order n
27.32.
I n ( x ) = i − n J n (ix ) = e − nπ i / 2 J n (ix )
=
27.33.
x4
( x / 2)n+ 2 k
xn
x2
⎫ ∞
⎧
1
+
+
⋅⋅⋅
+
=∑
⎬
⎨
n
2 Γ (n + 1) ⎩ 2(2n + 2) 2 i 4(2n + 2)(2n + 4)
⎭ k = 0 k ! Γ (n + k + 1)
I − n ( x ) = i n J − n (ix ) = e nπ i / 2 J − n (ix )
=
27.34.
⎧
⎫ ∞
x −n
x2
x4
( x / 2)2 k − n
1
+
+
+
⋅⋅⋅
=
⎨
⎬
∑
2 − n Γ (1 − n) ⎩ 2(2 − 2n) 2 i 4(2 − 2n)(4 − 2n)
⎭ k = 0 k ! Γ ( k + 1 − n)
I − n ( x ) = I n ( x ) n = 0, 1, 2, ...
156
BESSEL FUNCTIONS
If n ≠ 0, 1, 2, ..., then In(x) and I–n(x) are linearly independent.
For n = 0, 1, we have
x2
x4
x6
+
+
+ ⋅⋅⋅
22 22 i 4 2 22 i 4 2 i 62
27.35.
I 0 (x) = 1 +
27.36.
I1 ( x ) =
27.37.
I 0′ ( x ) = I1 ( x )
x
x3
x5
x7
+ 2 + 2 2 + 2 2 2 + ⋅⋅⋅
2 2 i 4 2 i 4 i6 2 i 4 i 6 i8
Modified Bessel Functions of the Second Kind of Order n
27.38.
⎧ π
⎪ 2 sin nπ {I − n ( x ) − I n ( x )}
⎪
K n (x) = ⎨
⎪lim π
{I ( x ) − I p ( x )}
⎪⎩p→n 2 sin pπ − p
n ≠ 0, 1, 2, ...
n = 0, 1, 2, ...
For n = 0, 1, 2, …, L’ Hospital’s rule yields
27.39.
K n ( x ) = (−1)n+1{ln ( x / 2) + γ )I n ( x ) +
+
(−1)n
2
∞
1 n−1
(−1) k (n − k − 1)!( x / 2)2 k − n
2∑
k =0
( x / 2)n+ 2 k
∑ k !(n + k )! {Φ(k ) + Φ(n + k )}
k =0
where Φ(p) is given by 27.10.
For n = 0,
27.40.
K 0 ( x ) = −{ln ( x / 2) + γ }I 0 ( x ) +
27.41.
K −n (x) = K n (x)
⎛ 1 1⎞
x2
x 4 ⎛ 1⎞
x6
+ 2 2 ⎜1 + ⎟ + 2 2 2 ⎜1 + + ⎟ + ⋅⋅⋅
2
2
2 i 4 ⎝ 2⎠ 2 i 4 i 6 ⎝ 2 3⎠
n = 0, 1, 2, ...
General Solution of Bessel’s Modified Equation
27.42.
y = AI n ( x ) + BI − n ( x )
n ≠ 0, 1, 2, ...
27.43.
y = AI n ( x ) + BK n ( x )
all n
27.44.
y = AI n ( x ) + BI n ( x )
dx
2
n (x)
∫ xI
where A and B are arbitrary constants.
Generating Function for In(x)
27.45.
e x ( t +1/ t ) / 2 =
∞
∑ I ( x )t
n
n =−∞
n
all n
BESSEL FUNCTIONS
157
Recurrence Formulas for Modified Bessel Functions
27.46.
I n+1 ( x ) = I n−1 ( x ) −
2n
I (x)
x n
27.52.
K n+1 ( x ) = K n−1 ( x ) +
2n
K (x)
x n
27.47.
I n′ ( x ) = 12 {I n−1 ( x ) + I n+1 ( x )}
27.53.
K n′ ( x ) = − 12 {K n−1 ( x ) + K n+1 ( x )}
27.48.
xI n′ ( x ) = xI n −1 ( x ) − nI n ( x )
27.54.
xK n′ ( x ) = − xK n−1 ( x ) − nK n ( x )
27.49.
xI n′ ( x ) = xI n+1 ( x ) + nI n ( x )
27.55.
xK n′ ( x ) = nK n ( x ) − xK n+1 ( x )
27.50.
d n
{x I n ( x )} = x n I n−1 ( x )
dx
27.56.
d n
{x K n ( x )} = − x n K n−1 ( x )
dx
27.51.
d −n
{x I n ( x )} = x − n I n+1 ( x )
dx
27.57.
d −n
{x K n ( x )} = − x − n K n+1 ( x )
dx
Modified Bessel Functions of Order Equal to Half an Odd Integer
In this case the functions are expressible in terms of hyperbolic sines and cosines.
27.58.
I1 / 2 ( x ) =
27.59.
I −1/ 2 ( x ) =
27.60.
I3/ 2 (x) =
2
sinh x
πx
2
cosh x
πx
2 ⎛
sinh x ⎞
cosh x −
x ⎟⎠
π x ⎜⎝
27.61.
I −3 / 2 ( x ) =
27.62.
I5/ 2 (x) =
27.63.
I −5 / 2 ( x ) =
2
πx
2
πx
2
πx
cosh x ⎞
⎛
⎜⎝sinh x − x ⎟⎠
3
⎫
⎧⎛ 3 ⎞
⎨⎜⎝ x 2 + 1⎟⎠ sinh x − x cosh x⎬
⎭
⎩
3
⎫
⎧⎛ 3 ⎞
⎨⎜⎝ x 2 + 1⎟⎠ cosh x − x sinh x⎬
⎭
⎩
For further results use the recurrence formula 27.46. Results for K1/2(x), K3/2(x), … are obtained from
27.38.
Ber and Bei Functions
The real and imaginary parts of J n ( xe3π i / 4 ) are denoted by Bern (x) and Bein(x) where
( x / 2)2 k + n
(3n + 2k )π
cos
1
k
!
(
n
+
k
+
)
Γ
4
k =0
∞
27.64.
Bern ( x ) = ∑
27.65.
Bei n ( x ) = ∑
( x / 2)2 k + n
(3n + 2k )π
sin
1
k
!
(
n
+
k
+
)
Γ
4
k =0
∞
If n = 0.
( x / 2)4 ( x / 2)8
+
− ⋅⋅⋅
2!2
4!2
27.66.
Ber ( x ) = 1 −
27.67.
Bei( x ) = ( x / 2)2 −
( x / 2)6 ( x / 2)10
+
− ⋅⋅⋅
3!2
5!2
BESSEL FUNCTIONS
158
Ker and Kei Functions
The real and imaginary parts of e − nπ i / 2 K n ( xeπ i / 4 ) are denoted by Kern(x) and Kein(x) where
27.68.
27.69.
Kern ( x ) = −{ln ( x / 2) + γ }Bern ( x ) + 14 π Bei n ( x )
+
1 n −1 (n − k − 1)!( x / 2)2 k − n
(3n + 2k )π
cos
2∑
k
!
4
k =0
+
1
2
∞
( x / 2)n+ 2 k
∑ k !(n + k )! {Φ(k ) + Φ(n + k )}cos
k =0
(3n + 2k)π
4
Kei n ( x ) = −{ln ( x / 2) + γ }Bei n ( x ) − 14 π Bern ( x )
−
1 n −1 (n − k − 1)!( x / 2)2 k − n
(3n + 2k )π
sin
∑
2 k =0
k!
4
+
1
2
∞
∑
k =0
( x / 2)n+ 2 k
(3n + 2k)π
{Φ(k ) + Φ(n + k )}sin
k !(n + k )!
4
and Φ is given by 27.10.
If n = 0,
27.70.
Ker ( x ) = −{ln ( x / 2) + γ }Ber ( x ) +
π
( x / 2)4
( x / 2)8
1
Bei( x ) + 1 −
(
1
+
)
+
(1 + 12 + 13 + 14 ) − ⋅⋅⋅
2
4
2!2
4!2
27.71.
Kei( x ) = −{ln ( x / 2) + γ }Bei( x ) −
π
( x / 2)6
(1 + 12 + 13 ) + ⋅⋅⋅
Ber(x ) + ( x / 2)2 −
3!2
4
Differential Equation For Ber, Bei, Ker, Kei Functions
27.72.
x 2 y′′ + xy′ − (ix 2 + n 2 ) y = 0
The general solution of this equation is
27.73.
y = A{Bern ( x ) + i Bei n ( x )} + B{Kern ( x ) + i Kei n ( x )}
Graphs of Bessel Functions
Fig. 27-1
Fig. 27-2
BESSEL FUNCTIONS
159
Fig. 27-3
Fig. 27-4
Fig. 27-5
Fig. 27-6
Indefinite Integrals Involving Bessel Functions
27.74.
∫ xJ ( x )dx = xJ ( x )
27.75.
∫ x J ( x )dx = x J ( x ) + xJ ( x ) − ∫ J ( x )dx
27.76.
∫x
27.77.
∫
J0 ( x )
J (x)
dx = J1 ( x ) − 0
− ∫ J 0 ( x )dx
2
x
x
27.78.
∫
J0 ( x )
J0 ( x )
1
J1 ( x )
−
−
dx =
(m − 1)2 x m − 2 (m − 1) x m −1 (m − 1)2
xm
27.79.
∫ J ( x )dx = − J ( x)
27.80.
∫ xJ ( x)dx = − xJ ( x) + ∫ J ( x)dx
27.81.
∫x
0
1
2
2
0
m
1
0
0
J 0 ( x )dx = x m J1 ( x ) + (m − 1) x m −1J 0 ( x ) − (m − 1)2
1
0
1
m
0
0
J1 ( x )dx = − x m J 0 ( x ) + m
∫x
m −1
J 0 ( x )dx
∫x
∫
m−2
J 0 ( x )dx
J0 ( x )
dx
x m−2
BESSEL FUNCTIONS
160
27.82.
∫
J1 ( x )
dx = − J1 ( x ) + ∫ J 0 ( x )dx
x
27.83.
∫
J1 ( x )
J1 ( x ) 1
+
m dx = −
x
mx m −1 m
27.84.
∫x
27.85.
∫x
27.86.
∫x
27.87.
∫ xJ (α x ) J (β x )dx =
27.88.
∫ xJ
n
∫
J0 ( x )
dx
x m −1
J n−1 ( x )dx = x n J n ( x )
−n
m
J n+1 ( x )dx = − x − n J n ( x )
J n ( x )dx = − x m J n−1 ( x ) + (m + n − 1)
n
2
n
n
(α x )dx =
∫x
m −1
J n−1 ( x )dx
x{α J n (β x ) J n′ (α x ) − β J n (α x ) J n′ (β x )}
β2 −α2
x2
x2 ⎛
n2 ⎞
{J n′ (α x )}2 + ⎜1 − 2 2 ⎟ {J n (α x )}2
2
2 ⎝ α x ⎠
The above results also hold if we replace Jn(x) by Yn(x) or, more generally, AJn(x) + BYn(x) where A and B
are constants.
Definite Integrals Involving Bessel Functions
27.89.
∫
27.90.
∫
∞
0
∞
0
∞
∫
0
27.92.
∫
0
27.93.
∫
0
27.94.
∫
27.95.
∫
0
27.96.
∫
0
27.97.
∫
27.91.
∞
∞
∞
0
1
1
1
0
1
e − ax J 0 (bx )dx =
e − ax J n (bx )dx =
a + b2
2
( a 2 + b 2 − a) n
bn a2 + b2
⎧
1
⎪ 2
cos ax J 0 (bx )dx = ⎨ a − b 2
⎪
0
⎩
J n (bx )dx =
1
,
b
n > −1
a>b
a<b
n > −1
J n (bx )
1
dx = ,
x
n
n = 1, 2, 3, ...
e− b / 4a
a
2
e − ax J 0 (b x ) dx =
xJ n (α x ) J n (β x )dx =
α J n (β ) J n′ (α ) − β J n (α ) J n′ (β )
β2 − α2
xJ n2 (α x )dx = 12 {J n′ (α )}2 + 12 (1 − n 2 /α 2 ){J n (α )}2
xJ 0 (α x )I 0 (β x )dx =
β J 0 (α )I 0′ (β ) − α J 0′ (α )I 0 (β )
α2 + β2
BESSEL FUNCTIONS
161
Integral Representations for Bessel Functions
π
27.98.
J0 ( x ) =
1
π
∫
27.99.
Jn ( x ) =
1
π
∫
27.100.
Jn ( x ) =
27.101.
Y0 ( x ) = −
27.102.
I 0 (x) =
2
cos( x sin θ )dθ
0
π
0
xn
π Γ (n + 12 )
n
2
π
1
π
cos(nθ − x sin θ )dθ
∫
π
π
0
cos( x sin θ ) cos 2 n θ dθ ,
cos( x cosh u)du
cosh ( x sin θ )dθ =
0
n > − 12
∞
0
∫
∫
n = integer
1
2π
∫
2π
0
e x sinθ dθ
Asymptotic Expansions
27.103.
Jn ( x ) ~
2
nπ π ⎞
⎛
cos ⎜ x −
−
2 4 ⎟⎠
πx
⎝
where x is large
27.104.
Yn ( x ) ~
2
nπ π ⎞
⎛
sin ⎜ x −
−
2 4 ⎟⎠
πx
⎝
where x is large
27.105.
Jn ( x ) ~
1 ⎛ ex ⎞
⎜ ⎟
2π n ⎝ 2n⎠
27.106.
Yn ( x ) ~ −
27.107
I n (x) ~
ex
2π x
where x is large
27.108
K n (x) ~
e− x
2π x
where x is large
2 ⎛ ex ⎞
π n ⎜⎝ 2n⎟⎠
n
where n is large
−n
where n is large
Orthogonal Series of Bessel Functions
Let λ1 , λ2 , λ3 , ... be the positive roots of RJ n ( x ) + SxJ n′ ( x ) = 0, n > −1. Then the following series expansions
hold under the conditions indicated.
S = 0, R ≠ 0, i.e., λ1 , λ2 , λ3 , . . . are positive roots of JN(x) = 0
27.109.
f ( x ) = A1 J n (λ1 x ) + A2 J n (λ2 x ) + A3 J n (λ3 x ) + ⋅⋅⋅
where
27.110.
Ak =
2
J (λ k )
2
n +1
∫
1
0
xf ( x ) J n (λ k x )dx
In psarticular if n = 0,
27.111.
f ( x ) = A1J 0 (λ1x ) + A2 J 0 (λ2 x ) + A3 J 0 (λ3 x ) + ⋅⋅⋅
where
27.112.
Ak =
2
J12 (λ k )
∫
1
0
xf ( x ) J 0 (λ k x )dx
BESSEL FUNCTIONS
162
R/S > −n
27.113.
f ( x ) = A1J n (λ1x ) + A2 J n (λ2 x ) + A3 J n (λ3 x ) + ⋅⋅⋅
where
27.114.
Ak =
2
J (λ k ) − J n−1 (λ k ) J n+1 (λ k )
2
n
∫
1
xf ( x ) J n (λ k x )dx
0
In particular if n = 0.
27.115.
f ( x ) = A1J 0 (λ1x ) + A2 J 0 (λ2 x ) + A3 J 0 (λ3 x ) + ⋅⋅⋅
where
27.116.
Ak =
2
J 02 (λ k ) + J12 (λ k )
∫
1
0
xf ( x ) J 0 (λ k x )dx
The next formulas refer to the expansion of Bessel functions where S ≠ 0.
R/S = −n
27.117.
f ( x ) = A0 x n + A1J n (λ1x ) + A2 J n (λ2 x ) + ⋅⋅⋅
where
27.118.
⎧A = 2(n + 1) 1 x n+1 f ( x )dx
∫0
⎪⎪ 0
⎨
2
⎪A = 2
⎪⎩ k J n (λ k ) − J n−1 (λ k ) J n+1 (λ k )
1
∫
0
xf ( x ) J n (λ k x )dx
In particular if n = 0 so that R = 0 [i.e., l1, l2, l3, … are the positive roots of J1 (x) = 0],
27.119.
f ( x ) = A0 + A1J 0 (λ1x ) + A2 J 0 (λ2 x ) + ⋅⋅⋅
where
27.120.
⎧A = 2 1 xf ( x )dx
∫0
⎪⎪ 0
⎨
1
⎪A = 2 2
xf ( x ) J 0 (λ k x )dx
k
∫
J 0 (λ k ) 0
⎪⎩
R/S < −N
In this case there are two pure imaginary roots ±il0 as well as the positive roots l1, l2, l3, … and we have
27.121.
f ( x ) = A0 I n (λ0 x ) + A1J n (λ1 x ) + A2 J n (λ2 x ) + ⋅⋅⋅
where
27.122.
2
⎧
⎪A0 = I n2 (λ0 ) + I n−1 (λ0 )I n+1 (λ0 )
⎪
⎨
2
⎪A =
⎪⎩ k J n2 (λ k ) − J n−1 (λ k ) J n+1 (λ k )
∫
1
0
∫
1
0
xf ( x )I n (λ0 x )dx
xf ( x ) J n (λ k x )dx
BESSEL FUNCTIONS
163
Miscellaneous Results
27.123.
cos ( x sin θ ) = J 0 ( x ) + 2 J 2 ( x ) cos 2θ + 2 J 4 ( x ) cos 4θ + ⋅⋅ ⋅
27.124.
sin ( x sin θ ) = 2 J1 ( x ) sin θ + 2 J3 ( x ) sin 3θ + 2 J5 ( x ) sin 5θ + ⋅⋅⋅
27.125.
J n ( x + y) =
∞
∑ J ( x)J
k =−∞
k
n−k
n = 0, ± 1, ± 2, ...
( y)
This is called the addition formula for Bessel functions.
27.126.
1 = J 0 ( x ) + 2 J 2 ( x ) + ⋅⋅⋅ + 2 J 2 n ( x ) + ⋅⋅⋅
27.127.
x = 2{J1 ( x ) + 3J3 ( x ) + 5J5 ( x ) + ⋅⋅⋅ + (2n + 1) J 2 n+1 ( x ) + ⋅ ⋅⋅}
27.128.
x 2 = 2{4 J 2 ( x ) + 16 J 4 ( x ) + 36 J6 ( x ) + ⋅⋅⋅ + (2n)2 J 2 n ( x ) + ⋅⋅⋅}
27.129.
xJ1 ( x )
= J 2 ( x ) − 2 J 4 ( x ) + 3J6 ( x ) − ⋅⋅⋅
4
27.130.
1 = J 02 ( x ) + 2 J12 ( x ) + 2 J 22 ( x ) + 2 J32 ( x ) + ⋅⋅⋅
27.131.
J n′′( x ) = 14 {J n− 2 ( x ) − 2 J n ( x ) + J n+ 2 ( x )}
27.132.
J n′′′( x ) = 81 {J n −3 ( x ) − 3J n −1 ( x ) + 3J n+1 ( x ) − J n+3 ( x )}
Formulas 27.131 and 27.132 can be generalized.
2 sin nπ
πx
27.133.
J n′ ( x ) J − n ( x ) − J −′ n J n ( x ) =
27.134.
J n ( x ) J − n+1 ( x ) + J − n ( x ) J n−1 ( x ) =
27.135.
J n+1 ( x )Yn ( x ) − J n ( x )Yn+1 ( x ) = J n ( x )Yn′( x ) − J n′ ( x )Yn ( x ) =
27.136.
sin x = 2{J1 ( x ) − J3 ( x ) + J5 ( x ) − ⋅⋅⋅}
27.137.
cos x = J 0 ( x ) − 2 J 2 ( x ) + 2 J 4 ( x ) − ⋅⋅⋅
27.138.
sinh x = 2{I1 ( x ) + I 3 ( x ) + I 5 ( x ) + ⋅⋅⋅}
27.139.
cosh x = I 0 ( x ) + 2{I 2 ( x ) + I 4 ( x ) + I 6 ( x ) + ⋅⋅⋅}
2 sin nπ
πx
2
πx
LEGENDRE and ASSOCIATED LEGENDRE
FUNCTIONS
28
Legendre’s Differential Equation
28.1.
(1 − x 2 ) y′′ − 2 xy′ + n(n + 1) y = 0
Solutions of this equation are called Legendre functions of order n.
Legendre Polynomials
If n = 0, 1, 2, …, a solution of 28.1 is the Legendre polynomial Pn(x) given by Rodrigues’ formula
28.2.
Pn ( x ) =
1 dn 2
( x − 1)n
2n n! dx n
Special Legendre Polynomials
28.3.
P0 ( x ) = 1
28.7.
P4 ( x ) = 81 (35x 4 − 30 x 2 + 3)
28.4.
P1 ( x ) = x
28.8.
P5 ( x ) = 81 (63x 5 − 70 x 3 + 15x )
28.5.
P2 ( x ) = 12 (3x 2 − 1)
28.9.
P6 ( x ) = 161 (231x 6 − 315x 4 + 105x 2 − 5)
28.6.
P3 ( x ) = 12 (5x 3 − 3x )
28.10.
P7 ( x ) = 161 (429 x 7 − 693x 5 + 315x 3 − 35x )
Legendre Polynomials in Terms of U where x ⴝ cos U
28.11.
P0 (cos θ ) = 1
28.12.
P1 (cos θ ) = cos θ
28.13.
P2 (cos θ ) = 14 (1 + 3 cos 2θ )
28.14.
P3 (cos θ ) = 81 (3 cos θ + 5 cos 3θ )
28.15.
P4 (cos θ ) =
164
1
64
(9 + 20 cos 2θ + 35 cos 4θ )
LEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS
28.16.
1
P5 (cos θ ) = 128
(30 cos θ + 35 cos 3θ + 63 cos 5θ )
28.17.
P6 (cos θ ) =
28.18.
1
P7 (cos θ ) = 1024
(175 cos θ + 189 cos 3θ + 231 cos 5θ + 429 cos 7θ )
1
512
(50 + 105 cos 2θ + 126 cos 4θ + 231 cos 6θ )
Generating Function for Legendre Polynomials
∞
1
=
Pn ( x )t n
∑
1 − 2tx + t 2 n= 0
28.19.
Recurrence Formulas for Legendre Polynomials
28.20.
(n + 1)Pn+1 ( x ) − (2n + 1) x Pn ( x ) + nPn−1 ( x ) = 0
28.21.
Pn′+1 ( x ) − xPn′( x ) = (n + 1)Pn ( x )
28.22.
xPn′( x ) − Pn′−1 ( x ) = nPn ( x )
28.23.
Pn′+1 ( x ) − Pn′−1 ( x ) = (2n + 1)Pn ( x )
28.24.
( x 2 − 1)Pn′( x ) − nxPn ( x ) − nPn−1 ( x )
Orthogonality of Legendre Polynomials
28.25.
∫
1
28.26.
