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Mellin Transform: { } 1 0 () () () s M x f x dx Fs fx - = = Inverse Mellin Transform: - 1 () () 2 ci s ci fx x Fsd i s π - - - = Q.1: Find the Mellin transform of x e - .

yashoverdhan-vyas
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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

= ∫

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)