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inttrans

  

invmellin

  

inverse Mellin transform

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

invmellin(expr, t, s)

Parameters

expr

-

expression, equation, or set of expressions or equations to be transformed

t

-

variable expr is transformed with respect to t

s

-

parameter of transform

ran

-

range for Re(t) (optional)

opt

-

option to run transform under (optional)

Description

• 

The invmellin function computes the inverse Mellin transform (F(s)) of expr (f(t)), a linear transformation from CCC0, defined by the contour integral:

Fs=I2cIc+Iftstⅆtπ

• 

In this integral, c is assumed to be real. Also note that Maple currently does not handle general contour integrals. The above contour integral definition is only used to provide the information below on properties of the inverse Mellin transform.

• 

The function Fs returned is defined only on the positive real axis.

• 

There are multiple transforms Fs for a given ft, corresponding to the cases where c1<t <c2 for various boundaries c1 and c2.  The range is specified by the parameter ran.  This parameter is optional. If the range parameter is not given, it is assumed to be ...

• 

All constants are assumed to be complex unless otherwise specified.

• 

The invmellin function attempts to simplify an expression according to a set of heuristics, and then to match the result against internal lookup tables of patterns.  These tables are of expressions containing algebraic, Bessel, exponential, GAMMA, trigonometric, as well as other functions. The user can add their own functions to invmellin's lookup tables with the function addtable.

• 

Other functions that can be transformed are linear combinations of products of integer powers of t; rational polynomials; terms of the form at where 0<a; some definite integrals of functions whose transforms are known; derivatives of functions whose transforms are known; convolutions of two functions ft and gt whose transforms are known; and functions of the form fat&plus;b and fatn with 0<a&comma;b complex, and n a positive integer where the transforms of ft and ftn are known.

• 

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.

• 

invmellin recognizes the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).

• 

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

Examples

withinttrans&colon;

assume0<a

assumeb&comma;complex&colon;

assumec&comma;complex&colon;

assumen&comma;posint&colon;

Inversion of mellin

Gmellingx&comma;x&comma;y

G:=mellingx&comma;x&comma;y

(1)

invmellinG&comma;y&comma;z

gz

(2)

Adding to the table

withinttrans&colon;

addtableinvmellin&comma;Ft&comma;fs&comma;t&comma;s&comma;invmellin&equals;&infin;..&infin;&colon;

addtableinvmellin&comma;F1t&comma;f1s&comma;t&comma;s&comma;invmellin&equals;&infin;..&infin;&colon;

addtableinvmellin&comma;F2t&comma;f2s&comma;t&comma;s&comma;invmellin&equals;&infin;..&infin;&colon;

invmellinFx&comma;x&comma;y

fy

(3)

invmellinF1x&comma;x&comma;y

f1y

(4)

invmellinF2x&comma;x&comma;y

f2y

(5)

General properties

invmellinbF1z&plus;cF2z&comma;z&comma;x

b~f1x&plus;c~f2x

(6)

invmellinFaz&plus;b&comma;z&comma;x

xb~a~fx1a~a~

(7)

invmellinFazn&comma;z&comma;x

invmellinFzn~&comma;z&comma;x1a~1n~a~1n~

(8)

invmellinDFz&comma;z&comma;x

lnxfx

(9)

invmellinzFz&comma;z&comma;x

x&DifferentialD;&DifferentialD;xfx

(10)

invmellinazFz&comma;z&comma;x

invmellin&ExponentialE;zlna~Fz&comma;z&comma;x

(11)

invmellinFz&plus;1z&comma;z&comma;x

&int;0&infin;invmellin1z&comma;z&comma;x_U&comma;&infin;..&infin;f_U&DifferentialD;_U

(12)

invmellin&Gamma;2zFz1&Gamma;1z&comma;z&comma;x

&DifferentialD;&DifferentialD;xfx

(13)

invmellinF1zF21z&comma;z&comma;x

&int;0&infin;f1x_Uf2_U&DifferentialD;_U

(14)

Some simple functions

invmellin1&comma;z&comma;x

Diracx1

(15)

invmellinz&comma;z&comma;x

xDirac1&comma;x1

(16)

invmellin&ExponentialE;az2&comma;z&comma;x

12&ExponentialE;14ln1x2a~&pi;a~

(17)

Specifying ranges

invmellin&Gamma;z&comma;z&comma;x

Invmellin can transform GAMMA(t) if Re(t)<>0, Re(t) > -1

(18)

invmellin&Gamma;z&comma;z&comma;x&comma;0..&infin;

&ExponentialE;x

(19)

invmellin&Gamma;z&comma;z&comma;x&comma;1..0

&ExponentialE;x1

(20)

See Also

dsolve

inttrans

inttrans[addtable]

inttrans[mellin]

 


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