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

    = =