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
sπ
−
−
∞
∞
−
= ∫
Q.1: Find the Mellin transform of x
e−
.

Soln:
1
0
( ); Re( ) 0{ }x s xM e x e d s sx
∞− − −= = Γ >∫
QBy the definition of Gamma function QBy the definition of Gamma function
1
0
( ) ; Re( ) 0z xx e dxz z
∞
− −Γ = >∫

Observe that
1
when 11
( ){ } { }
( )
x ssp
p
se
s
M L xp
− −
==
= =
Γ=
Γ
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
− − −∞
= =Γ
∫

Observe that
1
when } }
){
({ kx s
sp kp k
sM e L x
p
− −
==
= =
Γ
1Q.3: Find the Mellin transform of
1
1 x+.
Soln: { } 1
0
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
α β α βα β
α β− − Γ Γ
= − =Γ +∫

{ } ( ) (11
( ,1 ) )
; 0 Re( ) 1sin( )
1s s
s
M B s sx
s
π
π
∴ = − =
<
+Γ Γ −
= <
By a well-known result for Gamma
functions:

( ) (1 ) ; 0 Re( ) 1sin( )
z z zz
π
πΓ Γ − = < <
Q.4: Find the Mellin transform of cosnx
and sin nx .
( )sΓSoln: By Q.2, { }
(
( )
)
inx
sM
se
in
− Γ=

/2 2( ) ( )
( ) ( )( ) ( ) ( )
cos sin( ) 2
( )
2
( )is
s i s
s s s
s
i e en n n
s
s si
s s
s
n
ππ
π π
−− −Γ Γ
= = =
= −
Γ
Γ
( ) 2 2n
Separating real and imaginary parts, we
find that

( )cos2
( )s
{cos }
{sin } in2
s
s
M nx n
M
ss
ssnx n
π
π
−
−
= Γ
Γ
=
Above results can also be used to
calculate the Fourier cosine and Fourier
sine transform of 1p
x−
as follows:
By the definition of Mellin transform,

1
0
( )cos cos
2
s
s
s sx nxdx
n
π−∞
=
Γ
∫
Or equivalently
1 ( )2{ } cosp p p
F xπ− =
Γ
1 ( )2{ } cos
2
p
c p
p pF x
s
π
π− =
Γ
Similarly,
1 ( )2{ } sin
2
p
s p
p pF x
s
π
π− =
Γ

Properties of Mellin transforms:
If { ( )} ( )M f x F s= then
(i) Scaling Property:
{ ( )} ( ); 0sM f ax a F s a
−= >
(ii) Shifting Property: (ii) Shifting Property:
{ ( )} ( )aM x f x F s a= +
(iii) 1
{ ( )} ( )a sM f x F
a a=

(iv) {(log ) ( )} ( );
1,2,3,...
nn
n
dM x f x F s
dx
n
=
=
(v) Mellin transform of derivatives:
{ ( )} ( 1) ( 1)M f x s F s′ = − − −2
{ ( )} ( 1) ( 1)
{ ( )} ( 1) ( 1)( 2) ( 1)
M f x s F s
M f x s s F s
′ = − − −
′′ = − − − −
(vi) 2 2
{ ( )} ( )
{ ( )} ( 1) ( 1) ( )
M xf x s F s
M x f x s s F s
′ = −
′′ = − +

All the above properties can easily be
proved by the definition of Mellin
transform.







Alternative Notations for CWT(Continuous
Wavelet Transform)

Alternative Notations for CWT(Continuous
Wavelet Transform)