∫
1
−1
−1
Pm ( x )Pn ( x )dx = 0 m ≠ n
{Pn ( x )}2 dx =
2
2n + 1
Because of 28.25, Pm(x) and Pn(x) are called orthogonal in –1 x 1.
Orthogonal Series of Legendre Polynomials
28.27.
f ( x ) = A0 P0 ( x ) + A1P1 ( x ) + A2 P2 ( x ) + where
28.28.
Ak =
2k + 1 1
f ( x )Pk ( x )dx
2 ∫−1
165
LEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS
166
Special Results Involving Legendre Polynomials
28.29.
Pn (1) = 1
28.30.
Pn (−1) = (−1)n
28.31.
Pn (− x ) = (−1)n Pn ( x )
28.32.
⎧0
⎪
Pn (0) = ⎨
1 i 3 i 5 (n − 1)
⎪(−1)n / 2
2 i 4 i 6 n
⎩
28.33.
Pn ( x ) =
28.34.
∫ P ( x )dx =
28.35.
| Pn ( x ) | 1
28.36.
Pn ( x ) =
1
π
π
∫ (x +
0
1
2
n even
x 2 − 1 cos φ) dφ
n
Pn+1 ( x ) − Pn−1 ( x )
2n + 1
n
n +1
n odd
(z 2 − 1)n
π i ∫ (z − x )
c
n +1
dz
where C is a simple closed curve having x as interior point.
General Solution of Legendre’s Equation
The general solution of Legendre’s equation is
28.37.
y = AU n ( x ) + BVn ( x )
where
28.38.
Un (x) = 1 −
n(n + 1) 2 n(n − 2)(n + 1)(n + 3) 4
x +
x −
2!
4!
28.39.
Vn ( x ) = x −
(n − 1)(n + 2) 3 (n − 1)(n − 3)(n + 2)(n + 4) 5
x +
x −
3!
5!
These series converge for –1 < x < 1.
Legendre Functions of the Second Kind
If n = 0, 1, 2, … one of the series 28.38, 28.39 terminates. In such cases,
28.40.
⎧⎪U n ( x ) /U n (1)
Pn ( x ) = ⎨
⎪⎩Vn ( x ) /Vn (1)
n = 0, 2, 4, …
n = 1, 3, 5, …
where
28.41.
U n (1) = (−1)
n/2
⎡⎛ n ⎞ ⎤
2 ⎢⎜ ⎟ !⎥
⎣⎝ 2⎠ ⎦
n
2
n!
n = 0, 2, 4, …
LEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS
28.42.
⎡⎛ n − 1⎞ ⎤
Vn (1) = (−1)( n −1) / 2 2n −1 ⎢⎜
⎟ !⎥
⎣⎝ 2 ⎠ ⎦
167
2
n!
n = 1, 3, 5,…
The nonterminating series in such a case with a suitable multiplicative constant is denoted by Qn(x) and
is called Legendre’s function of the second kind of order n. We define
28.43.
⎧⎪U n (1)Vn ( x )
Qn ( x ) = ⎨
⎩⎪−Vn (1)U n ( x )
n = 0, 2, 4,…
n = 1, 3, 5,…
Special Legendre Functions of the Second Kind
28.44.
Q0 ( x ) =
1 ⎛1 + x⎞
ln
2 ⎜⎝ 1 − x ⎟⎠
28.45.
Q1 ( x ) =
x ⎛1 + x⎞
ln
−1
2 ⎜⎝ 1 − x ⎟⎠
28.46.
Q2 ( x ) =
3x 2 − 1 ⎛ 1 + x ⎞ 3x
ln ⎜
−
4
⎝ 1 − x ⎟⎠ 2
28.47.
Q3 ( x ) =
5x 3 − 3x ⎛ 1 + x ⎞ 5x 2 2
ln ⎜
−
+
4
2
3
⎝ 1 − x ⎟⎠
The functions Qn(x) satisfy recurrence formulas exactly analogous to 28.20 through 28.24.
Using these, the general solution of Legendre’s equation can also be written as
28.48.
y = APn ( x ) + BQn ( x )
Legendre’s Associated Differential Equation
28.49.
m2 ⎫
⎧
(1 − x 2 ) y′′ − 2 xy′ + ⎨n(n + 1) −
y=0
1 − x 2 ⎬⎭
⎩
Solutions of this equation are called associated Legendre functions. We restrict ourselves to the important
case where m, n are nonnegative integers.
Associated Legendre Functions of the First Kind
28.50.
Pnm ( x ) = (1 − x 2 )m / 2
dm
(1 − x 2 )m / 2 d m + n 2
P
(
x
)
=
( x − 1)n
dx m n
2n n! dx m + n
where Pn(x) are Legendre polynomials (page 164). We have
28.51.
Pn0 ( x ) = Pn ( x )
28.52.
Pnm ( x ) = 0
if m > n
LEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS
168
Special Associated Legendre Functions of the First Kind
28.53.
P11 ( x ) = (1 − x 2 )1/ 2
28.56.
P31 ( x ) = 32 (5x 2 − 1)(1 − x 2 )1/ 2
28.54.
P21 ( x ) = 3x (1 − x 2 )1/ 2
28.57.
P32 ( x ) = 15x (1 − x 2 )
28.55.
P22 ( x ) = 3(1 − x 2 )
28.58.
P33 ( x ) = 15(1 − x 2 )3 / 2
Generating Function for Pnm ( x)
28.59.
∞
(2m)!(1 − x 2 )m / 2 t m
=
Pnm ( x )t n
2m m !(1 − 2tx + t 2 )m +1/ 2 n∑
=m
Recurrence Formulas
28.60.
(n + 1 − m)Pnm+1 ( x ) − (2n + 1) x Pnm ( x ) + (n + m) Pnm−1 ( x ) = 0
28.61.
Pnm + 2 ( x ) −
2(m + 1) x m +1
P ( x ) + (n − m)(n + m + 1)Pnm ( x ) = 0
(1 − x 2 )1/ 2 n
Orthogonality of Pnm ( x)
1
28.62.
∫
28.63.
∫ {P
−1
Pl m ( x )P1m ( x )dx = 0
1
−1
m
n
( x )} dx =
2
if n ≠ l
2 (n + m)!
2n + 1 (n − m)!
Orthogonal Series
28.64.
f ( x ) = Am Pmm ( x ) + Am +1Pmm+1 ( x ) + Am + 2 Pmm+ 2 ( x ) + where
28.65.
Ak =
2k + 1 (k − m)! 1
f ( x )Pkm ( x )dx
2 (k + m)! ∫−1
Associated Legendre Functions of the Second Kind
28.66.
Qnm ( x ) = (1 − x 2 )m / 2
dm
Q (x)
dx m n
where Qn(x) are Legendre functions of the second kind (page 166).
These functions are unbounded at x = ±1, whereas Pnm ( x ) are bounded at x = ± 1.
The functions Qnm ( x ) satisfy the same recurrence relations as Pnm ( x ) (see 28.60 and 28.61).
General Solution of Legendre’s Associated Equation
28.67.
y = APnm ( x ) + BQnm ( x )
29
HERMITE POLYNOMIALS
Hermite’s Differential Equation
29.1.
y′′ − 2 xy′ + 2ny = 0
Hermite Polynomials
If n = 0, 1, 2, …, then a solution of Hermite’s equation is the Hermite polynomial Hn(x) given by Rodrigue’s
formula.
29.2.
H n ( x ) = (−1)n e x
2
d n −x
(e )
dx n
2
Special Hermite Polynomials
29.3.
H0 (x) = 1
29.7.
H 4 ( x ) = 16 x 4 − 48 x 2 + 12
29.4.
H1 ( x ) = 2 x
29.8.
H 5 ( x ) = 32 x 5 − 160 x 3 + 120 x
29.5.
H2 (x) = 4 x 2 − 2
29.9.
H 6 ( x ) = 64 x 6 − 480 x 4 + 720 x 2 − 120
29.6.
H 3 ( x ) = 8 x 3 − 12 x
29.10.
H 7 ( x ) = 128 x 7 − 1344 x 5 + 3360 x 3 − 1680 x
Generating Function
∞
29.11.
e 2 tx − t = ∑
2
n= 0
H n ( x )t n
n!
Recurrence Formulas
29.12.
H n+1 ( x ) = 2 xH n ( x ) − 2nH n−1 ( x )
29.13.
H n′ ( x ) = 2nH n−1 ( x )
Orthogonality of Hermite Polynomials
29.14.
∫
29.15.
∫
∞
−∞
∞
−∞
e − x H m ( x ) H n ( x )dx = 0
2
m≠n
e − x {H n ( x )}2 dx = 2n n! π
2
169
HERMITE POLYNOMIALS
170
Orthogonal Series
29.16.
f ( x ) = A0 H 0 ( x ) + A1H1 ( x ) + A2 H 2 ( x ) + where
29.17.
Ak =
1
2 k! π
∫
∞
−∞
k
e − x f ( x ) H k ( x )dx
2
Special Results
n(n − 1)
n(n − 1)(n − 2)(n − 3)
(2 x )n− 4 −
(2 x ) n − 2 +
2!
1!
29.18.
H n ( x ) = (2 x ) n −
29.19.
H n (− x ) = (−1)n H n ( x )
29.20.
H 2 n−1 (0) = 0
29.21.
H 2 n (0) = (−1)n 2n i 1 i 3 i 5 (2n − 1)
29.22.
∫
29.23.
d −x
{e H n ( x )} = −e − x H n+1 ( x )
dx
29.24.
∫
x
29.25.
∫
∞
29.26.
H n ( x + y) = ∑
x
H n (t )dt =
0
H n+1 ( x ) H n+1 (0)
−
2(n + 1) 2(n + 1)
2
0
2
e − t H n (t )dt = H n−1 (0) − e − x H n−1 ( x )
−∞
2
2
t ne − t H n ( xt )dt = π n! Pn ( x )
2
1 ⎛ n⎞
n / 2 ⎜ ⎟ H k ( x 2 ) H n− k ( y 2 )
2
⎝ k⎠
k =0
n
This is called the addition formula for Hermite polynomials.
n
29.27.
∑
k =0
H k ( x ) H k ( y) H n+1 ( x ) H n ( y) − H n ( x ) H n+1 ( y)
=
2k k !
2n+11 n!( x − y)
30
LAGUERRE and ASSOCIATED LAGUERRE
POLYNOMIALS
Laguerre’s Differential Equation
30.1.
xy ′′ + (1 − x ) y ′ + ny = 0
Laguerre Polynomials
If n = 0, 1, 2, …, then a solution of Laguerre’s equation is the Laguerre polynomial Ln(x) given by
Rodrigues’ formula
30.2.
Ln ( x ) = e x
d n n −x
(x e )
dx n
Special Laguerre Polynomials
30.3.
L0 ( x ) = 1
30.4.
L1 ( x ) = − x + 1
30.5.
L2 ( x ) = x 2 − 4 x + 2
30.6.
L3 ( x ) = − x 3 + 9 x 2 − 18 x + 6
30.7.
L4 ( x ) = x 4 − 16 x 3 + 72 x 2 − 96 x + 24
30.8.
L5 ( x ) = − x 5 + 25x 4 − 200 x 3 + 600 x 2 − 600 x + 120
30.9.
L6 ( x ) = x 6 − 36 x 5 + 450 x 4 − 2400 x 3 + 5400 x 2 − 4320 x + 720
30.10.
L7 ( x ) = − x 7 + 49 x 6 − 882 x 5 + 7350 x 4 − 29, 400 x 3 + 52, 920 x 2 − 35, 280 x + 5040
Generating Function
30.11.
∞
L ( x )t n
e − xt /(1− t )
=∑ n
n!
1− t
n= 0
171
LAGUERRE AND ASSOCIATED LAGUERRE POLYNOMIALS
172
Recurrence Formulas
30.12.
Ln+1 ( x ) − (2n + 1 − x ) Ln ( x ) + n 2 Ln−1 ( x ) = 0
30.13.
Ln′ ( x ) − nLn′−1 ( x ) + nLn −1 ( x ) = 0
30.14.
xLn′ ( x ) = nLn ( x ) − n 2 Ln−1 ( x )
Orthogonality of Laguerre Polynomials
30.15.
∫
∞
30.16.
∫
∞
0
0
e − x Lm ( x ) Ln ( x )dx = 0
m≠n
e − x {Ln ( x )}2 dx = (n!)2
Orthogonal Series
30.17.
f ( x ) = A0 L0 ( x ) + A1L1 ( x ) + A2 L2 ( x ) + where
30.18.
Ak =
1
(k !)2
∫
∞
0
e − x f ( x ) Lk ( x )dx
Special Results
30.19.
Ln ( 0 ) = n !
30.20.
∫
30.21.
n 2 x n−1 n 2 (n − 1)2 x n− 2
⎫
⎧
Ln ( x ) = (−1)n ⎨x n −
+
− (−1)n n!⎬
1
!
2
!
⎭
⎩
30.22.
⎧⎪ 0
p −x
(
)
x
e
L
x
dx
=
⎨
∫0
n
n
2
⎩⎪(−1) (n !)
30.23.
∑
x
0
Ln (t )dt = Ln ( x ) −
Ln+1 ( x )
n +1
∞
n
k =0
if p < n
if p = n
Lk ( x ) Lk ( y) Ln ( x ) Ln+1 ( y) − Ln+1 ( x ) Ln ( y)
=
(k !)2
(n!)2 ( x − y)
∞
t k Lk ( x )
= e t J 0 (2 xt )
(k !)2
k =0
30.24.
∑
30.25.
Ln ( x ) = ∫ u ne x −u J 0 (2 xu )du
∞
0
LAGUERRE AND ASSOCIATED LAGUERRE POLYNOMIALS
173
Laguerre’s Associated Differential Equation
30.26.
xy′′ + (m + 1 − x ) y′ + (n − m) y = 0
Associated Laguerre Polynomials
Solutions of 30.26 for nonnegative integers m and n are given by the associated Laguerre polynomials
30.27.
Lmn ( x ) =
dm
L (x)
dx m n
where Ln(x) are Laguerre polynomials (see page 171).
30.28.
L0n ( x ) = Ln ( x )
30.29.
Lmn ( x ) = 0
if m > n
Special Associated Laguerre Polynomials
30.30.
L11 ( x ) = −1
30.35.
L33 ( x ) = −6
30.31.
L12 ( x ) = 2 x − 4
30.36.
L14 ( x ) = 4 x 3 − 48 x 2 + 144 x − 96
30.32.
L22 ( x ) = 2
30.37.
L24 ( x ) = 12 x 2 − 96 x + 144
30.33.
L13 ( x ) = −3x 2 + 18 x − 18
30.38.
L34 ( x ) = 24 x − 96
30.34.
L23 ( x ) = −6 x + 18
30.39.
L44 ( x ) = 24
Generating Function for Lmn ( x)
30.40.
∞
Lmn ( x ) n
(−1)m t m − xt /(1− t )
e
=
t
∑
n!
(1 − t )m +1
n= m
Recurrence Formulas
30.41.
n − m +1 m
Ln+1 ( x ) + ( x + m − 2n − 1) Lmn ( x ) + n 2 Lmn−1 ( x ) = 0
n +1
30.42.
d m
{L ( x )} = Lmn +1 ( x )
dx n
30.43.
d m −x m
{x e Ln ( x )} = (m − n − 1) x m −1e − x Lmn −1 ( x )
dx
30.44.
x
d m
{L ( x )} = ( x − m) Lmn ( x ) + (m − n − 1) Lmn −1 ( x )
dx n
LAGUERRE AND ASSOCIATED LAGUERRE POLYNOMIALS
174
Orthogonality
30.45.
∫
∞
30.46.
∫
∞
0
0
x me − x Lmn ( x ) Lmp ( x )dx = 0
x me − x {Lmn ( x )}2 dx =
p≠n
(n!)3
(n − m)!
Orthogonal Series
30.47.
f ( x ) = Am Lmm ( x ) + Am +1Lmm +1 ( x ) + Am + 2 Lmm + 2 ( x ) + where
30.48.
Ak =
(k − m)! ∞ m − x m
x e Lk ( x ) f ( x )dx
(k !)3 ∫0
Special Results
30.49.
30.50.
Lmn ( x ) = (−1)n
∫
∞
0
{
}
n!
n(n − m) n− m −1 n(n − 1)(n − m)(n − m − 1) n− m − 2
x
+
x n−m −
x
+
2!
(n − m)!
1!
x m +1e − x {Lmn ( x )}2 dx =
(2n − m + 1)(n!)3
(n − m)!
31
CHEBYSHEV POLYNOMIALS
Chebyshev’s Differential Equation
31.1.
(1 − x 2 ) y n − xy ′ + n 2 y = 0
n = 0, 1, 2, ...
Chebyshev Polynomials of the First Kind
A solution of 31.1 is given by
31.2.
⎛ n⎞
⎛ n⎞
Tn ( x ) = cos (n cos −1 x ) = x n − ⎜ ⎟ x n − 2 (1 − x 2 ) + ⎜ ⎟ x n − 4 (1 − x 2 )2 − ⎝ 4⎠
⎝ 2⎠
Special Chebyshev Polynomials of The First Kind
31.3.
T0 ( x ) = 1
31.7.
T4 ( x ) = 8 x 4 − 8 x 2 + 1
31.4.
T1 ( x ) = x
31.8.
T5 ( x ) = 16 x 5 − 20 x 3 + 5x
31.5.
T2 ( x ) = 2 x 2 − 1
31.9.
T6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1
31.6.
T3 ( x ) = 4 x 3 − 3x
31.10.
T7 ( x ) = 64 x 7 − 112 x 5 + 56 x 3 − 7 x
Generating Function for Tn ( x)
31.11.
∞
1 − tx
n
2 = ∑ Tn ( x )t
1 − 2tx + t
n= 0
Special Values
31.12.
Tn (− x ) = (−1)n Tn ( x )
31.14.
Tn (−1) = (−1)n
31.13.
Tn (1) = 1
31.15.
T2 n (0) = (−1)n
31.16.
T2 n+1 (0) = 0
Recursion Formula for Tn ( x)
31.17.
Tn+1 ( x ) − 2 xTn ( x ) + Tn−1 ( x ) = 0
175
CHEBYSHEV POLYNOMIALS
176
Orthogonality
31.18.
31.19.
∫
1
∫
1
−1
Tm ( x )Tn ( x )
dx = 0
1− x2
⎧ π
dx = ⎨
1− x
⎩π / 2
{Tn ( x )}2
−1
2
m≠n
if n = 0
if n = 1, 2, ...
Orthogonal Series
31.20.
f ( x ) = 12 A0T0 ( x ) + A1T1 ( x ) + A2T2 ( x ) + where
31.21.
Ak =
2
π
∫
1
−1
f ( x )Tk ( x )
dx
1− x2
Chebyshev Polynomials of The Second Kind
31.22.
Un (x) =
sin{(n + 1) cos −1 x}
sin (cos −1 x )
⎛ n + 1⎞ n ⎛ n + 1⎞ n − 2
⎛ n + 1⎞ n − 4
x −⎜
x (1 − x 2 ) + ⎜
x (1 − x 2 )2 − =⎜
⎝ 3 ⎟⎠
⎝ 1 ⎠⎟
⎝ 5 ⎟⎠
Special Chebyshev Polynomials of The Second Kind
31.23.
U0 (x) = 1
31.27.
U 4 ( x ) = 16 x 4 − 12 x 2 + 1
31.24.
U1 ( x ) = 2 x
31.28.
U 5 ( x ) = 32 x 5 − 32 x 3 + 6 x
31.25.
U2 (x) = 4 x 2 − 1
31.29.
U 6 ( x ) = 64 x 6 − 80 x 4 + 24 x 2 − 1
31.26.
U3 ( x ) = 8 x 3 − 4 x
31.30.
U 7 ( x ) = 128 x 7 − 192 x 5 + 80 x 3 − 8 x
Generating Function for Un ( x)
31.31.
∞
1
n
2 = ∑ U n ( x )t
1 − 2tx + t
n= 0
Special Values
31.32.
U n (− x ) = (−1)nU n ( x )
31.34.
U n (−1) = (−1)n (n + 1)
31.33.
U n (1) = n + 1
31.35.
U 2 n (0) = (−1)n
31.36.
U 2 n+1 (0) = 0
CHEBYSHEV POLYNOMIALS
177
Recursion Formula for Un ( x)
31.37.
U n+1 ( x ) − 2 xU n ( x ) + U n−1 ( x ) = 0
Orthogonality
31.38.
∫
31.39.
∫
1
1 − x 2 U m ( x )U n ( x )dx = 0
−1
1
−1
1 − x 2 {U n ( x )}2 dx =
m≠n
π
2
Orthogonal Series
31.40.
f ( x ) = A0U 0 ( x ) + AU
1 1 ( x ) + A2U 2 ( x ) + where
31.41.
Ak =
2
π
∫
1
1 − x 2 f ( x )U k ( x )dx
−1
Relationships Between Tn ( x) and Un ( x)
31.42.
Tn ( x ) = U n ( x ) − xU n−1 ( x )
31.43.
(1 − x 2 )U n−1 ( x ) = xTn ( x ) − Tn+1 ( x )
31.44.
Un (x) =
1
π
∫
31.45.
Tn ( x ) =
1
π
∫
1
−1
1
−1
Tn+1 ( )d
( − x ) 1 − 2
1 − 2 U n−1 ( )
d
x−
General Solution of Chebyshev’s Differential Equation
31.46.
⎧AT ( x ) + B 1 − x 2 U ( x )
⎪ n
n −1
y=⎨
−1
⎩⎪A + B sin x
if n = 1, 2, 3, …
if n = 0
32
HYPERGEOMETRIC FUNCTIONS
Hypergeometric Differential Equation
32.1.
x (1 − x ) y n + {c − (a + b + 1) x}y′ − aby = 0
Hypergeometric Functions
A solution of 32.1 is given by
32.2.
F (a, b; c; x ) = 1 +
aib
a(a + 1)b(b + 1) 2 a(a + 1)(a + 2)b(b + 1)(b + 2) 3
x +
x+
x +
1i 2 i 3 i c(c + 1)(c + 2)
1i c
1i 2 i c(c + 1)
If a, b, c are real, then the series converges for –1 < x < 1 provided that c – (a + b) > –1.
Special Cases
32.3.
F (− p,1;1; − x ) = (1 + x ) p
32.8.
F ( 12 , 12 ; 32 ; x 2 ) = (sin −1 x ) /x
32.4.
F (1,1; 2; − x ) = [ln (1 + x )]/x
32.9.
F ( 12 ,1; 32 ; − x 2 ) = (tan −1 x ) /x
32.5.
lim F (1, n;1; x /n) = e x
32.10.
F (1, p; p; x ) = 1/ (1 − x )
32.6.
F ( 12 , − 12 ; 12 ; sin 2 x ) = cos x
32.11.
