Page 762 - Handbook Of Integral Equations

Supplement 8

Tables of Mellin Transforms

8.1. General Formulas

∞ s–1

ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x dx
0
ˆ
ˆ
1 af 1 (x)+ bf 2 (x) af 1 (s)+ bf 2 (s)
–s ˆ
2 f(ax), a >0 a f(s)
a
ˆ
3 x f(x) f(s + a)
ˆ
4 f(1/x) f(–s)
1
s
β
5 f x , β >0 f ˆ
β β
1 s
ˆ
–β
6 f x , β >0 f –
β β
β
β ˆ
λ

a
f
7 x f ax , a, β >0 1 – s+λ s + λ
β β
1 s+λ s + λ
–β
λ

f –
8 x f ax , a, β >0 a β ˆ
β β
ˆ

9 f (x) –(s – 1)f(s – 1)
x
ˆ

10 xf (x) –sf(s)
x
Γ(s)
ˆ
n
11 f x (n) (x) (–1) f(s – n)
Γ(s – n)
d n n ˆ
n
12 x f(x) (–1) s f(s)
dx
d n n ˆ
n
13 x f(x) (–1) (s – 1) f(s)
dx
∞

β
ˆ
ˆ
14 x α t f 1 (xt)f 2 (t) dt f 1 (s + α)f 2 (1 – s – α + β)
0

x
∞
β
ˆ
ˆ
15 x α t f 1 f 2 (t) dt f 1 (s + α)f 2 (s + α + β +1)
0 t
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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