# Table of Laplace Transforms

```Table of Laplace Transforms
469
Table of Laplace Transforms
y = f (t), t &gt; 0
[y = f (t) = 0, t &lt; 0]
Y = L(y) = F (p) =
∞
e−pt f (t) dt
0
L1
1
1
p
L2
e−at
1
p+a
Re (p + a) &gt; 0
L3
sin at
a
p 2 + a2
Re p &gt; | Im a|
L4
cos at
p
+ a2
Re p &gt; | Im a|
L5
tk , k &gt; −1
L6
tk e−at , k &gt; −1
L7
e−at − e−bt
b−a
1
(p + a)(p + b)
L8
ae−at − be−bt
a−b
p
(p + a)(p + b)
L9
sinh at
a
p 2 − a2
Re p &gt; | Re a|
L10
cosh at
p
p 2 − a2
Re p &gt; | Re a|
L11
t sin at
L12
p2
k!
Γ(k + 1)
or
pk+1
pk+1
k!
Γ(k + 1)
or
k+1
(p + a)
(p + a)k+1
Re p &gt; 0
Re p &gt; 0
Re (p + a) &gt; 0
Re (p + a) &gt; 0
Re (p + b) &gt; 0
Re (p + a) &gt; 0
Re (p + b) &gt; 0
2ap
+ a2 )2
Re p &gt; | Im a|
t cos at
p 2 − a2
(p2 + a2 )2
Re p &gt; | Im a|
L13
e−at sin bt
b
(p + a)2 + b2
Re (p + a) &gt; | Im b|
L14
e−at cos bt
p+a
(p + a)2 + b2
Re (p + a) &gt; | Im b|
L15
1 − cos at
a2
p(p2 + a2 )
Re p &gt; | Im a|
L16
at − sin at
a3
+ a2 )
Re p &gt; | Im a|
L17
sin at − at cos at
(p2
p2 (p2
2a3
(p2 + a2 )2
Re p &gt; | Im a|
470
Ordinary Differential Equations
Chapter 8
Table of Laplace Transforms (continued)
y = f (t), t &gt; 0
[y = f (t) = 0, t &lt; 0]
L18
e−at (1 − at)
L19
sin at
t
L20
1
sin at cos bt,
t
Y = L(y) = F (p) =
L22
L23
L24
L25
e−pt f (t) dt
0
p
(p + a)2
Re (p + a) &gt; 0
a
p
Re p &gt; | Im a|
arc tan
1
2
arc tan
a+b
a−b
+ arc tan
p
p
a &gt; 0, b &gt; 0
Re p &gt; 0
e−at − e−bt
t
L21
∞
ln
p+b
p+a
a
√ , a&gt;0
2 t
(See Chapter 11, Section 9)
1 −a√p
e
p
J0 (at)
(See Chapter 12, Section 12)
(p2 + a2 )−1/2
1, t &gt; a &gt; 0
0, t &lt; a
(unit step, or Heaviside function)
1 −pa
e
p
1 − erf
u(t − a) =
f (t) = u(t − a) − u(t − b)
Re (p + a) &gt; 0
Re (p + b) &gt; 0
Re p &gt; 0
Re p &gt; | Im a|;
or Re p ≥ 0
for real a = 0
Re p &gt; 0
e−ap − e−bp
p
All p
1
tanh 12 ap
p
Re p &gt; 0
1
a
0
L26
b
t
f (t)
1
t
−1
L27
a
2a 3a 4a
δ(t − a), a ≥ 0
(See Section 11)
L28
L29
g(t − a), t &gt; a &gt; 0
0,
t&lt;a
= g(t − a)u(t − a)
f (t) =
e−at g(t)
e−pa
e−pa G(p)
[G(p) means L(g).]
G(p + a)
Table of Laplace Transforms
Table of Laplace Transforms (continued)
y = f (t), t &gt; 0
[y = f (t) = 0, t &lt; 0]
L30
L31
L33
G(u) du
(−1)n
t
L34
0
0
g(t − τ )h(τ ) dτ =
0
dn G(p)
dpn
1
G(p)
p
g(τ ) dτ
t
∞
p
tn g(t)
1 p
G
a
a
(if integrable)
L32
L35
0
g(at), a &gt; 0
g(t)
t
∞
Y = L(y) = F (p) =
t
g(τ )h(t − τ ) dτ
G(p)H(p)
(convolution of g and h, often
written as g ∗ h; see Section 10)
Transforms
L(y ) =
L(y ) =
L(y ) =
L(y (n) ) =
of derivatives of y (see Section 9):
pY − y0
p2 Y − py0 − y0
p3 Y − p2 y0 − py0 − y0 , etc.
(n−1)
pn Y − pn−1 y0 − pn−2 y0 − &middot; &middot; &middot; − y0
e−pt f (t) dt
471
```