F (n + 1, − n;1;(1 − x ) / 2) = Pn ( x )
32.7.
F ( 12 ,1;1; sin 2 x ) = sec x
32.12.
F (n, − n; 12 ;(1 − x ) / 2) = Tn ( x )
n→∞
General Solution of The Hypergeometric Equation
If c, a – b and c – a – b are all nonintegers, then the general solution valid for | x | < 1 is
32.13.
178
y = AF (a, b; c; x ) + Bx 1−c F (a − c + 1, b − c + 1; 2 − c; x )
HYPERGEOMETRIC FUNCTIONS
Miscellaneous Properties
Γ (c)Γ (c − a − b)
Γ (c − a)Γ (c − b)
32.14.
F (a, b; c;1) =
32.15.
d
ab
F (a, b; c; x ) =
F (a + 1, b + 1; c + 1; x )
dx
c
32.16.
F (a, b; c; x ) =
32.17.
F (a, b; c; x ) = (1 − x )c− a− b F (c − a, c − b; c; x )
1
Γ (c)
u b−1 (1 − u)c− b−1 (1 − ux )− a du
∫
Γ (b)Γ (c − b) 0
179
Section VIII: Laplace and Fourier Transforms
33
LAPLACE TRANSFORMS
Definition of the Laplace Transform of F(t)
33.1.
{F (t )} =
∫
∞
0
e − st F (t )dt = f (s)
In general f (s) will exist for s > a where a is some constant. is called the Laplace transform operator.
Definition of the Inverse Laplace Transform of f(s)
If {F(t)} = f (s), then we say that F (t) = –1{f (s)} is the inverse Laplace transform of f (s). –1 is called the
inverse Laplace transform operator.
Complex Inversion Formula
The inverse Laplace transform of f (s) can be found directly by methods of complex variable theory. The
result is
33.2.
F (t ) =
1
2π i
∫
c + i∞
c − i∞
e st f (s)ds =
1
lim
2π i T →∞
∫
c + iT
c − iT
e st f (s)ds
where c is chosen so that all the singular points of f (s) lie to the left of the line Re{s} = c in the complex s
plane.
180
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
LAPLACE TRANSFORMS
181
Table of General Properties of Laplace Transforms
f (s)
F(t)
33.3.
a f1 (s) + bf2 (s)
aF1 (t ) + bF2 (t )
33.4.
f (s /a)
a F (at )
33.5.
f (s – a)
eatF(t)
33.6.
e–asf (s)
33.7.
sf (s) – F (0)
F ′ (t )
33.8.
s 2 f (s) − sF (0) − F ′(0)
F ′′(t )
33.9.
s n f (s) − s n−1 F (0) − s n− 2 F ′(0) − − F ( n−1) (0)
F(n)(t)
33.10.
f ′ (s )
–tF(t)
33.11.
f ′′(s)
t2F(t)
33.12.
f (n)(s)
(–1)nt nF(t)
33.13.
f (s )
s
33.14.
f (s )
sn
33.15.
f (s)g(s)
(t − a) =
∫
∫
t
0
t
{
F (t − a) t > a
0
t<a
t
0
F (u)du
∫ F (u)du n =
0
∫
t
0
(t − u)n−1
∫0 (n − 1)! F (u)du
t
F (u)G (t − u)du
LAPLACE TRANSFORMS
182
33.16.
∫
33.17.
1
1 − e − sT
F(t)
f (u)du
F (t )
t
∫
T
0
e − su F (u)du
F(t) = F(t + T)
1
πt
f ( s)
s
33.18.
()
1 1
f
s s
33.19.
1
33.20.
s
n+1
f
∫
(1s)
1
2 π
∫
∞
0
u −3 / 2 e − s
2
/ 4u
∞
0
∫
∞
0
e−u
2
/ 4t
F (u)du
J 0 (2 ut )F (u)du
∞
t n / 2 ∫ u − n / 2 J n (2 ut )F (u)du
0
f (s + 1/s)
s2 + 1
33.21.
33.22.
∞
s
f (s)
∫
t
0
J 0 (2 u(t − u)) F (u)du
f (u)du
F(t2)
∞
t u F (u)
du
Γ (u + 1)
33.23.
f (ln s)
s ln s
∫
33.24.
P (s )
Q (s )
∑ Q′(α
P(s) = polynomial of degree less than n,
Q(s) = (s – a1)(s – a2) … (s – an)
where a1, a2, …, an are all distinct.
0
n
k =1
P(α k ) α t
e
k)
k
LAPLACE TRANSFORMS
183
Table of Special Laplace Transforms
f (s)
F(t)
33.25.
1
s
1
33.26.
1
s2
t
33.27.
33.28.
1
sn
n = 1, 2, 3,…
1
sn
1
s−a
33.29.
33.30.
33.31.
n>0
1
(s − a) n
t n−1
Γ(n)
eat
n = 1, 2, 3,…
1
(s − a) n
t n−1
, 0! = 1
(n − 1)!
n>0
t n−1e at
, 0! = 1
(n − 1)!
t n−1e at
Γ(n)
33.32.
1
s + a2
sin at
a
33.33.
s
s2 + a2
cos at
33.34.
1
(s − b) 2 + a 2
e bt sin at
a
33.35.
s−b
(s − b) 2 + a 2
e bt cos at
33.36.
1
s2 − a2
sinh at
a
33.37.
s
s2 − a2
cosh at
33.38.
1
(s − b) 2 − a 2
e bt sinh at
a
2
LAPLACE TRANSFORMS
184
33.39.
f (s)
F(t)
s−b
(s − b) 2 − a 2
e bt cosh at
33.40.
1
(s − a)(s − b)
a≠b
e bt − e at
b−a
33.41.
s
(s − a)(s − b)
a≠b
be bt − ae at
b−a
33.42.
1
(s 2 + a 2 ) 2
sin at − at cos at
2a 3
33.43.
s
(s 2 + a 2 ) 2
t sin at
2a
33.44.
s2
(s + a 2 ) 2
sin at + at cos at
2a
33.45.
s3
(s + a 2 ) 2
cos at − 12 at sin at
33.46.
s2 − a2
(s 2 + a 2 ) 2
t cos at
33.47.
1
(s 2 − a 2 ) 2
at cosh at − sinh at
2a 3
33.48.
s
(s 2 − a 2 ) 2
t sinh at
2a
33.49.
s2
(s 2 − a 2 ) 2
sinh at + at cosh at
2a
33.50.
s3
(s − a 2 ) 2
cosh at + 12 at sinh at
33.51.
s2
(s − a 2 ) 3 / 2
t cosh at
33.52.
1
(s + a 2 ) 3
(3 − a 2 t 2 ) sin at − 3at cos at
8a 5
33.53.
s
(s + a 2 )3
t sin at − at 2 cos at
8a 3
33.54.
s2
(s + a 2 )3
(1 + a 2 t 2 ) sin at − at cos at
8a 3
33.55.
s3
(s 2 + a 2 )3
3t sin at + at 2 cos at
8a
2
2
2
2
2
2
2
LAPLACE TRANSFORMS
185
f (s)
F(t)
33.56.
s4
(s + a 2 )3
(3 − a 2 t 2 ) sin at + 5at cos at
8a
33.57.
s5
(s + a 2 )3
(8 − a 2 t 2 ) cos at − 7at sin at
8
33.58.
3s 2 − a 2
(s 2 + a 2 )3
t 2 sin at
2a
33.59.
s 3 − 3a 2 s
(s 2 + a 2 )3
1
2
t 2 cos at
33.60.
s 4 − 6a 2 s 2 + a 4
(s 2 + a 2 ) 4
1
6
t 3 cos at
33.61.
s3 − a2s
(s 2 + a 2 ) 4
t 3 sin at
24 a
33.62.
1
(s − a 2 ) 3
(3 + a 2 t 2 ) sinh at − 3at cosh at
8a 5
33.63.
s
(s − a 2 ) 3
at 2 cosh at − t sinh at
8a 3
33.64.
s2
(s − a 2 ) 3
at cosh at + (a 2 t 2 − 1) sinh at
8a 3
33.65.
s3
(s − a 2 )3
3t sinh at + at 2 cosh at
8a
33.66.
s4
(s − a 2 )3
(3 + a 2 t 2 ) sinh at + 5at cosh at
8a
33.67.
s5
(s − a 2 )3
(8 + a 2 t 2 ) cosh at + 7at sinh at
8
33.68.
3s 2 + a 2
(s 2 − a 2 )3
t 2 sinh at
2a
33.69.
s 3 + 3a 2 s
(s 2 − a 2 )3
1
2
t 2 cosh at
33.70.
s 4 + 6a 2 s 2 + a 4
(s 2 − a 2 ) 4
1
6
t 3 cosh at
33.71.
s3 + a2s
(s 2 − a 2 ) 4
t 3 sinh at
24 a
33.72.
1
s + a3
⎫
e at / 2 ⎧
3at
3at
− cos
+ e −3at / 2 ⎬
2 ⎨ 3 sin
2
2
3a ⎩
⎭
2
2
2
2
2
2
2
2
3
LAPLACE TRANSFORMS
186
f (s)
F(t)
33.73.
s
3
s + a3
⎫
3at
e at / 2 ⎧
3at
cos
+ 3 sin
− e −3at / 2 ⎬
3a ⎨⎩
2
2
⎭
33.74.
s2
s3 + a3
1 ⎛ − at
3at ⎞
e + 2e at / 2 cos
3 ⎜⎝
2 ⎟⎠
33.75.
1
s − a3
e − at / 2
3a 2
33.76.
s
s − a3
⎫
e − at / 2 ⎧
3at
3at
− cos
+ e3at / 2 ⎬
3 sin
3a ⎨⎩
2
2
⎭
33.77.
s2
s − a3
1 ⎛ at
3at ⎞
e + 2e − at / 2 cos
3 ⎜⎝
2 ⎟⎠
33.78.
1
s 4 + 4a 4
1
(sin at cosh at − cos at sinh at )
4a3
33.79.
s
s 4 + 4a 4
sin at sinh at
2a 2
33.80.
s2
s + 4a 4
1
(sin at cosh at + cos at sinh at )
2a
33.81.
s3
s + 4a 4
cos at cosh at
33.82.
1
s4 − a4
1
(sinh at − sin at )
2a 3
33.83.
s
s − a4
1
(cosh at − cos at )
2a 2
33.84.
s2
s − a4
1
(sinh at + sin at )
2a
33.85.
s3
s − a4
33.86.
33.87.
3
3
3
4
⎧ 3at / 2
3at
3at ⎫
− cos
− 3 sin
⎨e
2
2 ⎬⎭
⎩
4
4
4
4
1
s+a + s+b
1
s s+a
1
2
(cosh at + cos at )
e − bt − e − at
2(b − a) π t 3
erf at
a
33.88.
1
s (s − a)
e at erf at
a
33.89.
1
s−a +b
⎫
⎧ 1
e at ⎨
− b e b t erfc(b t )⎬
⎭
⎩ πt
2
LAPLACE TRANSFORMS
187
f (s)
F(t)
33.90.
1
s2 + a2
J 0 (at )
33.91.
1
s2 − a2
I 0 (at )
33.92.
( s 2 + a 2 − s) n
s2 + a2
n > −1
a n J n (at )
33.93.
(s − s 2 − a 2 ) n
s2 − a2
n > −1
a n I n (at )
33.94.
e b(s− s +a )
s2 + a2
J 0 (a t (t + 2b))
33.95.
e− b s +a
s2 + a2
⎧J 0 (a t 2 − b 2 ) t > b
⎨
t<b
⎩0
33.96.
1
(s 2 + a 2 ) 3 / 2
tJ1 (at )
a
33.97.
s
(s 2 + a 2 )3 / 2
tJ 0 (at )
33.98.
s2
(s + a 2 ) 3 / 2
J 0 (at ) − atJ1 (at )
33.99.
1
(s − a 2 )3 / 2
tI1 (at )
a
33.100.
s
(s 2 − a 2 ) 3 / 2
tI 0 (at )
33.101.
s2
(s − a 2 ) 3 / 2
I 0 (at ) + atI1 (at )
1
e− s
=
s(e − 1) s(1 − e − s )
F (t ) = n, n t < n + 1, n = 0,1, 2,…
2
2
33.102.
2
2
2
2
2
s
See also entry 33.165.
33.103.
33.104.
1
e− s
=
s
s(e − r ) s(1 − re − s )
es − 1
1 − e− s
=
s
s(e − r ) s(1 − re − s )
[t ]
F (t ) = ∑ r k
k =1
where [t] = greatest integer t
F (t ) = r n , n t < n + 1, n = 0,1, 2,…
See also entry 33.167.
33.105.
e − a /s
s
cos 2 at
πt
LAPLACE TRANSFORMS
188
33.106.
33.107.
f (s)
F(t)
e − a /s
s3/ 2
sin 2 at
πa
e − a /s
s n+1
n > −1
33.108.
e− a s
s
33.109.
e− a
33.110.
1 − e− a
s
33.111.
e− a
s
33.113.
33.114.
e− a / s
s n+1
ln
J n (2 at )
e− a / 4t
πt
2
a
e− a
2 π t3
s
2
/ 4t
s
erf (a / 2 t )
s
erfc(a / 2 t )
e− a s
s ( s + b)
33.112.
n/2
⎛ t⎞
⎝ a⎠
n > −1
⎛ s + a⎞
⎝ s + b⎠
a ⎞
⎛
e b ( bt + a ) erfc ⎜ b t +
⎝
2 t ⎟⎠
1
π ta 2 n+1
∫
∞
0
une−u
2
/ 4 a2t
J 2 n (2 u )du
e − bt − e − at
t
33.115.
ln[(s 2 + a 2 ) /a 2 ]
2s
Ci(at )
33.116.
ln[(s + a) /a]
s
Ei(at )
33.117.
(γ + ln s)
s
γ = Euler’s constant = .5772156 …
ln t
33.118.
⎛ s2 + a2 ⎞
ln ⎜ 2
⎝ s + b 2 ⎟⎠
2(cos at − cos bt )
t
33.119.
π 2 (γ + ln s)2
+
6s
s
γ = Euler’s constant = .5772156 …
ln 2 t
33.120.
ln s
s
33.121.
ln 2 s
s
−
−(ln t + γ )
γ = Euler’s constant = .5772156 …
(ln t + γ )2 − 16 π 2
γ = Euler’s constant = .5772156 …
LAPLACE TRANSFORMS
189
f (s)
33.122.
F(t)
Γ ′(n + 1) − Γ (n + 1) ln s
s n+1
n > −1
t n ln t
33.123.
tan −1 (a /s)
sin at
t
33.124.
tan −1 (a /s)
s
Si(at )
33.125.
e a /s
erfc( a /s)
s
e −2 at
πt
33.126.
es
2
/ 4 a2
erfc(s / 2a)
2a − a t
e
π
es
2
/ 4 a2
erfc(s / 2a)
s
erf(at )
33.127.
2 2
33.128.
e as erfc as
s
1
π (t + a)
33.129.
e as Ei(as)
1
t+a
33.130.
1⎡
π
⎤
cos as
− Si(as) − sin as Ci(as)⎥
a ⎢⎣
2
⎦
33.131.
33.132.
33.133.
sin as
{
}
{
{
}
}
{
}
π
− Si(as) + cos as Ci(as)
2
t
t 2 + a2
π
− Si(as) − sin as Ci(as)
2
s
tan −1 (t /a)
cos as
sin as
1
t 2 + a2
π
− Si(as) − cos as Ci(as)
2
s
1 ⎛ t 2 + a2 ⎞
ln
2 ⎜⎝ a 2 ⎟⎠
33.134.
⎡π − Si(as)⎤ + Ci 2 (as)
⎢⎣ 2
⎥⎦
1 ⎛ t 2 + a2 ⎞
ln
t ⎜⎝ a 2 ⎟⎠
33.135.
0
(t) = null function
33.136.
1
δ (t) = delta function
33.137.
e − as
δ (t − a)
33.138.
e − as
s
See also entry 33.163.
(t − a)
2
LAPLACE TRANSFORMS
190
f (s)
F(t)
33.139.
sinh sx
s sinh sa
x 2 ∞ (−1)n
nπ x
nπ t
sin
cos
+
a π∑
n
a
a
n =1
33.140.
sinh sx
s cosh sa
4 ∞ (−1)n
(2n − 1)π x
(2n − 1)π t
sin
sin
2
−
1
2
2a
π∑
n
a
n =1
33.141.
cosh sx
s sinh as
t 2 ∞ (−1)n
nπ x
nπ t
cos
sin
+
a π∑
n
a
a
n =1
33.142.
cosh sx
s cosh sa
33.143.
sinh sx
s sinh sa
33.144.
sinh sx
s cosh sa
33.145.
cosh sx
s 2 sinh sa
33.146.
cosh sx
s cosh sa
33.147.
cosh sx
s 3 cosh sa
33.148.
33.149.
1+
(2n − 1)π x
(2n − 1)π t
4 ∞ (−1)n
cos
cos
2
−
1
2
2a
π∑
n
a
n =1
xt 2a ∞ (−1)n
nπ x
nπ t
sin
sin
+
2
a π2 ∑
a
a
n
n =1
2
2
2
x+
(2n − 1)π x
(2n − 1)π t
8a ∞ (−1)n
cos
2 sin
2
a
2a
(
)
π2 ∑
2
n
−
1
n=1
t 2 2a ∞ (−1)n
nπ x ⎛
nπ t ⎞
cos
+
1 − cos
2
a
a ⎠
2a π 2 ∑
⎝
n
n =1
t+
8a ∞ (−1)n
(2n − 1)π x
(2n − 1)π t
sin
2 cos
2
2a
a
(
2
1
)
π2 ∑
n
−
n =1
2π
a2
sinh x s
sinh a s
cosh x s
cosh a s
∞
1 2
16a 2
(t + x 2 − a 2 ) − 3
2
π
π
a2
∞
∑ (−1)
n −1
(−1)n
∑ (2n − 1)
3
cos
n=1
∞
∑ (−1)
n
(2n − 1)π x
(2n − 1)π t
cos
2a
2a
ne − n π t /a sin
2
2
2
n =1
nπ x
a
(2n − 1)e − ( 2 n−1) π t / 4 a cos
2
2
2
n =1
(2n − 1)π x
2a
33.150.
sinh x s
s cosh a s
2 ∞
(2n − 1)π x
(−1)n−1 e − ( 2 n−1) π t / 4 a sin
2a
a∑
n =1
33.151.
cosh x s
s sinh a s
1 2 ∞
nπ x
+
(−1)n e − n π t /a cos
a a∑
a
n =1
33.152.
sinh x s
s sinh a s
x 2 ∞ (−1)n − n π t /a
nπ x
sin
+ ∑
e
a π n=1 n
a
33.153.
cosh x s
s cosh a s
33.154.
sinh x s
2
s sinh a s
33.155.
cosh x s
2
s cosh a s
2
2
2
2
2
2
1+
2
2
2
(2n − 1)π x
4 ∞ (−1)n − ( 2 n−1) π t / 4 a
cos
e
2a
2
−
1
π∑
n
n =1
xt 2a 2
+
a π3
2
2
2
(−1)n
nπ x
(1 − e − n π t /a ) sin
3
a
n
n =1
∞
∑
1 2
16a 2
(x − a2 ) + t − 3
2
π
∞
2
(−1)n
∑ (2n − 1)
n =1
2
2
e − ( 2 n−1) π t / 4 a cos
2
3
2
2
(2n − 1)π x
2a
LAPLACE TRANSFORMS
191
f (s)
F(t)
e − λ t /a J 0 (λn x /a)
λn J1 (λn )
n =1
∞
J 0 (ix s )
s J 0 (ia s )
33.156.
1 − 2∑
2
n
2
where λl, λ2,… are the positive roots of J0(λ) = 0
∞
e − λ t /a J 0 (λn x /a)
1 2
( x − a 2 ) + t + 2a 2 ∑
4
λn3 J1 (λn )
n =1
2
n
33.157.
J 0 (ix s )
s 2 J 0 (ia s )
where λ1, λ2,… are the positive roots of J0(λ) = 0
Triangular wave function
33.158.
1
⎛ as⎞
tanh
⎝ 2⎠
as 2
Fig. 33-1
Square wave function
33.159.
1
⎛ as⎞
tanh
s
⎝ 2⎠
Fig. 33-2
Rectified sine wave function
33.160.
πa
⎛ as⎞
coth
⎝ 2⎠
a2s2 + π 2
Fig. 33-3
Half-rectified sine wave function
33.161.
πa
(a s + π 2 )(1 − e − as )
2 2
Fig. 33-4
Sawtooth wave function
33.162.
2
1
e − as
−
as 2 s(1 − e − as )
Fig. 33-5
LAPLACE TRANSFORMS
192
f (s)
F(t)
Heaviside’s unit function (t – a)
33.163.
e − as
s
See also entry 33.138.
Fig. 33-6
Pulse function
33.164.
e − as (1 − e − s )
s
Fig. 33-7
Step function
33.165.
1
s(1 − e − as )
See also entry 33.102.
Fig. 33-8
F(t) = n2, n t < n + 1, n = 0, 1, 2, …
33.166.
e − s + e −2 s
s(1 − e − s )2
Fig. 33-9
F(t) = rn, n t < n + 1, n = 0, 1, 2, …
33.167.
1 − e− s
s(1 − re − s )
See also entry 33.104.
Fig. 33-10
⎧sin (π t /a) 0 t a
F (t ) = ⎨
t>a
⎩0
33.168.
π a(1 + e − as )
a2s2 + π 2
Fig. 33-11
34
FOURIER TRANSFORMS
Fourier’s Integral Theorem
∞
f ( x ) = ∫ {A(α ) cos α x + B(α )sin α x}dα
34.1.
0
where
34.2.
1 ∞
⎧
⎪A(α ) = π ∫−∞ f ( x ) cos α x dx
⎨
1 ∞
⎪ B(α ) = ∫ f ( x ) sin α x dx
π −∞
⎩
Sufficient conditions under which this theorem holds are:
(i) f (x) and f ′(x) are piecewise continuous in every finite interval –L < x < L;
∞
(ii) ∫ | f ( x ) | dx converges;
−∞
(iii) f (x) is replaced by 12 { f ( x + 0) + f ( x − 0)} if x is a point of discontinuity.
Equivalent Forms of Fourier’s Integral Theorem
34.3.
f (x) =
1
2π
∫
f (x) =
1
2π
∫
=
1
2π
∫ ∫
2
π
∞
∞
0
0
34.4.
34.5.
f (x) =
∫
∞
α =−∞
∞
∫
∞
u =−∞
f (u) cos α ( x − u) du dα
∞
eiα x dα ∫ f (u)e − iα u du
−∞
−∞
∞
∞
−∞
−∞
f (u)e iα ( x −u ) du dα
sin α x dα ∫ f (u) sin α u du
where f (x) is an odd function [ f (−x) = −f(x)].
