Presentation Final
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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
= =
