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inttrans

  

mellin

  

Mellin transform

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

mellin(expr, x, s)

Parameters

expr

-

expression to be transformed

x

-

variable expr is transformed with respect to x

s

-

parameter of transform

opt

-

option to run this under (optional)

Description

• 

The function mellin computes the Mellin transform (M(s)) of expr (m(x)) with respect to x, using the definition

Ms=0mxxs1ⅆx

• 

Some expressions involving exponentials, polynomials, algebraic functions, trigonometrics (sin, cos, sinh, cosh) or various special functions can be transformed.  The procedure will be able to obtain the Mellin transforms of all the functions of the type Klnxnfaxbxc as long as the Mellin transform of fx is known.

• 

The mellin function attempts to reduce the expression according to a set of simplification rules and then tries to match the reduced expression against an internal table of basic Mellin transforms.

• 

Users can add their own functions to mellin's internal lookup table by using the addtable function.

• 

If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail.  This will increase the speed at which the transform will run.

• 

The command with(inttrans,mellin) allows the use of the abbreviated form of this command.

Examples

withinttrans:

assume0<s&colon;

mellinaxb&ExponentialE;x14&comma;x&comma;s

4a&Gamma;4s~&plus;4b

(1)

mellinxx2&plus;1&comma;x&comma;s

12&pi;sin12&pi;s~&plus;1

(2)

mellinlnxxx2&plus;1&comma;x&comma;s2

14&pi;2cos12&pi;s~1sin12&pi;s~12

(3)

mellin1x3x&plus;1&comma;x&comma;s

_&alpha;&equals;RootOf_Z3_Z&plus;11236_&alpha;2&plus;9_&alpha;4_&alpha;s~_&alpha;&pi;sin&pi;s~

(4)

mellin&ExponentialE;3x2&ExponentialE;x21&comma;x&comma;s

12&Gamma;12s~&zeta;12s~212s~312s~212s~312s~212s~312s~212s~312s~

(5)

mellinlnx&ExponentialE;3x2&ExponentialE;x21&comma;x&comma;s

14&Gamma;12s~&Psi;12s~&zeta;12s~212s~312s~&Psi;12s~212s~312s~&plus;&zeta;1&comma;12s~212s~312s~&Psi;12s~212s~&Psi;12s~312s~&plus;ln2312s~&plus;ln3212s~212s~312s~

(6)

addtablemellin&comma;ft&comma;Fs&comma;t&comma;s&colon;

mellinfx&comma;x&comma;s

Fs~

(7)

See Also

dsolve

inttrans

inttrans[addtable]

inttrans[invmellin]

 


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