34.6.
f (x) =
2
π
∫
∞
0
∞
cos α x dα ∫ f (u) cos α u du
0
where f (x) is an even function [ f (−x) = f (x)].
193
FOURIER TRANSFORMS
194
Fourier Transforms
The Fourier transform of f (x) is defined as
34.7.
{ f ( x )} = F (α ) =
∫
∞
−∞
f ( x )e − iα x dx
Then from 34.7 the inverse Fourier transform of F(a ) is
34.8.
−1{F (α )} = f ( x ) =
1
2π
∫
∞
−∞
F (α )e iα x dα
We call f (x) and F(a) Fourier transform pairs.
Convolution Theorem for Fourier Transforms
If F(a) = {f (x)} and G(a ) = {g(x)}, then
34.9.
1
2π
∫
∞
−∞
F (α )G (α )e iα x dα =
∫
∞
−∞
f (u)g( x − u) du = f * g
where f *g is called the convolution of f and g. Thus,
34.10. { f *g} = { f} {g}
Parseval’s Identity
If F(a) = { f (x)}, then
34.11.
∫
∞
| f ( x ) |2 dx =
−∞
1
2π
∫
∞
| F (α ) |2 dα
−∞
More generally if F(a) = { f (x)} and G(a) = {g(x)}, then
34.12.
∫
∞
−∞
f ( x )g( x ) dx =
1
2π
∫
∞
−∞
F (α )G (α ) dα
where the bar denotes complex conjugate.
Fourier Sine Transforms
The Fourier sine transform of f (x) is defined as
34.13.
FS (α ) = S { f ( x )} =
∫
∞
0
f ( x ) sin α x dx
Then from 34.13 the inverse Fourier sine transform of FS(a ) is
34.14.
f ( x ) = −S1{FS (α )} =
2
π
∫
∞
0
FS (α ) sin α x dα
FOURIER TRANSFORMS
195
Fourier Cosine Transforms
The Fourier cosine transform of f (x) is defined as
34.15.
FC (α ) = C { f ( x )} =
∫
∞
0
f ( x ) cos α x dx
Then from 34.15 the inverse Fourier cosine transform of FC(a) is
34.16.
f ( x ) = −C1{FC (α )} =
2
π
∫
∞
0
FC (α ) cos α x dα
Special Fourier Transform Pairs
f (x)
F(a )
1 |x|<b
0 |x|>b
2 sin bα
α
34.18.
1
x + b2
π e − bα
b
34.19.
x
x 2 + b2
−iπ e − bα
34.20.
f (n)(x)
inanF(a)
34.21.
x nf (x)
34.22.
f (bx)eitx
34.17.
{
2
in
d nF
dα n
1 ⎛ α − t⎞
F
b ⎝ b ⎠
FOURIER TRANSFORMS
196
Special Fourier Sine Transforms
f (x)
FC(a )
1 0<x<b
0
x>b
1 − cos bα
α
34.24.
x –1
π
2
34.25.
x
x + b2
π − bα
e
2
34.26.
e–bx
α
α 2 + b2
34.27.
xn – 1e–bx
Γ(n)sin(n tan −1 α / b)
(α 2 + b 2 )n / 2
34.28.
xe − bx
34.29.
x –1/2
π
2α
34.30.
x –n
πα n−1 csc (nπ / 2)
2Γ (n)
34.31.
sin bx
x
34.32.
sin bx
x2
34.33.
cos bx
x
α<b
⎧ 0
⎪
⎨π / 4 α = b
⎪⎩π / 2 α > b
34.34.
tan −1 ( x / b)
π − bα
e
2α
34.35.
csc bx
π
πα
tanh
2b
2b
34.36.
1
e −1
π
⎛ πα ⎞ 1
−
coth
4
⎝ 2 ⎠ 2α
34.23.
{
2
2x
2
π
α e −α
4 b 3/ 2
2
/4b
0<n<2
1 ⎛ α + b⎞
ln
2 ⎝ α − b⎠
{
πα / 2 α < b
πb / 2 α > b
FOURIER TRANSFORMS
197
Special Fourier Cosine Transforms
f (x)
FC (a )
1 0<x<b
0
x>b
sin bα
α
34.38.
1
x + b2
π e − bα
2b
34.39.
e − bx
b
α 2 + b2
34.40.
x n−1e − bx
Γ(n) cos(n tan −1 α /b)
(α 2 + b 2 )n / 2
34.41.
e − bx
34.42.
x −1/ 2
π
2α
34.43.
x −n
πα n −1 sec (nπ /2)
, 0 < n <1
2Γ (n)
34.44.
⎛ x 2 + b2 ⎞
ln ⎜ 2
⎝ x + c 2 ⎟⎠
e − cα − e − bα
πα
34.45.
sin bx
x
⎧π /2 α < b
⎪
⎨π /4 α = b
⎪⎩ 0 α > b
34.46.
sin bx 2
π ⎛ α2
α2⎞
cos
− sin ⎟
⎜
8b ⎝
4b
4b⎠
34.47.
cos bx 2
π
8b
34.48.
sech bx
34.37.
34.49.
34.50.
{
2
1 π −α
e
2 b
2
cosh ( π x/2)
2
/4b
α2
α2⎞
⎛
⎜⎝ cos 4 b + sin 4 b ⎟⎠
π
πα
sech
2b
2b
cosh ( π x )
π cosh ( πα /2)
2 cosh ( πα )
e− b x
x
π
{cos(2b α ) − sin(2b α )}
2α
Section IX: Elliptic and Miscellaneous Special Functions
35
ELLIPTIC FUNCTIONS
Incomplete Elliptic Integral of the First Kind
35.1.
u = F (k , φ ) =
∫
dθ
φ
1 − k sin θ
0
2
2
=
∫
d
x
(1 − )(1 − k 2 2 )
0
2
where f = am u is called the amplitude of u and x = sin f, and where here and below 0 < k < 1.
Complete Elliptic Integral of the First Kind
35.2.
K = F (k , π / 2) =
∫
dθ
π /2
0
1 − k sin θ
2
2
=
∫
d
1
(1 − )(1 − k 2 2 )
0
2
2
2
2
⎫⎪
π ⎧⎪ ⎛ 1⎞ 2 ⎛ 1 i 3 ⎞ 4 ⎛ 1 i 3 i 5 ⎞ 6
k + ⎬
= ⎨1 + ⎜ ⎟ k + ⎜
k +⎜
⎟
⎟
2 ⎪ ⎝ 2⎠
⎝ 2 i 4 i 6⎠
⎝ 2 i 4⎠
⎪⎭
⎩
Incomplete Elliptic Integral of the Second Kind
35.3.
E (k , φ ) =
∫
φ
1 − k 2 sin 2 θ dθ =
0
∫
x
1 − k 2 2
0
1− 2
d
Complete Elliptic Integral of the Second Kind
35.4.
E = E (k , π / 2) =
∫
π /2
0
1 − k 2 sin 2 θ dθ =
∫
1
1 − k 2 2
0
1− 2
d
2
2
2
⎫⎪
π ⎧⎪ ⎛ 1⎞ 2 ⎛ 1 i 3 ⎞ k 4 ⎛ 1 i 3 i 5 ⎞ k 6
= ⎨1 − ⎜ ⎟ k − ⎜
−
−
⎬
2 ⎪ ⎝ 2⎠
⎝ 2 i 4 ⎟⎠ 3 ⎜⎝ 2 i 4 i 6⎟⎠ 5
⎪⎭
⎩
Incomplete Elliptic Integral of the Third Kind
35.5.
Π(k , n, φ ) =
∫
φ
dθ
0
(1 + n sin θ ) 1 − k sin θ
2
2
2
=
∫
x
d
0
(1 + n ) (1 − 2 )(1 − k 2 2 )
2
198
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
ELLIPTIC FUNCTIONS
199
Complete Elliptic Integral of the Third Kind
35.6.
Π(k , n, π / 2) =
∫
dθ
π /2
0
(1 + n sin θ ) 1 − k sin θ
2
2
2
=
∫
1
d
0
(1 + n ) (1 − 2 )(1 − k 2 2 )
2
Landen’s Transformation
35.7.
tan φ =
sin 2φ1
k + cos 2φ1
k sin φ = sin (2φ1 − φ )
or
This yields
35.8.
F (k , φ ) =
∫
φ
0
2
dθ
=
2
2
+
k
1
1 − k sin θ
∫
dθ1
1 − k12 sin 2 θ1
φ1
0
where k1 = 2 k /(1 + k ). By successive applications, sequences k1 , k2 , k3 , … and φ1 , φ2 , φ3 , … are obtained such
that k < k1 < k2 < k3 < < 1 where lim kn = 1. It follows that
n→∞
35.9.
F ( k , Φ) =
k1 k2 k3 … Φ
dθ
∫0 1 − sin 2 θ =
k
k1 k2 k3 …
⎛ π Φ⎞
ln tan ⎜ + ⎟
k
⎝ 4 2⎠
where
35.10.
k1 =
2 k
,
1+ k
k2 =
2 k1
, … and
1 + k1
Φ lim φn
n→∞
The result is used in the approximate evaluation of F(k, f).
Jacobi’s Elliptic Functions
From 35.1 we define the following elliptic functions:
35.11.
x = sin (am u) = sn u
35.12.
1 − x 2 = cos (am u) = cn u
35.13.
1 − k 2 x 2 = 1 − k 2 sn 2 u = dn u
We can also define the inverse functions sn −1 x , cn −1 x , dn −1 x and the following:
35.14.
ns u =
1
sn u
35.17.
sc u =
sn u
cn u
35.20.
cs u =
cn u
sn u
35.15.
nc u =
1
cn u
35.18.
sd u =
sn u
dn u
35.21.
dc u =
dn u
cn u
35.16.
nd u =
1
dn u
35.19.
cd u =
cn u
dn u
35.22.
ds u =
dn u
dn u
Addition Formulas
35.23.
sn (u + ) =
sn u cn dn + cn u sn dn u
1 − k 2 sn 2 u sn 2
ELLIPTIC FUNCTIONS
200
35.24.
cn (u + ) =
cn u cn − sn u sn dn u dn 1 − k 2sn 2 u sn 2
35.25.
dn (u + ) =
dn u dn − k 2 sn u sn cn u cn 1 − k 2sn 2 u sn 2
Derivatives
35.26.
d
sn u = cn u dn u
du
35.28.
d
dn u = − k 2sn u cn u
du
35.27.
d
cn u = −sn u dn u
du
35.29.
d
sc u = dc u nc u
du
Series Expansions
35.30.
sn u = u − (1 + k 2 )
35.31.
cn u = 1 −
35.32.
dn u = 1 − k 2
u3
u5
u7
+ (1 + 14 k 2 + k 4 ) − (1 + 135k 2 + 135k 4 + k 6 ) + 3!
5!
7!
u2
u4
u6
+ (1 + 4 k 2 ) − (1 + 44 k 2 + 16 k 4 ) + 2!
4!
6!
u2
u4
u6
+ k 2 (4 + k 2 ) − k 2 (16 + 44 k 2 + k 4 ) + 2!
4!
6!
Catalan’s Constant
35.33.
1 1
1 1 π /2
K dk = ∫ ∫
∫
2 0
2 k =0 θ =0
1 1
1
dθ dk
= 2 − 2 + 2 − = .915965594 …
2
2
1
3
5
1 − k sin θ
Periods of Elliptic Functions
Let
35.34.
K=
∫
π /2
0
dθ
,
1 − k 2 sin 2 θ
K′ =
Then
35.35.
sn u has periods 4K and 2iK ′
35.36.
cn u has periods 4K and 2K + 2iK ′
35.37.
dn u has periods 2K and 4iK ′
∫
π /2
0
dθ
1 − k ′ 2 sin 2 θ
where k ′ = 1 − k 2
ELLIPTIC FUNCTIONS
201
Identities Involving Elliptic Functions
35.38.
sn 2 u + cn 2u = 1
35.40.
dn 2 u − k 2 cn 2 u = k ′ 2
35.42.
cn 2u =
35.44.
where k ′ = 1 − k 2
dn 2u + cn 2u
1 + dn 2u
1 − cn 2u sn u dn u
=
1 + cn 2 u
cn u
35.39.
dn 2 u + k 2 sn 2 u = 1
35.41.
sn 2u =
1 − cn 2u
1 + dn 2u
35.43.
dn 2u =
1 − k 2 + dn 2u + k 2cn u
1 + dn 2u
35.45.
1 − dn 2u k sn u cn u
=
1 + dn 2u
dn u
Special Values
35.46.
sn 0 = 0
35.47. cn 0 = 1
35.48. dn 0 = 1
Integrals
1
35.51.
∫ sn u du = k ln (dn u − k cn u)
35.52.
∫ cn u du = k cos
35.53.
∫ dn u du = sin
35.54.
∫ sc u du =
35.55.
∫ cs u du = ln (ns u − ds u)
35.56.
∫ cd u du = k ln (nd u + k sd u)
35.57.
∫ dc u du = ln (nc u + sc u)
35.58.
∫ sd u du = k
35.59.
∫ ds u du = ln (ns u − cs u)
35.60.
∫ ns u du = ln (ds u − cs u)
35.61.
∫ nc u du =
⎛
1
sc u ⎞
ln ⎜ dc u +
2
⎝
1− k
1 − k 2 ⎟⎠
35.62.
∫ nd u du =
1
cos −1 (cd u)
1 − k2
1
−1
−1
(dn u)
(sn u)
(
)
1
ln dc u + 1 − k 2 nc u
1 − k2
1
−1
sin −1 (k cd u)
1 − k2
35.49. sc 0 = 0
35.50. am 0 = 0
ELLIPTIC FUNCTIONS
202
Legendre’s Relation
35.63.
EK ′ + E ′K − KK ′ = π /2
where
35.64.
E=
35.65.
E′ =
∫
π /2
0
∫
π /2
0
1 − k 2 sin 2 θ dθ
1 − k ′ 2 sin 2 θ dθ
K=
∫
K′ =
∫
π /2
0
π /2
0
dθ
1 − k 2 sin 2 θ
dθ
1 − k ′ 2 sin 2 θ
MISCELLANEOUS and RIEMANN
ZETA FUNCTIONS
36
2
π
Error Function erf ( x) =
36.1.
∫
x
0
e− u du
2
x3
x5
x7
2 ⎛
⎞
⎜⎝ x − 3 i 1! + 5 i 2! − 7 i 3! + ⎟⎠
π
e− x ⎛
1
1i 3 1i 3 i 5
⎞
erf ( x ) ~ 1 −
⎜⎝1 − 2 x 2 + (2 x 2 )2 − (2 x 2 )3 + ⎟⎠
πx
erf ( x ) =
2
36.2.
36.3.
erf (− x ) = − erf ( x ),
erf (0) = 0,
erf (∞) = 1
Complementary Error Function erfc ( x) = 1 − erf ( x) =
36.4.
erfc ( x ) = 1 −
36.5.
erfc ( x ) ~
36.6.
erfc (0) = 1,
2
π
∫
∞
x
e− u du
2
x3
x5
x7
⎛
⎞
⎜⎝ x − 3 i 1! + 5 i 2! − 7 i 3! + ⎟⎠
2
π
e− x ⎛
1
1i 3
1i 3i 5
⎞
⎜1 − 2 + (2 x 2 )2 − (2 x 2 )3 + ⎟⎠
π x ⎝ 2x
2
erfc (∞) = 0
∞ −u
Exponential Integral Ei ( x) = ∫ e du
x u
36.7.
36.8.
36.9.
36.10.
Ei ( x ) = −γ − ln x + ∫
x
0
1 − e−u
du
u
x2
x3
⎛ x
⎞
Ei ( x ) = −γ − ln x + ⎜
−
+
− ⎟
⎝ 1 i 1! 2 i 2! 3 i 3!
⎠
e − x ⎛ 1! 2! 3!
⎞
Ei ( x ) ~
1 − + 2 − 3 + ⎟
x ⎜⎝
x x
x
⎠
Ei (∞) = 0
x
Sine Integral Si ( x) = ∫0
sin u
du
u
36.11.
Si ( x ) =
x
x3
x5
x7
−
+
−
+
1 i 1! 3 i 3! 5 i 5! 7 i 7!
36.12.
Si ( x ) ~
π sin x ⎛ 1 3! 5!
2! 4 !
⎞
⎞ cos x ⎛
+
− ⎟
−
−
+
− ⎟ −
1−
2
x ⎜⎝ x x 3 x 5
x ⎜⎝ x 2 x 4
⎠
⎠
36.13.
Si (− x ) = −Si ( x ),
Si (0) = 0,
Si (∞) = π / 2
203
MISCELLANEOUS AND RIEMANN ZETA FUNCTIONS
204
∞
Cosine Integral Ci( x) = ∫x
36.14.
Ci ( x ) = −γ − ln x + ∫
36.15.
Ci ( x ) = −γ − ln x +
36.16.
Ci ( x ) ~
36.17.
Ci (∞) = 0
x
0
cos u
du
u
1 − cos u
du
u
x2
x4
x6
x8
−
+
−
+
2 i 2! 4 i 4 ! 6 i 6! 8 i 8!
cos x ⎛ 1 3! 5!
2! 4 !
⎞ sin x ⎛
⎞
−
+
− ⎟ −
+
− ⎟
1−
x ⎜⎝ x x 3 x 5
x ⎜⎝ x 2 x 4
⎠
⎠
2
∫
Fresnel Sine Integral S( x) = π
36.18.
36.19.
36.20.
x
0
2 ⎛ x3
x7
x 11
x 15
⎞
−
+
−
+ ⎟
⎜
π ⎝ 3 i 1! 7 i 3! 11 i 5! 15 i 7!
⎠
i
i
i
i
1i 3i 5
1
1 ⎧
1
1
3
1
3
5
7
⎞
⎞⎫
2 ⎛
2 ⎛ 1
S(x) ~ −
⎨(cos x ) ⎜⎝ x − 22 x 5 + 24 x 9 − ⎟⎠ + (sin x ) ⎜⎝ 2 x 3 − 23 x 7 + ⎟⎠ ⎬
2
2π ⎩
⎭
S(x) =
S (− x ) = − S ( x ),
S(0) = 0,
S(∞) =
2
Fresnel Cosine Integral C( x) = π
36.21.
36.22.
36.23.
sin u2 du
∫
x
0
1
2
cos u2 du
x5
x9
x 13
2 ⎛x
⎞
−
+
−
+ ⎟
⎜
π ⎝ 1! 5 i 2! 9 i 4 ! 13 i 6!
⎠
1i 3i 5
1
1 ⎧
1i 3 1i 3i 5i 7
⎞
⎞⎫
2 ⎛1
2 ⎛ 1
C(x) ~ +
⎨(sin x ) ⎜⎝ x − 22 x 5 + 24 x 9 − ⎟⎠ − (cos x ) ⎜⎝ 2 x 3 − 23 x 7 + ⎟⎠ ⎬
2
2π ⎩
⎭
C(x) =
C (− x ) = −C ( x ),
C(0) = 0,
C(∞) =
1
1
1
2
1
Riemann Zeta Function ζ ( x) = 1x + 2 x + 3 x + ∞ u x −1
1
du,
Γ ( x ) ∫0 eu−1
36.24.
ζ (x) =
36.25.
ζ (1 − x ) = 21− x π − x Γ ( x ) cos(π x/ 2)ζ ( x ) (extension to other values)
36.26.
ζ (2 k ) =
x >1
22 k −1 π 2 k Bk
k = 1, 2, 3,…
(2k )!
Section X: Inequalities and Infinite Products
37
INEQUALITIES
Triangle Inequality
37.1.
|a1 | − | a2 | | a1 + a2 | | a1 | + | a2 |
37.2.
| a1 + a2 + + an | | a1 | + | a2 | + + | an |
Cauchy-Schwarz Inequality
37.3.
(a1b1 + a2 b2 + + an bn )2 (a12 + a22 + + an2 )(b12 + b22 + + bn2 )
The equality holds if and only if a1 /b1 = a2 /b2 = = an /bn .
Inequalities Involving Arithmetic, Geometric, and Harmonic Means
If A, G, and H are the arithmetic, geometric, and harmonic means of the positive numbers a1, a2, ..., an, then
37.4.
H G A
where
a1 + a2 + + an
n
37.5.
A=
37.6.
G = n a1 a2 … an
37.7.
1 1⎛1
1
1⎞
= ⎜ + ++ ⎟
H n ⎝ a1 a2
an ⎠
The equality holds if and only if a1 = a2 = = an .
Holder’s Inequality
37.8.
| a1b1 + a2b2 + + an bn | (| a1 | p+ | a2 | p + + | an | p)1/ p (| b1 |q + | b2 |q + + | bn |q)1/q
where
37.9.
1 1
+ =1
p q
p > 1, q > 1
The equality holds if and only if | a1 | p−1/| b1 | = | a2 | p−1/| b2 | = = | an | p−1/| bn | . For p = q = 2 it reduces to 37.3.
205
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INEQUALITIES
206
Chebyshev’s Inequality
If a1 a2 an and b1 b2 bn , then
37.10.
⎛ a1 + a2 + + an ⎞ ⎛ b1 + b2 + + bn ⎞ a1 b1 + a2 b2 + + an bn
⎟
⎟⎜
⎜
n
n
n
⎠
⎠⎝
⎝
or
37.11.
(a1 + a2 + + an )(b1 + b2 + + bn ) n(a1 b1 + a2 b2 + + an bn )
Minkowski’s Inequality
If a1, a2,…an, b1, b2,… bn are all positive and p > 1, then
37.12.
{(a1 + b1) p + (a2 + b2 ) p + + (an + bn ) p}1/ p (a1p + a2p + + anp )1/ p + (b1p + b2p + + bnp )1/ p
The equality holds if and only if a1 /b1 = a2 /b2 = = an /bn .
Cauchy-Schwarz Inequality for Integrals
2
{
}{
}
b
⎡ b f ( x )g( x ) dx⎤ b [ f ( x )]2 dx
∫a
∫a [g( x)]2 dx
⎢⎣∫a
⎥⎦
The equality holds if and only if f (x)/g(x) is a constant.
37.13.
Holder’s Inequality for Integrals
37.14.
∫
b
a
| f ( x )g( x )|dx {∫ | f (x)| dx} {∫ |g(x)| dx}
b
1/ p
1/q
b
p
q
a
a
where 1/p + 1/q = 1, p > 1, q >1. If p = q = 2, this reduces to 37.13.
The equality holds if and only if | f ( x )| p−1/| g( x )| is a constant.
Minkowski’s Inequality for Integrals
If p > 1,
37.15.
{∫ | f (x) + g(x)| dx} {∫ | f (x)| dx} + {∫ |g(x)| dx}
b
a
1/ p
p
b
a
1/ p
p
b
a
The equality holds if and only if f (x)/g(x) is a constant.
