Page 452 - Handbook Of Integral Equations

TABLE 1
Main properties of the Laplace transform

No Function Laplace Transform Operation
˜
˜
1 af 1 (x)+ bf 2 (x) af 1 (p)+ bf 2 (p) Linearity
˜
2 f(x/a), a >0 af(ap) Scaling
f(x – a), –ap ˜
3 e f(p) Shift of the argument
f(ξ) ≡ 0 for ξ <0
n
n ˜ (n)
4 x f(x); n =1, 2, ... (–1) f p (p) Differentiation
of the transform
1 ∞ ˜ Integration
5 f(x) f(q) dq
x p of the transform
Shift in
ax
˜
6 e f(x) f(p – a)
the complex plane
˜

7 f (x) pf(p) – f(+0) Differentiation
x
n n–k (k–1)
n ˜
(n)
8 f x (x) p f(p) – p f x (+0) Differentiation
k=1
d
m
n ˜
m (n)
9 x f x (x), m ≥ n – p f(p) Differentiation
dp
d n m m n d m ˜
10 x f(x) , m ≥ n (–1) p m f(p) Differentiation
dx n dp
˜
x f(p)

11 f(t) dt Integration
0 p

x
˜
˜
12 f 1 (t)f 2 (x – t) dt f 1 (p)f 2 (p) Convolution
0
7.3. The Mellin Transform
7.3-1. Definition. The Inversion Formula
Suppose that a function f(x)isdefined for positive x and satisfies the conditions

1 ∞
|f(x)|x σ 1 –1 dx < ∞, |f(x)|x σ 2 –1 dx < ∞
0 1
for some real numbers σ 1 and σ 2 , σ 1 < σ 2 .
The Mellin transform of f(x)isdefined by

∞
ˆ
f(s)= f(x)x s–1 dx, (1)
0
where s = σ + iτ is a complex variable (σ 1 < σ < σ 2 ).
For brevity, we rewrite formula (1) as follows:
ˆ
ˆ
f(s)= M{f(x)}, or f(s)= M{f(x), s}.
ˆ
Given f(s), the function can be found by means of the inverse Mellin transform
σ+i∞
1 –s
ˆ
f(x)= f(s)x ds, (σ 1 < σ < σ 2 ) (2)
2πi
σ–i∞
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 433