Page 763 - Handbook Of Integral Equations

8.2. Expressions With Power-Law Functions

∞ s–1

ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x dx
0
x if 0 < x <1, 2(2 – 1)
s
1 2 – x if 1 < x <2, s(s +1) if s ≠ 0, Re s > –1
0 if 2 < x 2ln2 if s =0,
1 πa s–1
2 , a >0 , 0<Re s <1
x + a sin(πs)
s–1 s–1
1 π a – b
3 , a, b >0 , 0 < Re s <2
(x + a)(x + b) (b – a) sin(πs)
x + a π b – a b s–1 + c – a c s–1 ,
4 , b, c >0 sin(πs) b – c c – b
(x + b)(x + c)
0<Re s <1
1 πa s–2
5 2 2 , a >0 1 , 0<Re s <2
x + a 2 sin πs
2

1 πa s–2 sin β(s – 1)
6 , a >0, |β| < π – , 0<Re s <2
2
x +2ax cos β +a 2 sin β sin(πs)
s–2 s–2
1 π a – b
7 2 2 2 2 , a, b >0 1 , 0<Re s <4
2
(x + a )(x + b ) 2(b – a ) sin πs
2
2
n
1 (–1) π n
8 n+1 , a >0, n =1, 2, ... s C s–1 , 0<Re s < n +1
(1 + ax) a sin(πs)
1 πa s–n
9 , a >0, n =1, 2, ... , 0 < Re s < n
n
x + a n n sin(πs/n)
1 – x π sin(π/n)
10 , n =2, 3, ... , 0 < Re s < n – 1
1 – x n n sin(πs/n) sin π(s +1)/n
x if 0 < x <1, 1
ν
11 , Re s > –ν
0 if 1 < x s + ν
1 – x ν π sin(π/n)
12 , n =2, 3, ... πs π(s+ν) , 0 < Re s <(n – 1)ν
1 – x nν nν sin sin
nν nν
8.3. Expressions With Exponential Functions
∞
ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x s–1 dx
0
–s
1 e –ax , a >0 a Γ(s), Re s >0
–bx
e if 0 < x < a, –s
2 b >0 b γ(s, ab), Re s >0
0 if a < x,

0 if 0 < x < a, –s
3 –bx b >0 b Γ(s, ab)
e if a < x,
e –ax ab s–1
4 , a, b >0 e b Γ(s)Γ(1 – s, ab), Re s >0
x + b
β –1 –s/β
5 exp –ax , a, β >0 β a Γ(s/β), Re s >0

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 748