1/ p
p
38
INFINITE PRODUCTS
38.1.
⎛ x2 ⎞ ⎛
x2 ⎞ ⎛
x2 ⎞
sin x = x ⎜1 − 2 ⎟ ⎜1 − 2 ⎟ ⎜1 − 2 ⎟ ⎝ x ⎠ ⎝ 4π ⎠ ⎝ 9π ⎠
38.2.
⎛ 4x2 ⎞ ⎛ 4x2 ⎞ ⎛
4x2 ⎞
cos x = ⎜1 − 2 ⎟ ⎜1 − 2 ⎟ ⎜1 −
π ⎠ ⎝ 9π ⎠ ⎝ 25π 2 ⎟⎠
⎝
38.3.
⎛
x2 ⎞ ⎛
x2 ⎞ ⎛
x2 ⎞
sinh x = x ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎝ π ⎠ ⎝ 4π ⎠ ⎝ 9π ⎠
38.4.
⎛ 4x2 ⎞ ⎛ 4x2 ⎞ ⎛
4x2 ⎞
cosh x = ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 +
π ⎠ ⎝ 9π ⎠ ⎝ 25π 2 ⎟⎠
⎝
38.5.
1
⎫ ⎧⎛ x
⎫ ⎧⎛ x
⎫
⎧⎛ x
= xeγ x ⎨ 1 + ⎞ e − x ⎬ ⎨ 1 + ⎞ e − x/2 ⎬ ⎨ 1 + ⎞ e − x/3 ⎬
1
2
3
Γ( x )
⎝
⎠
⎝
⎠
⎝
⎠
⎭⎩
⎭⎩
⎭
⎩
See also 25.11.
⎛ x2 ⎞ ⎛ x2 ⎞ ⎛ x2 ⎞
J 0 ( x ) = ⎜1 − 2 ⎟ ⎜1 − 2 ⎟ ⎜1 − 2 ⎟ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠
where 1, 2, 3,… are the positive roots of J0(x) = 0.
38.6.
⎛ x2 ⎞ ⎛ x2 ⎞ ⎛ x2 ⎞
J1( x ) = x ⎜1 − 2 ⎟ ⎜1 − 2 ⎟ ⎜1 − 2 ⎟ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠
where 1, 2, 3,… are the positive roots of J1(x) = 0.
38.7.
38.8.
sin x
x
x
x
x
= cos cos cos cos 2
4
8
16
x
38.9.
π 2 2 4 4 6 6
= i i i i i i
2 1 3 3 5 5 7
This is called Wallis’ product.
207
Section XI: Probability and Statistics
39
DESCRIPTIVE STATISTICS
The numerical data x1, x2,… will either come from a random sample of a larger population or from the larger
population itself. We distinguish these two cases using different notation as follows:
n = number of items in a sample,
N = number of items in the population,
x = (read: x-bar) = sample mean,
s2 = sample variance,
s = sample standard deviation,
m (read: mu) = population mean,
s 2 = population variance,
s = population standard deviation
Note that Greek letters are used with the population and are called parameters, whereas Latin letters are
used with the samples and are called statistics. First we give formulas for the data coming from a sample.
This is followed by formulas for the population.
Grouped Data
Frequently, the sample data are collected into groups (grouped data). A group refers to a set of numbers
all with the same value xi, or a set (class) of numbers in a given interval with class value xi. In such a case, we
assume there are k groups with fi denoting the number of elements in the group with value or class value xi.
Thus, the total number of data items is
39.1.
n = ∑ fi
As usual, Σ will denote a summation over all the values of the index, unless otherwise specified.
Accordingly, some of the formulas will be designated as (a) or as (b), where (a) indicates ungrouped data
and (b) indicates grouped data.
Measures of Central Tendency
Mean (Arithmetic Mean)
The arithmetic mean or simply mean of a sample x1, x2,…, xn, frequently called the “average value,” is the
sum of the values divided by the number of values. That is:
39.2(a). Sample mean:
39.2(b). Sample mean:
x=
x=
x1 + x 2 + + x n Σ xi
=
n
n
f1 x1 + f2 x 2 + + fk x k Σ fi xi
=
f1 + f2 + + fk
Σ fi
Median
Suppose that the data x1, x2,…, xn are now sorted in increasing order. The median of the data, denoted by
M or Median
is defined to be the “middle value.” That is:
208
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DESCRIPTIVE STATISTICS
209
when n is odd and n = 2k + 1,
⎧x k +1
⎪
39.3(a). Median = ⎨ x + x
k +1
⎪ k
when n is even and n = 2k.
⎩ 2
The median of grouped data is obtained by first finding the cumulative frequency function Fs. Specifically,
we define
Fs = f1 + f2 + + fs
that is, Fs is the sum of the frequencies up to fs. Then:
39.3(b.1).
⎧x j +1
⎪
Median = ⎨ x + x
j +1
⎪ j
⎩ 2
when n = 2k + 1 (odd) and Fj < k + 1 ≤ Fj +1
when n = 2k (even), and Fj = k.
Finding the median of data arranged in classes is more complicated. First one finds the median class m,
the class with the median value, and then one linearly interpolates in the class using the formula
39.3(b.2).
Median = Lm + c
(n/2) − Fm −1
fm
where Lm denotes the lower class boundary of the median class and c denotes its class width (length of the
class interval).
Mode
The mode is the value or values which occur most often. Namely:
39.4. Mode xm = numerical value that occurs the most number of times
The mode is not defined if every xm occurs the same number of times, and when the mode is defined it
may not be unique.
Weighted and grand means
Suppose that each xi is assigned a weight wi ≥ 0. Then:
39.5. Weighted Mean x w =
w1x1 + w2 x 2 + + wk x k Σ wi xi
=
Σ wi
w1 + w2 + + wk
Note that 39.2(b.1) is a special case of 39.4 where the weight wi of xi is its frequency.
Suppose that there are k sample sets and that each sample set has ni elements and a mean x. Then the
grand mean, denoted by xi is the “mean of the means” where each mean is weighted by the number of elements in its sample. Specifically:
39.6. Grand Mean x =
n1 x1 + n2 x 2 + + nk x k Σ ni xi
=
n1 + n2 + + nk
Σ ni
Geometric and Harmonic Means
The geometric mean (G.M.) and harmonic mean (H.M.) are defined as follows:
39.7(a). G.M. = n x1 x 2 x n
f
f
f
39.7(b). G.M. = n x1 x 2 x k
1
2
k
DESCRIPTIVE STATISTICS
210
39.8(a).
H.M. =
39.8(b).
H.M. =
n
n
=
1/x1 + 1/x 2 + + 1/x n Σ (1/xi )
n
n
=
f1 /x1 + f2 /x 2 + + fk /x k Σ ( fk /xi )
Relation Between Arithmetic, Geometric, and Harmonic Means
39.9. H.M. ≤ G.M. ≤ x
The equality sign holds only when all the sample values are equal.
Midrange
The midrange is the average of the smallest value x1 and the largest value xn. That is:
39.10. midrange: mid =
x1 + x n
2
Population Mean
The formula for the population mean m follows:
39.11(a). Population mean:
x1 + x 2 + + x N Σ xi
=
N
N
μ=
39.11(b). Population mean: μ =
f1 x1 + f2 x 2 + + fk x k Σ fi xi
=
f1 + f2 + + fk
Σ fi
(Recall that N denotes the number of elements in a population.)
Observe that the formula for the population mean m is the same as the formula for the sample mean x.
On the other hand, the formula for the population standard deviation s is not the same as the formula for the
sample standard deviation s. (This is the main reason we give separate formulas for m and x. )
Measures of Dispersion
Sample Variance and Standard Deviation
Here the sample set has n elements with mean x.
39.12(a). Sample variance:
s2 =
Σ( xi − x )2 Σ xi2 − (Σ xi )2 /n
=
n −1
n −1
39.12(b). Sample variance:
s2 =
Σfi ( xi − x )2 Σfi xi2 − (Σ fi xi )2 / Σ fi
=
( Σ fi ) − 1
( Σ fi ) − 1
s = Variance = s 2
39.13. Sample standard deviation:
EXAMPLE 39.1:
Consider the following frequency distribution:
xi
1
2
3
4
5
6
fi
8
14
7
12
3
1
Then n = Σ fi = 45 and Σ fi xi = 126. Hence, by 39.2(b),
Mean x =
Σ xi fi 126
=
= 2.8
Σ fi
45
DESCRIPTIVE STATISTICS
211
Also, n – 1 = 44 and Σ fi xi2 = 430. Hence, by 39.12(b) and 39.13,
s2 =
430 − (126)2 /45
≈ 1.75 and s = 1.32
44
We find the median M, first finding the cumulative frequencies:
F1 = 8,
F2 = 22,
F3 = 29,
F4 = 41,
F5 = 44,
F6 = 45 = n
Here n is odd, and (n + 1)/2 = 23. Hence,
Median M = 23rd value = 3
The value 2 occurs most often, hence
Mode = 2
M.D. and R.M.S.
Here M.D. stands for mean deviation and R.M.S. stands for root mean square. As previously, x is the
mean of the data and, for grouped data, n = Σ fi.
1
x −x
n i
39.14(a).
M.D. =
39.15(a).
R.M.S. =
1
(Σ xi2 )
n
1
f x −x
n i i
39.14(b).
M.D. =
39.15(b).
R.M.S. =
1
(Σ fi xi2 )
n
Measures of Position (Quartiles and Percentiles)
Now we assume that the data x1, x2,…, xn are arranged in increasing order.
39.16. Sample range: xn – x1.
There are three quartiles: the first or lower quartile, denoted by Q1 or QL; the second quartile or median,
denoted by Q2 or M; and the third or upper quartile, denoted by Q3 or QU. These quartiles (which essentially
divide the data into “quarters”) are defined as follows, where “half” means n/2 when n is even and (n-1)/2
when n is odd:
39.17. QL(= Q1) = median of the first half of the values.
M (= Q2 ) = median of the values.
QU (= Q3) = median of the second half of the values.
39.18. Five-number summary: [L, QL, M, QU, H] where L = x1 (lowest value) and H = xn (highest value).
39.19. Innerquartile range: QU – QL
39.20. Semi-innerquartile range:
Q=
QU − QL
2
The kth percentile, denoted by Pk, is the number for which k percent of the values are at most Pk and
(100–k) percent of the values are greater than Pk. Specifically:
39.21. Pk = largest xs such that Fs ≤ k/100. Thus, QL = 25th percentile, M = 50th percentile,
QU = 75th percentile.
DESCRIPTIVE STATISTICS
212
Higher-Order Statistics
39.22. The rth moment: (a) mr =
1
Σ xir ,
n
(b) mr =
1
Σ f xr
n i i
39.23. The rth moment about the mean x:
(a)
μr =
1
Σ ( x i − x )r ,
n
(b)
μr =
1
Σ ( fi x i − x ) r
n
μr =
1
Σ fi x i − x
n
39.24. The rth absolute moment about mean x:
(a)
μr =
r
1
Σ xi − x ,
n
(b)
r
39.25. The rth moment in standard z units about z = 0:
αr =
1 r
Σz ,
n i
x −x
1
Σ f z r where zi = i
σ
n i i
(b)
αr =
39.26. Coefficient of skewness:
γ1 =
μ3
= α3
σ3
39.27. Momental skewness:
μ3
2σ 3
39.28. Coefficient of kurtosis:
α4 =
39.29. Coefficient of excess (kurtosis):
α4 − 3 =
39.30. Quartile coefficient of skewness:
QU − 2 xˆ + QL Q3 − 2Q2 + Q1
=
QU − QL
Q3 − Q1
(a)
Measures of Skewness and Kurtosis
μ4
σ4
μ4
−3
σ4
Population Variance and Standard Deviation
Recall that N denotes the number of values in the population.
39.31. Population variance: σ 2 =
Σ ( xi − x )2 Σ xi2 − (Σ xi )2 /n
=
N
N
39.32. Population standard deviation: σ = Variance = σ 2
Bivariate Data
The following formulas apply to a list of pairs of numerical values:
( x1, y1), ( x 2 , y2 ), ( x3, y3),… , ( x n , yn )
where the first values correspond to a variable x and the second to a variable y. The primary objective is to
determine whether there is a mathematical relationship, such as a linear relationship, between the data.
The scatterplot of the data is simply a picture of the pairs of values as points in a coordinate plane.
DESCRIPTIVE STATISTICS
213
Correlation Coefficient
A numerical indicator of a linear relationship between variables x and y is the sample correlation coefficient r of x and y, defined as follows:
r=
39.33. Sample correlation coefficient:
Σ ( xi − x )( yi − y )
Σ ( xi − x )2 Σ ( yi − y )2
We assume that the denominator in Formula 39.33 is not zero. An alternative formula for computing
r follows:
39.34.
r=
Σ xi yi − (Σ xi )(Σ yi)/n
Σ xi2 − (Σ xi )2 /n Σ yi2 − (Σ yi)2 /n
Properties of the correlation coefficient r follow:
39.35. (1) –1 r 1 or, equivalently, r 1.
(2) r is positive or negative according as y tends to increase or decrease as x increases.
(3) The closer |r| is to 1, the stronger the linear relationship between x and y.
The sample covariance of x and y is denoted and defined as follows:
sxy =
39.36. Sample covariance:
Σ ( xi − x )( yi − y )
n −1
Using the sample covariance, Formula 39.33 can be written in the compact form:
39.37.
r=
sxy
sx s y
where sx and sy are the sample standard deviations of x and y, respectively.
EXAMPLE 39.2:
Consider the following data:
x
y
50
2.5
45
5.0
40
6.2
38
7.4
32
8.3
40
4.7
55
1.8
The scatterplot of the data appears in Fig. 39-1. The correlation coefficient r for the data may be obtained
by first constructing the table in Fig. 39-2. Then, by Formula 39.34 with n = 7,
r=
1431.8 − (300)(35.9) / 7
≈ −0.9562
13, 218 − (300)2 / 7 218.67 + (35.9)2 / 7
Here r is close to –1, and the scatterplot in Fig. 39-1 does indicate a strong negative linear relationship
between x and y.
DESCRIPTIVE STATISTICS
214
Fig. 39-1
Fig. 39-2
Regression Line
Consider a given set of n data points Pi (xi, yi). Any (nonvertical) line L may be defined by an equation
of the form
y = a + bx
Let yi * denote the y value of the point on L corresponding to xi; that is, let yi* = a + bxi . Now let
di = yi − yi* = yi − (a + bxi )
that is, di is the vertical (directed) distance between the point Pi and the line L. The squares error between
the line L and the data points is defined by
39.38.
Σ di2 = d12 + d 22 + + d n2
The least-squares line or the line of best fit or the regression line of y on x is, by definition, the line L
whose squares error is as small as possible. It can be shown that such a line L exists and is unique.
The constants a and b in the equation y = a + bx of the line L of best fit can be obtained from the following
two normal equations, where a and b are the unknowns and n is the number of points:
39.39.
⎧⎪
na + (Σ xi ) b = Σ yi
⎨
⎪⎩(Σ xi )a + (Σ xi2 )b = Σ xi yi
The solution of the above normal equations follows:
39.40.
b=
n Σ xi yi − (Σ xi )(Σ yi ) rs y
;
=
n Σ xi2 − (Σ xi )2
sx
a=
Σ yi
Σ xi
−b
= y − bx
n
n
The second equation tells us that the point ( x , y ) lies on L, and the first equation tells us that the point
( x + sx , y + rs y ) also lies on L.
EXAMPLE 39.3: Suppose we want the line L of best fit for the data in Example 39.2. Using the table in Fig. 39-2 and
n = 7, we obtain the normal equations
7a + 300 b = 35.9
300 a + 13, 218b = 1431.8
Substitution in 39.40 yields
b=
7(1431.8) − (300)(35.9)
= − 0.2
2959
7(13, 218) − (300)2
a=
35.9
300
− (−0.2959)
= 17.8100
7
7
DESCRIPTIVE STATISTICS
215
Thus, the line L of best fit is
y = 17.8100 – 0.2959x
The graph of L appears in Fig. 39-3.
Fig. 39-3
Curve Fitting
Suppose that n data points Pi (xi, yi) are given, and that the data (using the scatterplot or the correlation
coefficient r) do not indicate a linear relationship between the variables x and y, but do indicate that some
other standard (well-known) type of curve y = f (x) approximates the data. Then the particular curve C that
one uses to approximate that data, called the best-fitting or least-squares curve, is the curve in the collection
which minimizes the squares error sum
Σ di2 = d12 + d 22 + + d n2
where di = yi – f(xi). Three such types of curve are discussed as follows.
Polynomial function of degree m: y = a0 + a1 x + a2 x 2 + + am x m
The coefficients a0 , a1, a2 ,… , am of the best-fitting polynomial can be obtained by solving the following
system of m + 1 normal equations:
na0 + a1Σ xi + a2 Σ xi2 + + am Σ xim = Σ yi
39.41.
a0 Σ xi + a1 Σ xi2 + a2 Σ xi3 + + am Σ xim +1 = Σ xi yi
....................................................................................
a0 Σ xim + a1 Σ xim +1 + a2 Σ xim + 2 + + am Σ xi2 m = Σ xim yi
Exponential curve:
y = ab x or log y = log a + (log b) x
The exponential curve is used if the scatterplot of log y verses x indicates a linear relationship. Then log a and
log b are obtained from transformed data points. Namely, the best-fit line L for data points P′(xi, log yi) is
39.42.
⎧⎪
na ′ + (Σ xi ) b ′ = Σ (log yi )
⎨
⎪⎩(Σ xi ) a ′ + (Σ xi2 ) b ′ = Σ ( xi log yi )
Then a = antilog a′, b = antilog b′.
EXAMPLE 39.4:
Consider the following data which indicates exponential growth:
x
y
1
6
2
18
3
55
4
160
5
485
6
1460
DESCRIPTIVE STATISTICS
216
Thus, we seek the least-squares line L for the following data:
x
log y
1
0.7782
2
1.2553
3
1.7404
4
2.2041
5
2.6857
6
3.1644
Using the normal equation 39.42 for L, we get
a ′ = 0.3028, b ′ = 0.4767
The antiderivatives of a′ and b′ yield, approximately,
a = 2.0,
b = 3.0
Hence, y = 2(3 ) is the required exponential curve C. The data points and C are depicted in Fig. 39-4.
x
Fig. 39-4
Power function: y = axb or log y = log a + b log x
The power curve is used if the scatterplot of log y verses log x indicates a linear relationship. The
log a and b are obtained from transformed data points. Namely, the best-fit line L for transformed data
points P′(log xi, log yi) is
39.43.
⎧⎪
na ′ + Σ (log xi )b = Σ (log yi )
⎨
⎪⎩Σ (log xi )a ′ + Σ (log xi )2 b = Σ (log xi log yi )
Then a = antilog a′.
40
PROBABILITY
Sample Spaces and Events
Let S be a sample space which consists of the possible outcomes of an experiment where the events are
subsets of S. The sample space S itself is called the certain event, and the null set ∅ is called the impossible
event.
It would be convenient if all subsets of S could be events. Unfortunately, this may lead to contradictions
when a probability function is defined on the events. Thus, the events are defined to be a limited collection
C of subsets of S as follows.
DEFINITION 40.1:
properties:
The class C of events of a sample space S form a σ-field. That is, C has the following three
(i) S ∈ C.
(ii) If A1, A2, … belong to C, then their union A1 ∪ A2 ∪ A3 ∪ … belongs to C.
(iii) If A ∈ C, then its complement Ac ∈ C.
Although the above definition does not mention intersections, DeMorgan’s law (40.3) tells us that the
complement of a union is the intersection of the complements. Thus, the events form a collection that is
closed under unions, intersections, and complements of denumerable sequences.
If S is finite, then the class of all subsets of S form a σ-field. However, if S is nondenumerable, then only
certain subsets of S can be the events. In fact, if B is the collection of all open intervals on the real line R,
then the smallest σ-field containing B is the collection of Borel sets in R.
If Condition (ii) in Definition 40.1 of a σ-field is replaced by finite unions, then the class of subsets of S
is called a field. Thus a σ-field is a field, but not visa versa.
First, for completeness, we list basic properties of the set operations of union, intersection, and complement.
40.1.
Sets satisfy the properties in Table 40-1.
TABLE 40-1
Laws of the Algebra of Sets
Idempotent laws:
(1a) A ∪ A = A
Associative laws:
(2a)
(A ∪ B) ∪ C = A ∪ (B ∪ C)
Commutative laws: (3a) A ∪ B = B ∪ A
(1b) A ∩ A = A
(2b) (A ∩ B) ∩ C = A ∩ (B ∩ C)
(3b) A ∩ B = B ∩ A
Distributive laws: (4a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (4b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ B)
Identity laws:
(5a) A ∪ ∅ = A
(6a) A ∪ U = U
Involution law:
(7)
(5b) A ∩ U = A
(6b) A ∩ ∅ = ∅
(AC)C = A
Complement laws: (8a) A ∪ Ac = U
(9a) Uc = ∅
(8b) A ∩ Ac = ∅
(9b) ∅c = U
DeMorgan’s laws: (10a) (A ∪ B)c = Ac ∩ Bc
(10b) (A ∩ B)c = Ac ∪ Bc
217
PROBABILITY
218
40.2. The following are equivalent: (i) A ⊆ B, (ii) A ∩ B = A, (iii) A ∩ B = B.
Recall that the union and intersection of any collection of sets is defined as follows:
∪j Aj = {x | there exists j such that x ∈ Aj}
40.3.
and
∩j Aj = {x | for every j we have x ∈ Aj}
(Generalized DeMorgan’s Law) (10a)'(∪j Aj)c = ∩j Ajc; (10b)'(∩j Aj)c = ∪j Ajc
Probability Spaces and Probability Functions
DEFINITION 40.2: Let P be a real-valued function defined on the class C of events of a sample space S. Then P is
called a probability function, and P(A) is called the probability of an event A, when the following axioms hold:
Axiom [P1] For every event A, P(A) ≥ 0.
Axiom [P2] For the certain event S, P(S) = 1.
Axiom [P3] For any sequence of mutually exclusive (disjoint) events A1, A2, …,
P(A1 ∪ A2 ∪ …) = P(A1) + P(A2) + …
The triple (S, C, P), or simply S when C and P are understood, is called a probability space.
Axiom [P3] implies an analogous axiom for any finite number of sets. That is:
Axiom [P3'] For any finite collection of mutually exclusive events A1, A2, …, An,
P(A1 ∪ A2 ∪ … ∪ An) = P(A1) + P(A2) + … + P(An)
In particular, for two disjoint events A and B, we have P(A ∪ B ) = P(A) + P(B).
The following properties follow directly from the above axioms.
40.4.
(Complement rule) P(Ac) = 1 – P(A). Thus, P(∅) = 0.
40.5.
(Difference Rule) P(A\B) = P(A) – P(A ∩ B).
