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Poularikas A. D. The Mellin Transform

The Handbook of Formulas and Tables for Signal Processing.Ed. Alexander D. Poularikas

Boca Raton: CRC Press LLC,1999

1999 by CRC Press LLC

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1999 by CRC Press LLC

18The Mellin Transform18.1 The Mellin Transform

18.2 Properties of Mellin Transform

18.3 Examples of Mellin Transform

18.4 Special Functions Frequency Occuring in Mellin Transforms

18.5 Tables of Mellin Transform

References

18.1 The Mellin Transform

18.1.1 Definition

Example

18.1.2 Relation to Laplace Transform

By letting the transform becomes

18.1.3 Relation to Fourier Transform

By setting in 18.1.2 we obtain

which implies that

M{ ( ); } ( ) ( )f t s F s f t t dts= =

01

M{ ( )} ( )e u t e t dt a sat at s s

= =01

t e dt e dxx x= = , ,

M{ ( )} ( ) ( ) { ( )}f t F s f e e dx f ex sx x= = =

L

s j= + 2

F s f e e e dxx axj x( ) ( )=

2

M{ ( ); } { ( ) ; }f t s j f e ex ax= + = 2 F

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where

Fourier Transform

18.1.4 Inversion Formula

where c is within the strip of analyticity

18.2 Properties of Mellin Transform

18.2.1 Scaling Property

18.2.2 Multiplication by ta

18.2.3 Raising the Independent Variable to a Real Power

18.2.4 Inverse of Independent Variable

18.2.5 Multiplication by ln t

18.2.6 Multiplication by a Power of ln t

F{ ( ); } ( )f x f x e dxj x = =

2

f tj

F s t ds

c j

c j

s( ) ( )=

+

12a s b<

0

1

0

1 11 11

0

M{ ( ); } ( )t f t s F s = 1 1 1

M{ln ( ); } ( )t f t sd

dsF s=

M{(ln ) ( ); } ( )t f t sd

dsF sk

k

k=

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18.2.7 Derivat ive

18.2.8 Derivative Multiplied by Independent Variable

Example

18.2.9 Convolut ion

18.2.10 Multiplicative Convolution

Properties of the Multiplicative Convolution

1. commutative

2. associative

3. unit element

4.

5.

6.

Md

dsf t s s k F s k

k

k

k

k( ); ( ) ( ) ( )

= 1

( ) ( )( ) ( )( )!

( )!

( )

( )s k s k s k s

s

s k

s

s kk

+ =

=

1 1

1

1L

M td

dsf t s s F s

s k

sF s s s s s k k

k

k

k

k

k

k( ); ( ) ( ) ( ) ( )

( )

( )( ), ( ) ( ) ( )

= = + = + + 1 1 1 1

L

M td f t

dtt

df t

dts s F s

22

2

2( ) ( ) ; ( )+

=

M{ ( ) ( ); } ( ) ( )f t g t sj

F z G s z dz

c j

c j

=

+

12

M M

M

{ } ( ) ; ( ) ( )

{ ( ) ( )} ( )

f g ft

ug u

du

us F s G s

F s G s f t

ug u

du

u

=

=

=

0

1

0

0

f tu g udu

u( )

f g g f =

( ) ( )f g h f g h =

f t f =( )1

td

dtf g t

d

dtf g f t

d

dtg

k k k

=

=

( )

(ln )( ) [(ln ) ] [(ln ) ]t f g t f g f t g = + ( ) ( )t a f a f a t = 1 1

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18.2.11 Parsevals Formulas

where

18.3 Examples of Mellin Transform

18.3.1 Example

18.3.2 Example

.

Setting we obtain: Hence ,

18.3.3 Example

From with the setting we obtain

.

Hence,

.

18.3.4 Example

Using 18.2.3 and 18.3.3 we obtain

( ) ( ) ( ), ,

( )( )

t p t p t pp p p

d t

dtf

d

dst f

k

k

k

k

= >

=

0

1

0

0

2 1

1

21

+

+

=

=

f t g tj

f s g s ds

f t g t t dt f g d

c j

c j

r

( ) ( ) { ; } { ; }

( ) ( ) { }( ) { }( )

M M

M M

M{ }( ) ( )f f t t dt

j r

=

+

02

M{ ( )} , Re{ }t u t t t dt t

a ss aa

t

a s o

a s

= = +

<

+ +

0 10

M1

1

1

10

1

+

=+

t s tt dts;

tx

+ =

11

1x

t

tdx

dt

t=

+=

+1 1 2,

( ).

M f s xx

x

dx

xx x dx B s s s s

s

s

s

s

s s; ( )( ) ( )

( ) ( , ) ( ) ( )sin

Re{ } .

{ } =

= = = =

< > ( )

( ) ( )

( ), Re{ } , Re{ } ,

u x x= +/( ),1

0

1

1

+ + = +x

xdx

m s

m s

s

m s( )

( ) ( )

( )

M{( ) ; }( ) ( )

( ), Re{ } Re{ }1 0+ =

<

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.

