Mellin Transform:

{ } 1

0

( ) ( )( )s

M x f x dx F sf x

= =

Inverse Mellin Transform:

1( ) ( )

2

c i

s

c i

f x x F s di

s

=

Q.1: Find the Mellin transform of x

e

.

Soln:

1

0

( ); Re( ) 0{ }x s xM e x e d s sx

= = >

QBy the definition of Gamma function QBy the definition of Gamma function

1

0

( ) ; Re( ) 0z xx e dxz z

= >

Observe that

1

when 11

( ){ } { }

( )

x ssp

p

se

s

M L xp

==

= =

=

where L denotes Laplace transform.

Q.2: Find the Mellin transform of kx

e

,

where 0k > .

where 0k > .

Soln: { } 1

0

s kxkxM x e dxe

=

By putting kx t= , we get

1

0

1 ((

))kx s t

s sM e t e dt

k

s

k

= =

Observe that

1

when } }

){

({ kx s

sp kp k

sM e L x

p

==

= =

1Q.3: Find the Mellin transform of

1

1 x+.

Soln: { } 10

11.11

sM x dx

xx

=

++

By substituting 1

tx

t=

or

1

xt

x=

+.

{ }1

1 (1 ) 1

0

1(1 )

1

s sM t t dt

x

= + 0

( ,1 )B s s=

where, the Beta function is: 1

1 1

0

( ) ( )( , ) (1 )

( )B t t dt

= = +

{ } ( ) (11 ( ,1 ) ); 0 Re( ) 1

sin( )

1s s

s

M B s sx

s

= =