40.6.
(Addition Rule) P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
40.7. For n ≥ 2, P
40.8.
(∪
n
j =1
)
Aj ≤
∑
n
j =1
P( Aj )
(Monoticity Rule) If A ⊆ B, then P(A) ≤ P(B).
Limits of Sequences of Events
40.9. (Continuity) Suppose A1, A2, … form a monotonic increasing (decreasing) sequence of events; that is,
Aj ⊆ Aj+1 (Aj ⊇ Aj+1). Let A = ∪ j Aj (A = ∩ j Aj). Then lim P(An) exists and
lim P(An) = P(A)
For any sequence of events A1, A2, …, we define
lim inf An =
∪ ∩
+∞
+∞
k=1
j=k
Aj
and
lim sup An =
∩ ∪
+∞
+∞
k=1
j=k
Aj
If lim inf An = lim sup An, then we call this set lim An. Note lim An exists when the sequence is
monotonic.
PROBABILITY
40.10.
219
For any sequence Aj of events in a probability space,
P(lim inf An) ≤ lim inf P(An) ≤ lim sup P(An) ≤ P(lim sup An)
Thus, if lim An exists, then P(lim An) = lim P(An).
40.11.
For any sequence Aj of events in a probability space, P(∪j Aj) ≤
∑
j
P(Aj).
40.12. (Borel-Cantelli Lemma) Suppose Aj is any sequence of events in a probability space. Furthermore,
+∞
suppose ∑ n=1 P(An) < +∞. Then P(lim sup An) = 0.
40.13. (Extension Theorem) Let F be a field of subsets of S. Let P be a function on F satisfying Axioms P1,
P2, and P3¢. Then there exists a unique probability function P* on the smallest σ-field containing F
such that P* is equal to P on F.
Conditional Probability
DEFINITION 40.3: Let E be an event with P(E) > 0. The conditional probability of an event A given E is denoted and
defined as follows:
P(A|E) =
40.14.
P( A ∩ E )
P(E )
(Multiplication Theorem for Conditional Probability) P(A ∩ B) = P(A)P(B|A). This theorem can
be genealized as follows:
40.15. P(A1 ∩ … ∩ An) = P(A1)P(A2|A1)P(A3|A1 ∩ A2) … P(An|A1 ∩ … ∩ An-1)
EXAMPLE 40.1: A lot contains 12 items of which 4 are defective. Three items are drawn at random from the lot one
after the other. Find the probabiliy that all three are nondefective.
The probability that the first item is nondefective is 8/12. Assuming the first item is nondefective, the
probability that the second item is nondefective is 7/11. Assuming the first and second items are nondefective, the probability that the third item is nondefective is 6/10. Thus,
p=
8 7 6 14
⋅ ⋅ =
12 11 10 55
Stochastic Processes and Probability Tree Diagrams
A (finite) stochastic process is a finite sequence of experiments where each experiment has a finite number of outcomes with given probabilities. A convenient way of describing such a process is by means of a
probability tree diagram, illustrated below, where the multiplication theorem (40.14) is used to compute the
probability of an event which is represented by a given path of the tree.
EXAMPLE 40.2: Let X, Y, Z be three coins in a box where X is a fair coin, Y is two-headed, and Z is weighted so the
probability of heads is 1/3. A coin is selected at random and is tossed. (a) Find P(H), the probability that heads appears.
(b) Find P(X|H), the probability that the fair coin X was picked if heads appears.
PROBABILITY
220
The probability tree diagram corresponding to the two-step stochastic process appears in Fig. 40-1a.
(a) Heads appears on three of the paths (from left to right); hence,
P(H) =
1 1 1
1 1 11
⋅ + ⋅1 + ⋅ =
3 2 3
3 3 18
(b) X and heads H appear only along the top path; hence
P(X ∩ H) =
1 1 1
P( X ∩ H )
1/ 6
3
⋅ = and so P(X|H) =
=
=
3 2 6
P( H )
11 / 18 11
1/2
3%
H
A
X
1/2
1/3
1/3
o
D
50%
T
30%
1
H
Y
N
o
4%
D
B
N
1/3
1/3
H
2/3
T
20%
Z
5%
D
C
N
(b)
(a)
Fig. 40-1
Law of Total Probability and Bayes’ Theorem
Here we assume E is an event in a sample space S, and A1, A2, … An are mutually disjoint events whose
union is S; that is, the events A1, A2, …, An form a partition of S.
40.16.
(Law of Total Probability) P(E) = P(A1)P(E|A1) + P(A2)P(E|A2) + … + P(An)P(E|An)
40.17. (Bayes’ Formula) For k = 1, 2, …, n,
P(Ak|E) =
P( Ak )P(E | Ak )
P( Ak )P(E | Ak )
=
P(E )
P( A1 )P(E | A1 ) + P( A2 )P(E | A2 ) + ⋅ ⋅ ⋅ + P( An )P(E | An )
EXAMPLE 40.3: Three machines, A, B, C, produce, respectively, 50%, 30%, and 20% of the total number of items
in a factory. The percentages of defective output of these machines are, respectively, 3%, 4%, and 5%. An item is randomly selected.
(a) Find P(D), the probability the item is defective.
(b) If the item is defective, find the probability it came from machine: (i) A, (ii) B, (iii) C.
(a) By 40.16 (Total Probability Law),
P(D) = P(A)P(D|A) + P(B)P(D|B) + P(C)P(D|C)
= (0.50)(0.03) + (0.30)(0.04) + (0.20)(0.05) = 3.7%
P( A)P( D | A) (0.50)(0.03)
(b) By 40.17 (Bayes’ rule), (i) P(A|D) =
=
= 40.5%. Similarly,
0.037
P( D)
P ( B ) P ( D | B)
P(C )P( D | C )
= 32.5%; (iii) P(C|D) =
= 27.0%
(ii) P(B|D) =
P( D)
P( D)
PROBABILITY
221
Alternately, we may consider this problem as a two-step stochastic process with a probability tree diagram, as in Fig. 40-1(b). We find P(D) by adding the three probability paths to D:
(0.50)(0.03) + (0.30)(0.04) + (0.20)(0.05) = 3.7%
We find P(A|D) by dividing the top path to A and D by the sum of the three paths to D.
(0.50)(0.03)/0.037 = 40.5%
Similarly, we find P(B|D) = 32.5% and P(C|D) = 27.0%.
Independent Events
DEFINITION 40.4:
Events A and B are independent if P(A ∩ B) = P(A)P(B).
40.18. The following are equivalent:
(i) P(A ∩ B) = P(A)P(B), (ii) P(A|B) = P(A), (iii) P(B|A) = P(B).
That is, events A and B are independent if the occurrence of one of them does not influence the occurrence of the other.
EXAMPLE 40.4: Consider the following events for a family with children where we assume the sample space S is an
equiprobable space:
E = {children of both sexes},
F = {at most one boy}
(a) Show that E and F are independent events if a family has three children.
(b) Show that E and F are dependent events if a family has two children.
(a) Here S = {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}. So:
E = {bbg, bgb, bgg, gbb, gbg, ggb}, P(E) = 6/8 = 3/4,
F = {bgg, gbg, ggb, ggg}, P(F) = 4/8 = 1/2
E ∩ F = {bgg, gbg, ggb}, P(E ∩ F) = 3/8
Therefore, P(E)P(F) = (3/4)(1/2) = 3/8 = P(E ∩ F). Hence, E and F are independent.
(b) Here S = {bb, bg, gb, gg}. So:
E = {bg, gb}, P(E) = 2/4 = 1/2,
F = {bg, gb, gg}, P(F) = 3/4
E ∩ F = {bg, gb}, P(E ∩ F) = 2/4 = 1/2
Therefore, P(E)P(F) = (1/2)(3/4) = 3/8 ≠ P(E ∩ F). Hence, E and F are dependent.
DEFINITION 40.5:
and
For n > 2, the events A1, A2, …, An are independent if any proper subset of them is independent
P(A1 ∩ A2 ∩ … ∩ An) = P(A1)P(A2) … P(An)
Observe that induction is used in this definition.
DEFINITION 40.6: A collection {Aj | j ∈ J} of events is independent if, for any n > 0, the sets Aj , Aj , …, Aj are inn
1
2
dependent.
The concept of independent repeated trials, when S is a finite set, is formalized as follows.
PROBABILITY
222
DEFINITION 40.7: Let S be a finite probability space. The probability space of n independent trials or repeated trials,
denoted by Sn, consists of ordered n-tuples (s1, s2, …, sn) of elements of S with the probability of an n-tuple defined by
P((s1, s2, …, sn)) = P(s1)P(s2) … P(sn)
EXAMPLE 40.5: Suppose whenever horses a, b, c race together, their respective probabilities of winning are 20%,
30%, and 50%. That is, S = {a, b, c} with P(a) = 0.2, P(b) = 0.3, and P(c) = 0.5.
They race three times. Find the probability that
(a) the same horse wins all three times
(b) each horse wins once
(a) Writing xyz for (x, y, z), we seek the probability of the event A = {aaa, bbb, ccc}. Here,
P(aaa) = (0.2)3 = 0.008, P(bbb) = (0.3)3 = 0.027, P(ccc) = (0.5)3 = 0.125
Thus, P(A) = 0.008 + 0.027 + 0.125 = 0.160.
(b) We seek the probability of the event B = {abc, acb, bac, bca, cab, cba}. Each element in B has the same
probability (0.2)(0.3)(0.5) = 0.03. Thus, P(B) = 6(0.03) = 0.18.
41
RANDOM VARIABLES
Consider a probability space (S, C, P).
DEFINITION 41.1. A random variable X on the sample space S is a function from S into the set R of real numbers
such that the preimage of every interval of R is an event of S.
If S is a discrete sample space in which every subset of S is an event, then every real-valued function on
S is a random variable. On the other hand, if S is uncountable, then certain real-valued functions on S may
not be random variables.
Let X be a random variable on S, where we let RX denote the range of X; that is,
RX = {x | there exists s ∈S for which X(s) = x}
There are two cases that we treat separately. (i) X is a discrete random variable; that is, RX is finite or
countable. (ii) X is a continuous random variable; that is, RX is a continuum of numbers such as an interval
or a union of intervals.
Let X and Y be random variables on the same sample space S. Then, as usual, X + Y, X + k, kX, and XY
(where k is a real number) are the functions on S defined as follows (where s is any point in S):
(X + Y)(s) = X(s) + Y(s),
(X + k)(s) = X(s) + k,
(kX)(s) = kX(s),
(XY)(s) = X(s)Y(s).
More generally, for any polynomial, exponential, or continuous function h(t), we define h(X) to be the
function on S defined by
[h(X)](s) = h[X(s)]
One can show that these are also random variables on S.
The following short notation is used:
P(X = xi)
P(a ≤ X ≤ b
μ X or E(X) or simply μ
σ X or Var(X) or simply σ 2
σ X or simply σ
denotes the probability that X = xi.
denotes the probability that X lies in the closed interval [a, b].
denotes the mean or expectation of X.
denotes the variance of X.
denotes the standard deviation of X.
2
Sometimes we let Y be a random variable such that Y = g(X), that is, where Y is some function of X.
Discrete Random Variables
Here X is a random variable with only a finite or countable number of values, say
RX = {x1, x2, x3, …}where, say, x1 < x2, < x3 < …. Then X induces a function f(x) on RX as follows:
f(xi) = P(X = xi) = P({s ∈S | X(s) = xi})
The function f(x) has the following properties:
(i) f(xi) ≥ 0
and
(ii) Σ i f(xi) = 1
Thus, f defines a probability function on the range RX of X. The pair (xi, f(xi)), usually given by a table,
is called the probability distribution or probability mass function of X.
223
RANDOM VARIABLES
224
Mean
41.1.
μ X = E(X) = Σxif(xi)
Here, Y = g(X).
41.2.
μY = E(Y) = Σg(xi) f(xi)
Variance and Standard Deviation
41.3.
σ X 2 = Var(X) = Σ(xi – m)2 f(xi) = E((X – m)2)
Alternately, Var(X) = s 2 may be obtained as follows:
41.4. Var(X) = Sxi2f(xi) – m2 = E(X2) – m2
41.5. σX =
Var ( X ) =
E(X 2 ) − μ 2
REMARK: Both the variance Var(X) = s 2 and the standard deviation s measure the weighted spread of the values xi
about the mean m; however, the standard deviation has the same units as m.
EXAMPLE 41.1:
Suppose X has the following probability distribution:
x
2
f(x)
0.1
4
6
8
0.2
0.3
0.4
Then:
m = E(X) = Σxif(xi) = 2(0.1) + 4(0.2) + 6(0.3) + 8(0.4) = 6
E(X2) = Σxi2 f(xi) = 22(0.1) + 42(0.2) + 62(0.3) + 82(0.4) = 40
s 2 = Var(X) = E(X2) − m2 = 40 − 36 = 4
s=
Var ( X ) =
4 =2
Continuous Random Variable
Here X is a random variable with a continuum number of values. Then X determines a function f(x), called
the density function of X, such that
(i) f(x) ≥ 0
and
(ii)
∫
∞
−∞
f(x) dx =
Furthermore,
P(a ≤ X ≤ b) =
Mean
41.6.
μ X = E(X) =
∫
∞
−∞
xf(x) dx
Here, Y = g(X).
41.7.
μY = E(Y) =
∫
∞
−∞
g(x) f(x) dx
∫
b
a
f(x) dx
∫
R
f ( x ) dx = 1
RANDOM VARIABLES
225
Variance and Standard Deviation
∞
41.8. σ X 2 = Var(X) = ∫ (x − m)2 f(x)dx = E((X − m)2)
−∞
Alternately, Var(X) = s 2 may be obtained as follows:
41.9. Var(X) =
41.10. s X =
∫
∞
−∞
x2f(x)dx − m2 = E(X2) − m2
Var ( X ) =
E(X 2 ) − μ 2
Let X be the continuous random variable with the following density function:
EXAMPLE 41.2:
⎧(1 / 2) x
f(x) = ⎨
⎩ 0
if 0 ≤ x ≤ 2
elsewhere
Then:
E(X) = ∫
E(X2) =
∞
−∞
∫
∞
−∞
2
xf(x) dx =
∫
x2f(x) dx =
⎡x3 ⎤
1 2
4
x dx = ⎢ 6 ⎥ =
2
3
⎣ ⎦0
2
0
∫
2
0
1 3
x dx =
2
s 2 = Var(X) = E(X2) − m2 = 2 −
s=
Var ( X ) =
2
⎡x4 ⎤
⎢⎣ 8 ⎥⎦ = 2
0
16 2
=
9
9
2
1
2
=
9
3
Cumulative Distribution Function
The cumulative distribution function F(x) of a random variable X is the function F:R → R defined by
41.11. F(a) = P(X ≤ a)
The function F is well-defined since the inverse of the interval (−∞, a] is an event.
The function F(x) has the following properties:
41.12.
41.13.
F(a) ≤ F(b) whenever a ≤ b.
lim F(x) = 0
x→−∞
and
lim F(x) = 1
x→+∞
That is, F(x) is monotonic, and the limit of F to the left is 0 and to the right is 1.
If X is the discrete random variable with distribution f(x), then F(x) is the following step function:
41.14. F(x) =
∑ f(xi)
xi ≤ x
If X is a continuous random variable, then the density funcion f(x) of X can be obtained from the cummulative distribution function F(x) by differentiation. That is,
d
F(x) = F′(x)
dx
Accordingly, for a continuous random variable X,
41.15. f(x) =
41.16.
F(x) =
∫
x
−∞
f(t) dt
RANDOM VARIABLES
226
Standardized Random Variable
The standardized random variable Z of a random variable X with mean m and standard deviation s > 0 is
defined by
X−μ
σ
Properties of such a standardized random variable Z follow:
41.17.
Z=
μ Z = E(Z) = 0
σZ = 1
and
Consider the random variable X in Example 41.1 where μ X = 6 and σ X = 2.
EXAMPLE 41.3:
The distribution of Z = (X – 6)/2 where f(z) = f(x) follows:
Z
−2
−1
0
1
f(Z)
0.1
0.2
0.3
0.4
Then:
E(Z) = Σ zif(zi) = (−2)(0.1) + (−1)(0.2) + 0(0.3) + 1(0.4) = 0
E(Z2) = Σ zi2 f(zi) = (−2)2(0.1) + (−1)2(0.2) + 02(0.3) + 12(0.4) = 1
Var(Z) = 1 − 02 = 1
and
sZ = Var ( X ) = 1
Probability Distributions
∑
⎛ n⎞ t n-t
⎜⎝ t ⎟⎠ p q
41.18.
Binomial Distribution: Φ(x) =
41.19.
Poisson Distribution: Φ(x) =
41.20.
Hypergeometric Distribution: Φ(x) =
41.21.
Normal Distribution: Φ(x) =
41.22.
t≤x
∑
t≤x
λ t e −λ
t!
1
2π
Student’s t Distribution: Φ(x) =
∫
∑
t≤x
x
⎛r⎞ ⎛ s ⎞
⎜⎝ t⎟⎠ ⎜⎝ n − t⎟⎠
z
⎛r + s⎞
⎜⎝ n ⎟⎠
e − t / 2 dt
2
−∞
⎛ n + 1⎞
Γ⎜
⎝ 2 ⎟⎠
nπ Γ (n/2)
1
2
41.23. c (Chi Square) Distribution: Φ(x) =
2
n/2
∫
x
−∞
⎛ t 2⎞
⎜⎝1 + ⎟⎠
n
− ( n +1)/ 2
dt
x
1
t(n - 2)/2e-t/2 dt
∫
Γ (n/2) 0
⎛ n1 + n 2 ⎞ n /2 n
Γ⎜
⎟ n1 n 2
⎝ 2 ⎠
F Distribution: Φ(x) =
Γ (n 1 /2)Γ (n2 /2)
1
41.24.
p > 0, q > 0, p + q = 1
2
/2
∫
x
0
t (n
1
/ 2 ) −1
(n 2 + n 1 t ) − ( n + n
1
2
)/ 2
dt
Section XII: Numerical Methods
42
INTERPOLATION
Lagrange Interpolation
Two-point formula
42.1.
x − x0
x − x1
+ f ( x1 )
x 0 − x1
x1 − x 0
p ( x ) = f ( x0 )
where p (x) is a linear polynomial interpolating two points
( x 0 , f ( x 0 )), ( x1 , f ( x1 )), x 0 ≠ x1
General formula
42.2.
p ( x ) = f ( x 0 ) Ln , 0 ( x ) + f ( x1 ) Ln ,1 ( x ) + + f ( x n ) Ln , n ( x )
where
Ln , k =
n
∏
i=0,i≠ k
x − xi
x k − xi
and where p(x) is an nth-order polynomial interpolating n + 1 points
( x k , f ( x k )), k = 0, 1, … , n; and
xi ≠ x j for i ≠ j
Remainder formula
Suppose f ( x ) ∈ Cn+1[a, b]. Then there is a ξ ( x ) ∈ (a, b) such that:
42.3.
f (x) = p (x) +
f n+1 (ξ ( x ))
( x − x 0 )( x − x1 ) ( x − x n )
(n + 1)!
Newton’s Interpolation
First-order divided-difference formula
42.4.
f [ x 0 , x1 ] =
f ( x1 ) − f ( x 0 )
x1 − x 0
Two-point interpolatory formula
42.5.
p( x ) = f ( x 0 ) + f [ x 0 , x1 ]( x − x 0 )
where p(x) is a linear polynomial interpolating two points
( x 0 , f ( x 0 )), ( x1 , f ( x1 )), x 0 ≠ x1
227
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INTERPOLATION
228
Second-order divided-difference formula
f [ x1 , x 2 ] − f [ x 0 , x1 ]
42.6. f [ x 0 , x1 , x 2 ] =
x2 − x0
Three-point interpolatory formula
42.7.
p ( x ) = f ( x 0 ) + f [ x 0 , x1 ]( x − x 0 ) + f [ x 0 , x1 , x 2 ]( x − x 0 )( x − x1 )
where p(x) is a quadrant polynomial interpolating three points
( x 0 , f ( x 0 )), ( x1 , f ( x1 )), ( x 2 , f ( x3 ))
General kth-order divided-difference formula
42.8.
f [ x 0 , x1 ,… , x k } =
f [ x1 , x 2 ,… , x k ] − f [ x 0 , x1 ,… , x k −1 ]
xk − x0
General interpolatory formula
42.9.
p ( x ) = f ( x 0 ) + f [ x 0 , x1 ]( x − x 0 ) + + f [ x 0 , x1 ,… , x n ]( x − x 0 )( x − x1 ) ( x − x n−1 )
where p(x) is an nth-order polynomial interpolating n + 1 points
( x k , f ( x k )), k = 0, 1,… , n; and
xi ≠ x j for i ≠ j
Remainder formula
Suppose f ( x ) ∈ Cn+1[a, b]. Then there is a ξ ( x ) ∈ (a, b) such that
42.10.
f ( x ) = p( x ) +
f n+1 (ξ ( x ))
( x − x 0 )( x − x1 ) ( x − x n )
(n + 1)!
Newton’s Forward-Difference Formula
First-order forward-difference at x0
42.11.
Δf ( x 0 ) = f ( x1 ) − f ( x 0 )
Second-order forward difference at x0
42.12.
Δ 2 f ( x 0 ) = Δf ( x1 ) − Δf ( x 0 )
General kth-order forward difference at x0
42.13.
Δ k f ( x 0 ) = Δ k −1 f ( x1 ) − Δ k −1 f ( x 0 )
Binomial coefficient
42.14.
⎛ s⎞ s(s − 1) (s − k + 1)
⎜⎝ k⎟⎠ =
k!
Newton’s forward-difference formula
⎛ n⎞
p( x ) = ∑ ⎜ ⎟ Δ k f ( x 0 )
⎝ k⎠
k =0
n
42.15.
where p(x) is an nth-order polynomial interpolating n + 1 equal spaced points
( x k , f ( x k )), x k = x 0 + kh k = 0, 1, … , n
INTERPOLATION
229
Newton’s Backward-Difference Formula
First-order backward difference at xn
42.16.
∇f ( x n ) = f ( x n ) − f ( x n−1 )
Second-order backward difference at xn
42.17.
∇ 2 f ( x n ) = ∇f ( x n ) − ∇f ( x n−1 )
General kth-order backward difference at xn
42.18.
