Page 764 - Handbook Of Integral Equations

∞
ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x s–1 dx
0
–β –1 s/β
6 exp –ax , a, β >0 β a Γ(–s/β), Re s <0
β –1 –s/β
7 1 – exp –ax , a, β >0 –β a Γ(s/β), –β <Re s <0
–1 s/β

8 1 – exp –ax –β , a, β >0 –β a Γ(–s/β), 0<Re s < β
8.4. Expressions With Logarithmic Functions

∞ s–1

ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x dx
0
ln x if 0 < x < a, s ln a – 1

1 , Re s >0
0 if a < x s a
2 s
π
2 ln(1 + ax), a >0 s , –1<Re s <0
sa sin(πs)
π
3 ln |1 – x| cot(πs), –1<Re s <0
s
ln x πa s–1 ln a – π cot(πs)
4 , a >0 , 0 < Re s <1
x + a sin(πs)
s–1 s–1 s–1 s–1
ln x π a ln a – b ln b – π cot(πs)(a – b )
5 , a, b >0 (b – a) sin(πs) ,
(x + a)(x + b)
0<Re s <1
x ln x if 0 < x <1, 1
ν
6 – 2 , Re s > –ν
0 if 1 < x (s + ν)
3
2

2
ln x π 2 – sin (πs)
7 3 , 0<Re s <1
x +1 sin (πs)

ν–1
ln x if 0 < x <1, –ν
8 Γ(ν)(–s) , Re s <0, ν >0
0 if 1 < x
2π cos(βs)
2
9 ln x +2x cos β +1 , |β| < π , –1<Re s <0
s sin(πs)
π 1
10 1+ x tan 2 πs , –1<Re s <1
1 – x s
ln
d n
n
–x
11 e ln x, n =1, 2, ... Γ(s), Re s >0
ds n
8.5. Expressions With Trigonometric Functions
∞
ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x s–1 dx
0
–s

1 sin(ax), a >0 a Γ(s) sin 1 πs , –1<Re s <1
2
2

a Γ(s) cos
2 sin (ax), a >0 –2 –s–1 –s 1 2 πs , –2<Re s <0
1 Γ(s) cos 1 πs |b – a| – (b + a) –s ,
–s

3 sin(ax) sin(bx), a, b >0, a ≠ b 2 2
–2<Re s <1
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 749