Page 728 - Handbook Of Integral Equations

∞

˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
d m n ˜

m (n)
18 x f (x), m ≥ n – p f(p)
x dp
d n m m n d m
˜
19 x f(x) , m ≥ n (–1) p m f(p)
dx n dp
˜
x f(p)
20 f(t) dt
0 p
x
1
˜
21 (x – t)f(t) dt 2 f(p)
0 p
x
ν
22 (x – t) f(t) dt, ν > –1 Γ(ν +1)p –ν–1 ˜
f(p)
0
x
1
˜
23 e –a(x–t) f(t) dt f(p)
0 p + a
x
˜
af(p)
24 sinh a(x – t) f(t) dt 2 2
0 p – a
x
˜
af(p)
25 sin a(x – t) f(t) dt 2 2
0 p + a
x

˜
˜
26 f 1 (t)f 2 (x – t) dt f 1 (p)f 2 (p)
0
x
1 1 ˜
∞
27 f(t) dt f(q) dq
0 t p p
p
1 1
∞
˜
28 f(t) dt f(q) dq
x t p 0
√
1
∞ 1 √ π

29 √ sin 2 xt f(t) dt √ f ˜
0 t p p p
√
1 ∞ √ π

1
30 √ cos 2 xt f(t) dt √ f ˜
x p p
0

∞ 1 t 2 1 √
31 √ exp – f(t) dt √ f ˜ p
0 πx 4x p
∞
t t 2 √

32 √ exp – f(t) dt f ˜ p
0 2 πx 3 4x
x
√

2
2
33 f(x) – a f x – t 2 J 1 (at) dt f ˜ p + a 2
0
x √
2
2
34 f(x)+ a f x – t 2 I 1 (at) dt f ˜ p – a 2
0
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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