∇ k f ( x n ) = ∇ k −1 f ( x n ) − ∇ k −1 f ( x n−1 )
Newton’s backward-difference formula
⎛− n
p( x ) = ∑ (−1) k ⎜ ⎞⎟ ∇ k f ( x n )
⎝ k⎠
k =0
n
42.19.
where p(x) is an nth-order polynomial interpolating n + 1 equal spaced points
( x k , f ( x k )), x k = x 0 + kh
k = 0, 1, … , n
Hermite Interpolation
Two-point basis polynomials
42.20.
x − x 0 ⎞ ( x − x1 )2
x − x1 ⎞ ( x − x 0 )2
⎛
⎛
H1,1 = ⎜1 − 2
H1, 0 = ⎜1 − 2
⎟
2 ,
x 0 − x1 ⎠ ( x 0 − x1 )
x1 − x 0 ⎟⎠ ( x1 − x 0 )2
⎝
⎝
( x − x 0 )2
( x − x1 )2
Hˆ 1, 0 = ( x − x 0 )
Hˆ 1,1 = ( x − x1 )
2 ,
( x 0 − x1 )
( x1 − x 0 )2
Two-point interpolatory formula
42.21.
H 3 ( x ) = f ( x 0 ) H1, 0 + f ( x1 ) H1,1 + f ′( x 0 ) Hˆ 1, 0 + f ′( x1 ) Hˆ 1,1
where H3(x) is a third-order polynomial, agrees with f (x) and its first-order derivatives at two points, i.e.,
H 3 ( x 0 ) = f ( x 0 ), H 3′ ( x 0 ) = f ′( x 0 ),
H 3 ( x1 ) = f ( x1 ), H 3′ ( x1 ) = f ′( x1 )
General basis polynomials
42.22.
x − xj ⎞ 2
⎛
H n , j = ⎜1 − 2
L ( x ), Hˆ n , j = ( x − x j ) L2n , j ( x )
Ln′, j ( x j )⎟⎠ n , j
⎝
where
Ln , j =
n
∏
i=0,i≠ j
x − xi
x j − xi
INTERPOLATION
230
General interpolatory formula
42.23.
n
n
j=0
j=0
H 2 n+1 ( x ) = ∑ f ( x j ) H n , j ( x ) + ∑ f ′( x j ) Hˆ n , j ( x )
where H 2 n+1 ( x ) is a (2n + 1)th-order polynomial, agrees with f(x) and its first order derivatives at n + 1
points, i.e.,
H 2 n+1 ( x k ) = f ( x k ), H 2′n+1 ( x k ) = f ′( x k )
Remainder formula
Suppose f ( x ) ∈ C2 n+ 2 [a, b]. Then there is a ξ ( x ) ∈ (a, b) such that
42.24.
f ( x ) = H 2 n+1 ( x ) +
f 2 n+ 2 (ξ ( x ))
( x − x 0 )2 ( x − x1 )2 ( x − x n )2
(2n + 2)!
k = 0, 1, … , n
43
QUADRATURE
Trapezoidal Rule
Trapezoidal rule
43.1.
∫
b
a
f ( x ) dx ~
b−a
[ f (a) + f (b)]
2
Composite trapezoidal rule
43.2.
∫
b
a
f ( x ) dx ~
n −1
⎞
h⎛
f (a) + 2∑ f (a + ih) + f (b)⎟
⎜
2⎝
⎠
i =1
where h = (b − a)/n is the grid size.
Simpson’s Rule
Simpson’s rule
43.3.
∫
b
a
f ( x ) dx ~
b−a ⎡
⎤
⎛ a + b⎞
f (a) + 4 f
+ f (b)⎥
6 ⎢⎣
⎝ 2 ⎠
⎦
Composite Simpson’s rule
43.4.
∫
b
a
f ( x ) dx ~
n/2
n/2
⎞
h⎛
f
(
x
)
+
f
(
x
)
+
f ( x 2i −1 ) + f ( x n )⎟
2
4
∑
∑
0
2
i
−
2
3 ⎜⎝
⎠
i=2
i =1
where n even, h = (b − a)/n, xi = a + ih, i = 0, 1, … , n.
Midpoint Rule
Midpoint rule
43.5.
∫
b
a
f ( x ) dx ~ (b − a) f
⎛ a + b⎞
⎝ 2 ⎠
Composite midpoint rule
43.6.
∫
b
a
n/2
f ( x ) dx ~ 2h∑ f ( x 2i )
i=0
where n even, h = (b − a)/(n + 2), xi = a + (i − 1)h, i = −1, 0, … , n + 1.
231
QUADRATURE
232
Gaussian Quadrature Formula
Legendre polynomial
43.7.
Pn ( x ) =
1 dn
[( x 2 − 1)n ]
2n n ! dx n
Abscissa points and weight formulas
The abscissa points x k( n ) and weight coefficient ω k( n ) are defined as follows:
43.8.
x k( n ) = the kth zero of the Legendre polynomial Pn(x)
43.9.
ω k( n ) =
2Pn′( x k( n ) )2
1 − x k( n )
2
Tables for Gauss-Legendre abscissas and weights appear in Fig. 43-1.
Gauss-Legendre formula in interval (–1, 1)
43.10.
∫
1
−1
n
f ( x ) dx = ∑ ω k( n ) f ( x k( n ) ) + Rn
k =1
Gauss-Legendre formula in general interval (a, b)
43.11.
∫
b
a
f ( x ) dx =
b − a n (n) ⎛ a + b
b − a⎞
ωk f
+ x k( n )
+ Rn
2 ∑
2
2 ⎠
⎝
k =1
Remainder formula
43.12.
Rn =
(b − a)2 n+1 (n !)4 ( 2 n )
f (ξ )
(2n + 1)[(2n)!]3
for some a < ξ < b.
Fig. 43-1
44
SOLUTION of NONLINEAR EQUATIONS
Here we give methods to solve nonlinear equations which come in two forms:
44.1.
Nonlinear equation: f (x) = 0
44.2. Fixed point nonlinear equation: x = g(x)
One can change from 44.1 to 44.2 or from 44.2 to 44.1 by settting:
g( x ) = f ( x ) + x or
f ( x ) = g( x ) − x
Since the methods are iterative, there are two types of error estimates:
44.3.
| f ( x n ) | < or | x n+1 − x n | < for some preassigned > 0.
Bisection Method
The following theorem applies:
Intermediate Value Theorem: Suppose f is continuous on an interval [a, b] and f (a) f (b) < 0. Then there
is a root x* to f (x) = 0 in (a, b).
The bisection method approximates one such solution x*.
44.4.
Bisection method:
Initial step: Set a0 = a and b0 = b.
Repetitive step:
(a) Set cn = (an + bn )/2.
(b) If f (an ) f (cn ) < 0, then set an+1 = an and bn+1 = cn ; else set an+1 = cn and bn+1 = bn .
Newton’s Method
Newton method
44.5.
x n+1 = x n −
f ( xn )
f ′( x n )
Quadratic convergence
44.6.
| x n+1 − x ∗ |
f ′′( x ∗ )
=
∗
2
n→∞ | x − x |
2( f ′( x ∗ ))2
n
lim
where x* is a root of the nonlinear equation 44.1.
233
SOLUTION OF NONLINEAR EQUATIONS
234
Secant Method
Secant method
44.7.
x n+1 = x n −
( x n − x n−1 ) f ( x n )
f ( x n ) − f ( x n−1 )
Rate of convergence
44.8.
| x n+1 − x ∗ |
f ′′( x ∗ )
=
n→∞ | x − x ∗ || x
− x ∗ | 2( f ′( x ∗ ))2
n
n −1
lim
where x* is a root of the nonlinear equation 44.1.
Fixed-Point Iteration
The following definition and theorem apply:
Definition: A function g from (a, b) to (a, b) is called a contraction mapping if
| g ( x ) − g ( y) | L | x − y |
for any x , y ∈ (a, b)
where L < 1 is a positive constant.
Fixed-point theorem: Suppose that g is a contraction mapping on (a, b). Then g has a unique fixed point in
(a, b).
Given such a contraction mapping g, the following method may be used.
Fixed-point iteration
44.9.
x n+1 = g( x n )
45
NUMERICAL METHODS for ORDINARY
DIFFERENTIAL EQUATIONS
Here we give methods to solve the following initial-value problem of an ordinary differential equation:
45.1.
⎧⎪dx = f ( x , t )
⎨ dt
⎩⎪x (t0 ) = x 0
The methods will use a computational grid:
45.2.
tn = t0 + nh
where h is the grid size.
First-Order Methods
Forward Euler method (first-order explicit method)
45.3.
x (t + h) = x (t ) + hf ( x (t ), t )
Backward Euler method (first-order implicit method)
45.4.
x (t + h) = x (t ) + hf ( x (t + h), t + h)
Second-Order Methods
Mid-point rule (second-order explicit method)
45.5.
⎧ *
h
⎪⎪x = x (t ) + 2 f ( x (t ), t )
⎨
⎛
h⎞
⎪x (t + h) = x (t ) + hf ⎜ x * , t + ⎟
2⎠
⎝
⎪⎩
Trapezoidal rule (second-order implicit method)
45.6.
h
x (t + h) = x (t ) + { f ( x (t ), t ) + f ( x (t + h), t + h)}
2
Heun’s method (second-order explicit method)
45.7.
⎧x * = x (t ) + hf ( x (t ), t )
⎪
h
⎨
*
⎪⎩x (t + h) = x (t ) + 2 { f ( x (t ), t ) + f ( x , t + h)}
235
NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
236
Single-Stage High-Order Methods
Fourth-order Runge–Kutta method (fourth-order explicit method)
45.8.
x (t + h) = x (t ) +
1
(F + 2F2 + 2F3 + F4 )
6 1
where
F
⎛
F1 = hf ( x , t ), F2 = hf ⎜ x + 1 , t +
2
⎝
F
h⎞
⎛
, F3 = hf ⎜ x + 2 , t +
2⎟⎠
2
⎝
h⎞
, F4 = hf ( x + F3 , t + h)
2⎟⎠
Multi-Step High-Order Methods
Adams-Bashforth two-step method
45.9.
x (t + h) = x (t ) + h
1
⎛3
f ( x (t ), t ) − f ( x (t − h), t − h)⎞
2
⎝2
⎠
Adams-Bashforth three-step method
45.10.
x (t + h) = x (t ) + h
5
4
⎛ 23
f ( x (t − 2h), t − 2h)⎞
f ( x (t ), t ) − f ( x (t − h), t − h) +
12
3
⎝ 12
⎠
Adams-Bashforth four-step method
45.11.
⎛ 55
⎞
59
37
9
x (t + h) = x (t ) + h ⎜
f ( x (t ), t ) −
f ( x (t − h), t − h) +
f ( x (t − 2h), t − 2h) −
f ( x (t − 3h), t − 3h)⎟
24
24
24
⎝ 24
⎠
Milne’s method
45.12.
x (t + h) = x (t − 3h) + h
8
4
⎛8
f ( x (t ), t ) − f ( x (t − h), t − h) + f ( x (t − 2h), t − 2h)⎞
3
3
⎝3
⎠
Adams-Moulton two-step method
45.13.
x (t + h) = x (t ) + h
2
1
⎛5
f ( x (t + h), t + h) + f ( x (t ), t ) −
f ( x (t − h), t − h)⎞
3
12
⎝ 12
⎠
Adams-Moulton three-step method
45.14.
⎛3
⎞
5
1
19
x (t + h) = x (t ) + h ⎜ f ( x (t + h), t + h) +
f ( x (t ), t ) −
f ( x (t − h), t − h) +
f ( x (t − 2h), t − 3h)⎟
24
24
24
⎝8
⎠
46
NUMERICAL METHODS for PARTIAL
DIFFERENTIAL EQUATIONS
Finite-Difference Method for Poisson Equation
The following is the Poisson equation in a domain (a, b) × (c, d):
46.1.
∇ 2u = f ,
∇2 =
∂2
∂2
2 +
∂x
∂y 2
Boundary condition:
46.2.
u ( x , y) = g ( x , y)
for x = a, b
or
y = c, d
Computation grid:
46.3.
xi = a + iΔx
for i = 0, 1, … , n
y j = c + jΔy
for j = 0, 1, … , m
where Δx = (b − a)/n and Δy = (d − c)/m are grid sizes for x and y variables, respectively.
Second-order difference approximation
46.4.
( Dx2 + Dy2 )u( xi , y j ) = f ( xi , y j )
where
Dx2 u( xi , y j ) =
u( xi +1 , y j ) − 2u( xi , y j ) + u( xi −1 , y j )
Δx 2
Dy2 u( xi , y j ) =
u( xi , y j +1 ) − 2u( xi , y j ) + u( xi , y j −1 )
Δy 2
Computational boundary condition
46.5.
u( x 0 , y j ) = g(a, y j ),
u( x n , y j ) = g(b, y j )
for j = 1, 2, … , m
u( xi , y0 ) = g( xi , c),
u( xi , ym ) = g( xi , d )
for i = 1, 2, … , n
Finite-Difference Method for Heat Equation
The following is the heat equation in a domain (a, b) × (c, d ) × (0, T ) :
46.6.
∂u
= ∇ 2u
∂t
237
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
238
Boundary condition:
46.7.
u ( x , y , t ) = g ( x , y)
for x = a, b
or
y = c, d
Initial condition:
46.8.
u( x , y, 0) = u0 ( x , y)
Computational grid:
46.9.
xi = a + iΔx
for i = 0,1, … , n
y j = c + jΔy
for j = 0,1, … , m
t k = kΔt
for k = 0,1, … ,
where Δx = (b − a)/n, Δy = (d − c)/m, and Δt are grid sizes for x, y and t variables, respectively.
Computational boundary condition
46.10.
u( x 0 , y j ) = g(a, y j ), u( x n , y j ) = g(b, y j )
for j = 1, 2, … , m
u( xi , y0 ) = g( xi , c), u( xi , ym ) = g( xi , d )
for i = 1, 2, … , n
Computational initial condition
46.11.
u( xi , y j , 0) = u0 ( xi , y j )
for i = 1, 2, … , n; j = 0, 1, … , m
Forward Euler method with stability condition
46.12.
u( xi , y j , t k +1 ) = u( xi , y j , t k ) + Δt ( Dx2 + Dy2 )u( xi , y j , t k )
46.13.
2Δt 2Δt
+
1
Δx 2 Δy 2
Backward Euler method (unconditional stable)
46.14.
u( xi , y j , t k +1 ) = u( xi , y j , t k ) + Δt ( Dx2 + Dy2 )u( xi , y j , t k +1 )
Crank-Nicholson method (unconditional stable)
46.15.
u( xi , y j , t k +1 ) = u( xi , y j , t k ) + Δt ( Dx2 + Dy2 ){u( xi , y j , t k ) + u( xi , y j , t k +1 )}/2
Finite-Difference Method for Wave Equation
The following is a wave equation in a domain (a, b) × (c, d ) × (0, T ):
46.16.
∂2u
= A2 ∇ 2 u
∂t 2
where A is a constant representing the speed of the wave.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
Boundary condition:
46.17.
u ( x , y, t ) = g ( x , y)
for x = a, b or y = c, d
Initial condition:
46.18.
u( x , y, 0) = u0 ( x , y),
∂u
u( x , y, 0) = u1 ( x , y)
∂t
Computational grids:
46.19.
xi = a + iΔx
for i = 0, 1, … , n
y j = c + jΔy
for j = 0, 1, … , m
t k = kΔt
for k = −1, 0, 1, …
where Δx = (b − a)/n, Δy = (d − c)/m, and Δt are the grid sizes for x, y, and t variables, respectively.
A second-order finite-difference approximation
46.20.
u( xi , y j , t k +1 ) = 2u( xi , y j , t k ) − u( xi , y j , t k −1 ) + Δt 2 A2 ( Dx2 + Dy2 )u( xi , y j , t k )
Computational boundary condition
46.21.
u( x 0 , y j ) = g(a, y j ), u( x n , y j ) = g(b, y j )
for j = 1, 2, … , m
u( xi , y0 ) = g( xi , c), u( xi , ym ) = g( xi , d )
for i = 1, 2, … , n
Computational initial condition
46.22.
u( xi , y j , t0 ) = u0 ( xi , y j )
for i = 1, 2, … , n; j = 0, 1, … , m
u( xi , y j , t −1 ) = u0 ( xi , y j ) + Δt 2 u1 ( xi , y j )
Stability condition
46.23.
Δt A min(Δx , Δx )
for i = 1, 2, … , n; j = 0, 1, … , m
239
ITERATION METHODS for LINEAR
SYSTEMS
47
Iteration Methods for Poisson Equation
The finite-difference approximation to the Poisson equation follows:
47.1.
⎧ui +1, j + ui −1, j + ui , j +1 + ui , j −1 − 4ui , j = fi , j
⎪⎪
for j = 1, 2, … , n − 1
⎨u0 , j = un , j = 0
⎪
for i = 1, 2, … , n − 1
⎪⎩ui ,0 = ui ,n = 0
for i, j = 1, 2, … , n − 1
Three iteration methods for solving the system follow:
Jacobi method
47.2.
uik,+j 1 =
1 k
(u + uik−1, j + uik, j +1 + uik, j −1 − fi , j )
4 i +1, j
Gauss-Seidel method
47.3.
uik,+j 1 =
1 k
(u + uik−+11, j + uik, j +1 + uik,+j −11 − fi , j )
4 i +1, j
Successive-overrelaxation (SOR) method
47.4.
1 k
⎧ *
*
k
*
⎪ui , j = (ui +1, j + ui −1, j + ui , j +1 + ui , j −1 − fi , j )
4
⎨
⎪⎩uik,+j 1 = (1 − ω )uik, j + ω ui*, j
Iteration Methods for General Linear Systems
Consider the linear system
47.5. Ax = b
where A is an n × n matrix and x and b are n-vectors. We assume the coefficient matrix A is partitioned as
follows:
47.6. A = D – L – U
where D = diag (A), L is the negative of the strictly lower triangular part of A, and U is the negative of the
strictly upper triangular part of A.
240
ITERATION METHODS FOR LINEAR SYSTEMS
Four iteration methods for solving the system follow:
Richardson method
47.7.
x k +1 = (I − A) x k + b
Jacobi method
47.8.
Dx k +1 = ( L + U ) x k + b
Gauss-Seidel method
47.9.
( D − L ) x k +1 = Ux k + b
Successive-overrelaxation (SOR) method
47.10.
( D − ω L ) x k +1 = ω (Ux k + b) + (1 − ω ) Dx k
241
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PART B
TABLES
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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Section I: Logarithmic, Trigonometric, Exponential Functions
1
FOUR PLACE COMMON LOGARITHMS
log 10 N or log N
245
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
1
246
FOUR PLACE COMMON LOGARITHMS
log 10 N or log N (Continued)
2
Sin x
(x in degrees and minutes)
247
3
248
Cos x
(x in degrees and minutes)
4
Tan x
(x in degrees and minutes)
249
5
250
CONVERSION OF RADIANS TO DEGREES, MINUTES,
AND SECONDS OR FRACTIONS OF DEGREES
6
CONVERSION OF DEGREES, MINUTES, AND
SECONDS TO RADIANS
251
7
NATURAL OR NAPIERIAN LOGARITHMS
log e x or ln x
ln 10 = 2.30259
2 ln 10 = 4.60517
3 ln 10 = 6.90776
252
4 ln 10 = 9.21034
5 ln 10 = 11.51293
6 ln 10 = 13.81551
7 ln 10 = 16.11810
8 ln 10 = 18.42068
9 ln 10 = 20.72327
7
NATURAL OR NAPIERIAN LOGARITHMS
log e x or ln x(Continued)
253
8
254
EXPONENTIAL FUNCTIONS
ex
9
EXPONENTIAL FUNCTIONS
e –x
255
10
256
EXPONENTIAL, SINE, AND COSINE INTEGRALS
Ei( x ) =
∫
∞ −u
x
e
du, Si( x ) =
u
∫
x
0
sin u
du, Ci( x ) =
u
∫
∞
x
cos u
du
u
Section II: Factorial and Gamma Function, Binomial Coefficients
11
FACTORIAL n
n ! = 1 i 2 i 3 ii n
257
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
12
258
GAMMA FUNCTION
Γ( x ) =
∫
∞
x
t x −1e − t dt
for 1 x 2
[For other values use the formula Γ(x + 1) = x Γ(x)]
13
BINOMIAL COEFFICIENTS
n!
n(n − 1) (n − k + 1) ⎛ n ⎞
⎛ n⎞
=⎜
, 0! = 1
⎜⎝ k⎟⎠ = k !(n − k )! =
k!
⎝ n − k⎟⎠
Note that each number is the sum of two numbers in the row above; one of these numbers is in the same column and the other is in the preceding column (e.g., 56 = 35 + 21). The arrangement is often called Pascal’s
triangle (see 3.6, page 8).
259
13
BINOMIAL COEFFICIENTS
n!
n(n − 1) (n − k + 1) ⎛ n ⎞
⎛ n⎞
=⎜
, 0 ! = 1 (Continued)
⎜⎝ k⎟⎠ = k !(n − k )! =
k!
⎝ n − k⎟⎠
⎛ n⎞ ⎛ n ⎞
.
For k > 15 use the fact that ⎜ ⎟ = ⎜
⎝ k⎠ ⎝ n − k⎟⎠
260
Section III: Bessel Functions
14
15
BESSEL FUNCTIONS
J 0 (x)
BESSEL FUNCTIONS
J 1 (x)
261
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16
17
262
BESSEL FUNCTIONS
Y 0 (x)
BESSEL FUNCTIONS
Y 1 (x)
18
19
BESSEL FUNCTIONS
I 0 (x)
BESSEL FUNCTIONS
I 1 (x)
263
20
21
264
BESSEL FUNCTIONS
K 0 (x)
BESSEL FUNCTIONS
K 1 (x)
22
23
BESSEL FUNCTIONS
Ber(x)
BESSEL FUNCTIONS
Bei(x)
265
24
25
266
BESSEL FUNCTIONS
Ker(x)
BESSEL FUNCTIONS
Kei(x)
26
VALUES FOR APPROXIMATE ZEROS
OF BESSEL FUNCTIONS
The following table lists the first few positive roots of various equations. Note that for all cases listed the
successive large roots differ approximately by π = 3.14159. . . .
267
Section IV: Legendre Polynomials
27
LEGENDRE POLYNOMIALS P n (x)
[P 0 (x)=1, P 1 (x)=x]
268
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28
LEGENDRE POLYNOMIALS P n (cos ␪)
[P 0 (cos ␪)=1]
269
Section V: Elliptic Integrals
29
COMPLETE ELLIPTIC INTEGRALS
OF FIRST AND SECOND KINDS
K=
∫
π /2
0
dθ
1 − k sin θ
2
2
, E =∫
π /2
0
1 − k 2 sin 2 θ dθ , k = sin ψ
270
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
30
31
INCOMPLETE ELLIPTIC INTEGRAL
OF THE FIRST KIND
F (k , φ ) =
∫
φ
0
dθ
, k = sin ψ
1 − k 2 sin 2 θ
INCOMPLETE ELLIPTIC INTEGRAL
OF THE SECOND KIND
E (k , φ ) =
∫
φ
0
1 − k 2 sin 2 θ dθ , k = sin ψ
271
Section VI: Financial Tables
32
COMPOUND AMOUNT : (1 + r) n
If a principal P is deposited at interest rate r (in decimals) compounded annually, then
at the end of n years the accumulated amount A = P(1 + r)n.