18.3.5 Example

(see 5.3.1)

18.3.6 Example

From 18.3.1 and 18.3.5 and hence,

18.4 Special Functions Frequency Occurring in Mellin

Transforms

18.4.1 Definition

The gamma function (s) is defined on the complex half-plane Re(s) > 0 by the integral

18.4.2 Analytic Continuation

The analytical continued gamma function is holomorphic in the whole plane except at the points s =

where it has a simple pole.

18.4.3 Residues at the Poles

18.4.4 Relation to the Factorial

18.4.5 Functional Relations

M{( ) ; }

( / )

( ), Re{ } Re{ }1 0+ =

<

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(Legendres duplication formula)

(Gauss-Legendres multiplication formula)

(Stirling asymptotic formula)

18.4.6 The Beta Function

Definition:

Relation to the gamma function:

18.4.7 The psi Function (logarithmic derivative of the gamma function)

Definition:

Eulers constant , also called C, is defined by

and has value ....

18.4.8 Riemanns Zeta Function

The function (z) is analytic in the whole complex z-plane except inz = 1 where it has a simple pole

with residue equal to +1.

1

2

=

( ) ( ) ( / )/2 2 1 21 2 2 1s s ss= +

( ) ( ) ( / ), , , ,/ ( ) /ms m s k m mms m

k

m

= + =

=

1 2 1 20

1

2 2 3 L

( ) ~ exp ( ) , , arg( )/s s ss

O s s ss2 11

12

1 2 2 +

+

>

=

=

0

0

11 0 1 2

11

L

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18.5 Tables of Mellin Transform

18.5.1 Tables of Mellin Transform

TABLE 18.1 Some Standard Mellin Transform Pairs

Original Function Mellin Transform

Strip of holomor phy

Algebraic Functions

Exponential Functions

f t t( ), > 0 M f s f t t dts[ ; ] ( )

10

u t a t ab( ) , > 0 +

+a

b s

b s

Re( ) Re( )s b<

( ( ) ( ))u t a u t t b +

+a

b s

b s

Re( ) Re( )s b>

( )1 1+ t sin( )s

0 1<

t t

t

h

h

h

s

h

sh

+

1

( )

Re( )s > 0

( )( )1 1 1 t ta na na n

s

na

s a

nasin csc csc

+

0 1< < Re( ) ( )s n a

e ppt >, 0 p ss ( ) Re( )s > 0

( )e t 1 1 ( ) ( )s s Re( )s > 1

( ( ) s = zeta function)

( ) , Re( )e aat + >1 01 a s ss s ( )( ) ( )1 21 Re( )s > 0

( ) , Re( )e a

at

>

1 0

1

a s s

s

( ) ( ) Re( )s > 1( )( ) , Re( )e e aat t >1 01 ( ) ( , )s s a Re( )s > 1

( )et 1 2 ( )[ ( ) ( )]s s s 1 Re( )s > 2

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Logarithmic Functions

Trigonometric Functions

TABLE 18.1 Some Standard Mellin Transform Pairs (continued)


Strip of holomor phyf t t( ), > 0 M f s f t t dt s[ ; ] ( )

1

0

e a hath > >, Re( ) ,0 0 h a s hs h 1 / ( / ) Re( )s > 0

t e t 1 1 ( )1 s < >e a hath

, Re( ) , h a s hs h1 / ( / ) < 0p

s

s

2Re( )s > 0

sin( ),at a > 0 a s ss ( )sin( / ) 2 < sin( ), Re( ) Im ( ) ( )sin tan/a s sa

s2 2 2 1+

Re( )s > 1

sin ( ),2 0at a > 2 21s sa s s( )cos( / ) < 0 a s ss ( )cos( / ) 2 0 1<

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References

Bertrand, Jacqueline, Pierre Bertrand, and Jean-Philippe Ovarlez, The Mellin Transform, in Transforms

and Applications Handbook, ed. Alexander Poularikas, CRC Press, Boca Raton, Florida, 1996.

Davies, G.,Integral Transforms and Their Applications, 2nd ed., Springer-Verlag, New York, NY, 1984.

Erdelyi, A., W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transfer, McGraw-Hill

Book Co., New York, NY, 1954.

Oberhettinger, F., Tables of Mellin Transform, 2nd ed., Springer-Verlag, New York, NY, 1974.

Sneddon, Ian N., The Use of Integral Transform, McGraw-Hill Book Co., New York, NY, 1972.

Other Functions

whole plane

none (analytic functional)

TABLE 18.1 Some Standard Mellin Transform Pairs (continued)


Strip of holomor phyf t t( ), > 0 M f s f t t dt s[ ; ] ( )

1

0

J at a ( ), > 02

2 2

2 21

1s

s

s

as

+

+

< 02

1

2

1

2 2 2

1 12 2

1v

s

ss

a v ss

+ +

+

( )

< < 0 ps1

( ),t pn pn

>=

01

p ss 1 1( ) Re( )s < 0

J t ( )2 2

1 2 1

1s s

s

+ +

( ) /

[( / )( ) ]

< =0, real = Im( )s

tb ( )b s+

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