
Chapter 2 8
Integral Transforms
28.1 Tables of Laplace Transforms
28.1.1 G eneral Formulas
No. Original function, f(x)
Laplace transform,
e
f(p) =
Z
∞
0
e
−px
f(x) dx
1 af
1
(x) + bf
2
(x)
a
e
f
1
(p) + b
e
f
2
(p)
2 f(x/a), a > 0
a
e
f
(ap)
3
0 if 0 < x < a,
f(x − a) if x > a
e
−ap
e
f
(p)
4 x
n
f(x); n = 1, 2, . . .
(−1)
n
d
n
dp
n
e
f
(p)
5
1
x
f(x)
Z
∞
p
e
f(q) dq
6 e
ax
f(x)
e
f(p − a)
7 sinh(ax)f(x)
1
2
e
f(p − a) −
e
f(p + a)
8 cosh(ax)f (x)
1
2
e
f(p − a) +
e
f(p + a)
9 sin(ωx)f(x)
−
i
2
e
f(p − iω) −
e
f(p + iω)
, i
2
= −1
10 cos(ωx)f(x)
1
2
e
f(p − iω) +
e
f(p + iω)
, i
2
= −1
11
f(x
2
)
1
√
π
Z
∞
0
exp
−
p
2
4t
2
e
f
(t
2
) dt
12
x
a−1
f
1
x
, a > −1
Z
∞
0
(t/p)
a/2
J
a
2
p
pt
e
f(t) dt
13 f(a sinh x), a > 0
Z
∞
0
J
p
(at)
e
f(t) dt
14
f(x + a) = f(x)
(periodic function) ...