272
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33
PRESENT VALUE OF AN AMOUNT : (1 ⴙ r) ⴚn
The present value P which will amount to A in n years at an interest rate of r (in decimals)
compounded annually is P = A(1 + r)n.
273
AMOUNT OF AN ANNUITY :
34
274
(1 + r ) n - 1
r
If a principal P is deposited at the end of each year at interest rate r (in decimals)
compounded annually, then at the end of n years the accumulated amount is
⎡(1 − r )n − 1⎤ . The process is often called an annuity.
P⎢
r
⎣
⎦⎥
PRESENT VALUE OF AN ANNUITY :
35
1 - (1 + r ) - n
r
An annuity in which the yearly payment at the end of each of n years is A
at an interest rate r (in decimals) compounded annually has present value
⎡1 − (1 + r ) − n ⎤ .
A⎢
⎥⎦
r
⎣
275
Section VII: Probability and Statistics
36
NOTE:
AREAS UNDER THE
STANDARD NORMAL CURVE
from −∞ to x
x
1
Φ( x ) =
e − t / 2 dt
∫
−∞
2π
2
erf (x) = 2Φ(x 2 ) − 1
276
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37
ORDINATES OF THE
STANDARD NORMAL CURVE
y=
1 −x
e
2π
2
/2
277
38
278
PERCENTILE VALUES (t p )
FOR STUDENT'S t
DISTRIBUTION
with n degrees of freedom (shaded area = p)
39
PERCENTILE VALUES (␹ 2p )
FOR ␹ 2 (CHI-SQUARE)
DISTRIBUTION
with n degrees of freedom (shaded area = p)
279
40
280
95th PERCENTILE VALUES
FOR THE F DISTRIBUTION
n1 = degrees of freedom for numerator
n2 = degrees of freedom for denominator
(shaded area = .95)
41
99th PERCENTILE VALUES
FOR THE F DISTRIBUTION
n1 = degrees of freedom for numerator
n2 = degrees of freedom for denominator
(shaded area = .99)
281
42
282
RANDOM NUMBERS
Index of Special Symbols
and Notations
The following list show special symbols and notations together with pages on which they are defined or first
appear. Cases where a symbol has more than one meaning will be clear from the context.
Symbols
Bern(x), Bein(x)
B(m, n)
Bb
C(x)
Ci(x)
e1, e2, e3
erf(x)
erfc(x)
E = E(k, p /2)
E = E(k, f)
Ei(x)
En
E(X)
f [x0, x1, ..., xk]
F(a), F(x)
F(a, b; c; x)
F(k, f )
g, g−1
G. M.
h1, h2, h3
Hn(x)
Hn(1)(x), Hn(2)(x)
H. M.
i, j, k
In(x)
Jn(x)
K = F(k, p /2)
Kern(x), Kein(x)
Kn(x)
ln x or loge x
log x or log10 x
Ln(x)
Lnm(x)
l, l−1
M.D.
P(A/E)
Pn(x)
Pnm(x)
QU, M, QL
Qn(x)
Qnm(x)
r
R.M.S.
s
s2
sxy
Si(x)
S(x)
Tn(x)
Ber and Bei functions, 157
beta function, 152
Bernoulli numbers, 142
Fresnel cosine integral, 204
cosine integral, 204
unit vectors in curvilinear coordinates, 127
error function, 203
complementary error function, 203
complete elliptic integral of the second kind, 198
incomplete elliptic integral of the second kind, 198
exponential integral, 203
Euler number, 142
mean or expectation of random variable X, 223
divided distance formula, 287, 288
cumulative distribution function, 209
hypergeometric function, 178
incomplete elliptic integral of the first kind, 198
Fourier transform and inverse Fourier transform, 194
geometric mean, 209
scale factors in curvilinear coordinates, 127
Hermite polynomial, 169
Hankel functions of the first and second kind, 155
harmonic mean, 210
unit vectors in rectangular coordinates, 120
modified Bessel function of the first kind, 155
Bessel function of the first kind, 153
complete elliptic integral of the first kind, 198
Ker and Kei functions, 158
modified Bessel function of the second kind, 156
natural logarithm of x, 53
common logarithm of x, 53
Laguerre polynomials, 171
associated Laguerre polynomials, 173
Laplace transform and inverse Laplace transform, 180
mean deviation
conditional probability of A given E, 219
Legendre polynomials, 164
associated Legendre polynomials, 173
quartiles, 211
Legendre functions of second kind, 167
associated Legendre functions of second kind, 168
sample correlation coefficient, 213
root-mean-square, 211
sample standard deviation, 208
sample variance, 210
sample covariance, 213
Sine integral, 203
Fresnel sine integral, 204
Chebyshev polynomials of first kind, 175
283
Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use.
INDEX OF SPECIAL SYMBOLS AND NOTATIONS
284
Un(x)
Var(X)
x, x
xk(n)
Yn(x)
Z
Chebyshev polynomials of second kind, 176
variance of random variable X, 224
sample mean, grand mean, 208, 209
kth zero of Legendre polynomial Pn(x), 232
Bessel function of second kind, 153
standardized random variable, 226
Greek Symbols
ar
g
Γ(x)
ζ(x)
m
q
rth moment in standard units, 212
Euler’s constant, 4
gamma function, 149
Rieman zeta function, 204
population mean, 208
coordinate: cylindrical 37,
polar, 11, 24; spherical, 38
p
f
Φ (p)
Φ (x)
s
s2
pi, 3
spherical coordinate, 38
1 1
1
sum 1 + + + + , Φ(0) = 0, 154
p
2 3
probability distribution function, 226
population standard deviation, 223
population variance, 223
Notations
A~B
|A|
A is asymptotic to B or A/B approaches 1, 151
⎧ A if A ≥ 0
absolute value of A = ⎨
⎩− A if A < 0
n!
factorial n, 7
⎛ n⎞
⎜⎝ k⎟⎠
dy
⎫
= f ( x ) ⎪
dx
⎬
d2y
y = 2 = f ( x ), etc.⎪
dx
⎭
p
d
Dp = p
dx
∂f ∂f ∂2 f
,
,
, etc.
∂x ∂x ∂x ∂y
∂( x , y, z )
∂(u1 , u2 , u3 )
binomial coefficients, 8
y =
∫ f ( x )dx
b
∫a f ( x )dx
∫C A i dr
AiB
A×B
∇
∇2 = ∇ i ∇
∇4 = ∇2 (∇2)
derivatives of y or f(x) with respect to x, 62
pth derivative with respect to x, 64
partial derivatives, 65
Jacobian, 128
indefinite integral, 67
definite integral, 108
line integral of A along C, 124
dot product of A and B, 120
cross product of A and B, 121
del operator, 122
Laplacian operator, 123
biharmonic operator, 123
Index
Adams-Bashforth methods, 236
Adams-Moulton methods, 236
Addition formula:
Bessel functions, 163
Hermite polynomials, 170
Addition rule (probability) 208
Addition of vectors, 119
Algebra of sets, 217
Algebraic equations, solutions of, 13
Alphabet, Greek, 3
Analytic geometry, plane, 22–33
solid, 34–40
Annuity table, 274
Anti-derivative, 67
Anti-logarithms, 53
Arithmetic:
mean, 208
series, 134
Arithmetic-geometric series, 134
Associated Laguerre polynomials, 173
(See also Laguerre polynomials)
Associated Legendre functions, 164
(See also Legendre functions)
of the first kind, 168
of the second kind, 168
Asymptotic expansions or formulas:
Bernoulli numbers, 143
Bessel functions, 160
Backward difference formulas, 228
Her and Bei functions, 157
Bayes formula, 220
Bernoulli numbers, 142
asymptotic formula, 143
series, 143
Bernoulli’s differential equation, 116
Bessel functions, 153–164
graphs, 159
integral representation, 161
modified, 155
recurrence formulas, 154, 157
series, orthogonal, 161
tables, 261–267
Bessel’s differential equation,
118, 153
general solution, 154
modified differential equation, 155
Best fit, line of, 214
Beta function, 152
Biharmonic operator, 123
Binomial:
coefficients, 7, 228, 259
distribution, 226
formula, 7
series, 136
Bipolar coordinates, 131
Bisection method, 223
Bivariate data, 212
Carioid, 29
Cassini, ovals of, 32
Catalan’s constant, 200
Catenary, 29
Cauchy or Euler differential equation, 117
Cauchy’s form of remainder in Taylor series, 134
Cauchy-Schwarz inequality, 205
for integrals, 206
Central tendency, 208
Chain rule for derivatives, 67
Chebyshev polynomials, 175
of the first kind, 175
of the second kind, 176
recurrence formula, 175
Chebyshev’s differential equation, 175
general solution, 177
Chebyshev’s inequality, 206
Chi-square distribution, 226
table of values, 279
Circle, 17, 25
Coefficient:
of excess (kurtosis), 212
of skewness, 212
Coefficients:
binomial, 7
multinomial, 9
Complementary error function, 203
Complex:
conjugate, 10
numbers, 10
logarithm of, 55
plane, 10
Components of a vector, 120
Compound amount, 262
Confocal:
ellipsdoidal coordinates, 133
paraboloidal coordinates, 133
Conical coordinates, 129
Conics, 25 (See also Ellipse, Parabola, Hyperbola)
Conjugate, complex, 10
Constant of integration, 67
Constants, 3
series of, 134
Continuous random variable, 224
Convergence, interval of, 138.
Conversion factors, 15
Convolution theorem, Fourier transform, 194
Coordinates, 127
bipolar, 131
confocal ellipsoidal, 133
confocal paraboloidal, 133
conical, 132
curvilinear, 127
cylindrical, 129
elliptic cylindrical, 130
oblate spheroidal, 131
paraboloidal, 130
prolate spheroidal, 131
spherical, 129
toroidal, 132
Correlation coefficient, 213
Cosine, 43
graph of, 46
table of values, 245
Cosine integral, 203, 256
Cosines, law of, 51
Covariance, 213
Cross or vector product, 121
Cubic equation, solution of, 13
285
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INDEX
286
Cumulative distribution function, 225
Curl, 123
Curve fitting, 215
Curvilinear coordinates, 134
Cycloid, 28
Cylindrical coordinates, 37, 129
Definite integrals, 108–116
approximate formula, 109
definition of, 108
Degrees, conversion to radians, 251
Del operator, 122
DeMoivre’s theorem, 11
Derivatives, 62–66
chain rule for, 62
higher, 64
Leibniz’s rule, 64
of vectors, 122
Deviation:
mean, 210
standard, 210
Differential equations, numerical methods for solution:
ordinary, 235–236
partial, 237–240
Differentials, 65, 66
Differentiation, 62–66 (See also Derivatives)
Direction numbers, 34
cosines, 34
Discrete random variable, 223
Distributions, probability, 226
Divergence, 122, 128
theorem, 126
Divided-difference formula (general), 228
Dot or scalar product, 120
Double integrals, 125
Eccentricity, 25
Ellipse, 18, 25
Ellipsoid, 39
Elliptic cylinder, 41
Elliptic cylindrical coordinates, 130
Elliptic functions, 198–202
Jacobi’s, 199
series expansion, 200
Elliptic integrals, 198–199
table of values, 270–271
Epicycloid, 30
Equality of vectors, 119
Equations, algebraic, 13
Error functions, 203
Euler:
constant, 4
differential equation, 117
methods, 235
numbers, 142
Euler-Maclaurin summation formula,
137
Exact differential equation, 116
Excess, coefficient of kurtosis, 212
Exponential curve (least-squares), 215
Exponential function, 53–54
series for, 139
table of values, 254–255
Exponential integral, 203, 256
Exponents, 53
F distribution, 226
table of values, 280–281
Factorial n, 7
table of values, 257
Factors, special, 5
Financial tables, 272–275
Finite-difference methods for solution of:
heat equation, 237
Poisson equation, 237
wave equation, 238
First-order divided-difference formula, 227
Five number summary [L, QL, M, QH, H ], 211
Fixed-point iteration, 234
Folium of Descartes, 31
Forward difference formulas, 228
Fourier series, 144–146
Fourier transform, 193
convolution of, 194
cosine, 194, 197
Parseval’s identity for, 193
sine, 194, 196
tables, 195–199
Fourier’s integral theorem, 193
Fresnel sine and cosine integral, 204
Frullani’s integral, 115
Gamma function, 149, 150
relation to beta function, 152
table of values, 258
Gauss’ theorem, 126
Gauss-Legendre formula, 232
Gauss-Seidel method, 230
Gaussian quadrature formula, 231
Generating functions, 157, 165, 168, 169, 171, 173, 175, 176
Geometric:
mean (G.M.), 209
series, 134
Geometry, 16–21
analytic, 22–40
Gradient, 122, 128
Grand mean, 209
Greek alphabet, 3
Green’s theorem, 126
Griggsian logarithms, 53
Half angle formulas, 48
Half rectified sine wave function, 191
Hankel functions, 155
Harmonic mean, 209
Heat equation, 237
Heaviside’s unit function, 192
Hermite:
interpolation,229
polynomials, 169–170
Hermite’s differential equation, 169
Heun’s method, 235
Holder’s inequality, 205
for integrals, 206
Homogeneous differential equation, 116
linear second order, 117
Hyperbola, 25
Hyperbolic functions, 56–61
graphs of, 59
inverse, 59–61
series for, 140
Hyperboloid, 39
Hypergeometric:
differential equation, 178
distribution, 226
functions, 178
Hypocycloid, 28, 30
Imaginary part of a complex number, 10
Indefinite integrals, 67–107
definition of, 67
tables of, 71–107
transformation of, 69
Independent events, 221
INDEX
Inequalities, 205
Infinite products, 207
Integral calculus, fundamental theorem, 108
Integrals:
definite (see Definite integrals)
improper, 108
indefinite (see Indefinite integrals)
line, 124
multiple, 125
surface, 125
Integration, 64 (See also Integrals)
constant of, 67
general rules, 67–69
Integration by parts, 67
generalized, 69
Intercepts, 22
Interest, 272–275
Intermediate Value Theorem, 233
Interpolation, 227
Hermite, 229
Interpolatory formula (general), 228
Interquartile range, 211
Interval of convergence, 138
Inverse:
hyperbolic functions, 59–61
Laplace transforms, 180
trigonometric functions, 49–51
Iteration methods, 240
for general linear systems, 240
for Poisson equation, 240
Jacobi method, 240
Jacobi’s elliptic functions, 199
Jacobian, 128
Ker and Kei functions, 158–159
Kurtosis, 212
Lagrange:
form of remainder, 138
interpolation, 227
Laguerre polynomials, 172
generating function for, 173
recurrence formula, 192
Laguerre’s associated differential equation,
170
Laguerre’s differential equation, 172
Landen’s transformation, 199
Laplace transform, 180–192
complex inversion formula for, 180
definition of, 180
inverse, 180
tables of, 181–192
Laplacian, 123, 128
Least-squares:
curve, 215
line, 214
Legendre functions, 164–168
of the second kind, 166
Legendre polynomial, 164–165, 232
generating function for, 164
recurrence formula for, 166
tables of values for, 269
Legendre’s associated differential equation, 168
Legendre’s differential equation, 118, 164
Leibniz’s rule, 64
Lemniscate, 28
Limacon of Pascal, 32
Line, 22, 35
of best fit, 214
regression, 214
Line integral, 124
287
Logarithmic functions, 53–55 (See also Logarithms)
series for, 139
table of values, 245–246, 252–253
Logarithms, 53–55
of complex numbers, 55
Griggsian, 53
Maclaurin series, 138
Mean, 208
continuous random variable, 224
deviation (M.D.), 211
discrete random variable, 223
geometric, 209
grand, 209
harmonic, 209
population, 212
weighted. 209
Mean value theorem,
for definite integrals, 108
generalized, 109
Median, 208
Midpoint rule, 231, 235
Midrange, 210
Milne’s method, 236
Minkowski’s inequality, 206
for integrals, 206
Mode, 209
Modified Bessel functions,
155–157
generating function for, 157
graphs of, 159
recurrence formulas for, 157
Modulus of a complex number, 11
Moment, rth, 212
Momental skewness, 212
Moments of inertia, 41
Monoticity Rule (Probability), 218
Mutinomial coefficients, 9
Multiple, integrals, 125
Napier’s rules, 52
Natural logarithms and antilogarithms,
53
tables of, 252–253
Neumann’s function, 153
Newton’s:
backward-difference formula, 228
forward-difference formula, 228
interpolation, 227
method, 233
Nonhomogeneous differential equation, linear
second order, 117
Nonlinear equations, solution of, 233
Normal curve, 276–277
distribution, 226
Normal equations for least-squares line, 214
Null function, 189
Numbers:
Bernoulli, 142
Euler, 142
Numerical methods for partial differential equations,
237–239
Oblate spheroidal coordinates, 131
Orthogonal curvilinear coordinates, 127–128
formulas involving, 128
Orthogonality:
Chebyshev’s polynomials, 176
Laguerre polynomials, 172
Legendre polynomials, 165
Ovals of Cassini, 32
INDEX
288
Parabola, 25
segment of, 18
Parabolic cylindrical coordinates, 129
Paraboloid, 40
Paraboloidal coordinates, 130
Parallelepiped, 19
Parallelogram, 7
Parameter, 208
Parseval’s identity for:
Fourier series, 144
Fourier transform, 194
Partial:
derivatives, 65
differential equations, numerical methods, 237
Pascal’s triangle, 8
Percentile, kth, 211
Periods of elliptic functions, 200
Plane analytic geometry, formulas from, 22–27
Plane, complex, 10
Poisson:
distribution, 226
equation, 237
summation formula, 137
Polar:
coordinates, 24
form of a complex number, 11
Polygon, regular, 17
Polynomial function (least-squares), 214
Polynomials:
Chebyshev’s, 175
Laguerre, 171
Legendre, 164
Population, 208
mean 210
standard deviation, 212
variance, 212
Power function (least-squares), 214
Power series, 138–141
reversion of 141
Powers, sums of, 134
Present value, of an amount, 273
of an annuity, 275
Probability, 217
distribution, 223
function, 218
tables, 276
Products, infinite, 207
special, 5
Pulse function, 192
Pyramid, volume of, 20
Quadrants, 43
Quadratic convergence, 233
Quadratic equation, solution of, 103
Quadrature, 231–232
Quartic equation, solution of, 13
Quartile coefficient of skewness, 212
Quartiles [QL, M, QU], 211
Radians, 4, 44
table of conversion to degrees, 250
Random numbers table, 282
Random variable, 223–226
standardized, 226
Range, sample, 210
Real part of a complex number, 10
Reciprocals of powers, sums of, 135
Rectangle, 13
Rectangular coordinate system, 120
Rectangular coordinates, 24
transformation to polar coordinates, 24
Rectangular formula, 109
Rectified sine wave function, 191
Recurrence or recursion formulas:
Bessel functions, 154
Chebyshev’s polynomials, 175
gamma function, 149
Hermite polynomials, 169
Laguerre polynomials, 171
Legendre polynomials, 165
Regression line, 214
Regular polygon, 17
Remainder:
Cauchy’s form, 13
Lagrange form, 138
Remainder formula:
Gauss-Legendre interpolation, 232
Hermite interpolation, 230
Lagrange interpolation, 227
Reversion of power series, 141
Richardson method, 240
Riemann zeta function, 204
Right circular cone, 20
Rochigue’s formula:
Laguerre polynomials, 171
Legendre’s polynomials, 164
Root mean square (R.M.S.), 211
Roots of complex numbers, 11
Rose, 29
Rotation, 24, 37
Runge-Kutta method, 236
Sample, 208
covariance, 213
Saw tooth wave function, 191
Scalar, 119
multiplication of vectors, 119
Scalar or dot product, 120
Scale factors, 127
Scatterplot, 212
Schwarz (Cauchy-Schwarz) inequality, 205
for integrals, 206
Secant method, 233
Second-order differential equation, 117
Second-order divided-difference formula, 228
Sector of a circle, 17
Segment:
of circle, 18
of parabola, 18
Semi-interquartile range, 211
Separation of variables, 116
Series, arithmetic, 134
arithmetic-geometric, 134
binomial, 188
of constants, 134
Fourier, 144–148
geometric, 134
power, 138
of sums of powers, 134
Taylor, 138–141
Simpson’s formula, 109, 231
Sine, 43
graph of, 46
table of values, 247
Sine integral, 88
table of values, 264
Sines, law of, 51
Skewness, 212
Solid analytic geometry, 34–40
Solutions of algebraic equations, 13–14
SOR (successive-overrelaxation) method, 240
Sphere, equations of, 38
surface area, 19
volume, 21
Spherical coordinates, 38, 129
Spherical triangle, 51
INDEX
Spiral of Archimedes, 33
Square wave function, 191
Squares error, 215
Standard deviation, 210
continuous random variable, 225
discrete random variable, 224
population, 212
sample, 210
Standardized random variable, 215
Statistics, 208–216
tables, 276–281
Step function, 192
Stirling’s formula, 150
Stochastic process, 219
Stokes’ theorem, 126
Student’s t distribution, 226
table of, 298
Successive-overrelaxation (SOR) method, 240
Summation formula:
Euler-Maclaurin, 137
Poisson, 137
Surface integrals, 125
Tangent function, 43
graph of, 46
table of values, 249
Tangents, law of, 51, 52
Taylor series, 138–141
two variables, 141
Three-point interpolatory formula, 228
Toroidal coordinates, 132
Torus, surface area, volume, 18
Total probability, Law of, 220
Tractrix, 31
Transformation:
Jacobian of, 128
of coordinates, 24, 36–37, 128
of integrals, 70, 128
Translation of coordinates:
in a plane, 24
in space, 36
Trapezoid, area, perimeter, 16
Trapezoidal rule (formula), 109, 231, 235
Tree diagrams, Probability, 219
289
Triangle inequality, 205
Triangular wave function, 191
Trigonometric functions, 43–52
definition of, 43
graphs of, 46
inverse, 49–50
series for, 139
tables of, 247–249
Triple integrals, 125
Trochoid, 30
Two-point formula, 228
Two-point interpolatory
formula, 228
Unit function, Heaviside’s, 192
Unit normal to the surface, 125
Unit vector, 120
Variance, 210
continuous random variable, 225
discrete random variable, 224
population, 210
sample, 210
Vector analysis, 119–133
Vector or cross-product, 121
Vectors, 119
derivatives of, 122
integrals involving, 124
unit, 119
Volume integrals, 125
Wallis’ product, 207
Wave equation, 238
Weber’s function, 153
Weighted mean, 209
Witch of Agnesi, 31
x-intercept, 22
y-intercept, 22
Zero vector, 119
Zeros of Bessel functions, 267
Zeta function of Riemann, 204