Page 727 - Handbook Of Integral Equations

Supplement 4

Tables of Laplace Transforms

4.1. General Formulas

∞

˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
˜
˜
1 af 1 (x)+ bf 2 (x) af 1 (p)+ bf 2 (p)
˜
2 f(x/a), a >0 af(ap)

0 if 0 < x < a, –ap ˜
3 e f(p)
f(x – a)if a < x,
d n
˜
n
n
4 x f(x); n =1, 2, ... (–1) n f(p)
dp
∞
1 ˜
5 f(x) f(q) dq
x p
˜
ax
6 e f(x) f(p – a)

f(p – a) – f(p + a)
7 sinh(ax)f(x) 1 ˜ ˜
2

f(p – a)+ f(p + a)
8 cosh(ax)f(x) 1 ˜ ˜
2

f(p – iω) – f(p + iω) , i = –1
9 sin(ωx)f(x) – i ˜ ˜ 2
2

f(p – iω)+ f(p + iω) , i = –1
10 cos(ωx)f(x) 1 ˜ ˜ 2
2
1 ∞ p 2
˜ 2
2
11 f(x ) √ exp – 2 f(t ) dt
π 0 4t

1 a/2 ˜
∞ √
a–1
12 x f , a > –1 (t/p) J a 2 pt f(t) dt
x 0
∞

˜
13 f(a sinh x), a >0 J p (at)f(t) dt
0
1 a –px
14 f(x + a)= f(x) (periodic function) ap f(x)e dx
1 – e 0
a

f(x + a)= –f(x) 1 –px
15 f(x)e dx
(antiperiodic function) 1+ e –ap 0
˜

16 f (x) pf(p) – f(+0)
x
n

n ˜
f
17 f x (n) (x) p f(p) – p n–k (k–1) (+0)
x
k=1
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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