Page 735 - Handbook Of Integral Equations

Supplement 5

Tables of Inverse Laplace Transforms

5.1. General Formulas

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
˜
1 f(p + a) e –ax f(x)
x
1
˜
2 f(ap), a >0 f
a a
x
1 b
˜
3 f(ap + b), a >0 exp – x f
a a a
˜
˜
4 f(p – a)+ f(p + a) 2f(x) cosh(ax)
˜
˜
5 f(p – a) – f(p + a) 2f(x) sinh(ax)

0 if 0 ≤ x < a,
–ap ˜
6 e f(p), a ≥ 0
f(x – a)if a < x.
df(x)
˜
7 pf(p) , if f(+0) = 0
dx
1 x
˜
8 f(p) f(t) dt
p 0
1 –ax x at
˜
9 f(p) e e f(t) dt
p + a
0
x

1
˜
10 2 f(p) (x – t)f(t) dt
p 0
˜
f(p) 1 x a(x–t)
11 1 – e f(t) dt
p(p + a) a 0
˜
f(p) x –a(x–t)
12 (x – t)e f(t) dt
(p + a) 2 0
˜
f(p) 1 x –a(x–t) –b(x–t)
13 e – e f(t) dt
(p + a)(p + b) b – a 0
˜
f(p) 1 x –a(x–t)
14 e sin b(x – t) f(t) dt
2
(p + a) + b 2 b 0
x
1 1 n–1

˜
15 n f(p), n =1, 2, ... (x – t) f(t) dt
p (n – 1)!
0
x

˜
˜
16 f 1 (p)f 2 (p) f 1 (t)f 2 (x – t) dt
0
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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