# MellinTransform

MellinTransform[expr,x,s]

gives the Mellin transform of expr.

MellinTransform[expr,{x1,x2,},{s1,s2,}]

gives the multidimensional Mellin transform of expr.

# Details and Options • The Mellin transform of a function is defined to be .
• The multidimensional Mellin transform of a function is given by .
• The Mellin transform of exists only for complex values of such that . In some cases, this strip of definition may extend to a half-plane.
• • The following options can be given:
•  Assumptions \$Assumptions assumptions on parameters GenerateConditions False whether to generate results that involve conditions on parameters Method Automatic what method to use
• In TraditionalForm, MellinTransform is output using .

# Examples

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## Basic Examples(2)

Compute the Mellin transform of a function:

Compute a multivariate Mellin transform:

## Scope(16)

### Basic Uses(3)

Compute the Mellin transform of a function for a symbolic parameter s:

Use an exact value for the parameter:

Use an inexact value for the parameter:

Obtain the condition for validity of a Mellin transform:

The result is valid in the half-plane : ### Elementary Functions(3)

Exponential function:

Gaussian function:

General exponential functions:

Composition of logarithmic and exponential functions:

Rational functions:

Mellin transforms of polynomials are given in terms of DiracDelta:

### Special Functions(3)

Product of Bessel functions:

Exponential integral function ExpIntegralE:

MellinTransform of the error function Erf:

Complementary error function Erfc:

### Piecewise Functions(3)

Products of functions with UnitStep:

MellinTransform of a Piecewise function:

### Generalized Functions(2)

MellinTransform of functions involving HeavisideTheta:

### Multivariate Functions(2)

Multivariate rational function:

Multivariate exponential function:

## Options(5)

### Assumptions(1)

Compute the Mellin transform of a function depending on a parameter a:

Obtain a simpler result by specifying assumptions on the parameter:

### GenerateConditions(1)

Obtain conditions for validity of the result given by MellinTransform:

GenerateConditions is set to False by default in this case:

### Method(3)

Compute a Mellin transform using the default method:

This example is done using table lookup:

Attempting to evaluate this example by a conversion to MeijerG fails:

Evaluate the example using the definition of MellinTransform in terms of Integrate:

The default method uses a conversion to MeijerG for this example:

This is faster than using the definition of MellinTransform in terms of Integrate:

Here, the symbolic method fails because the input is purely numerical:

This example is evaluated using a numerical method based on NIntegrate:

## Applications(3)

Use MellinTransform to evaluate , which may be regarded as a Mellin convolution of the following functions:

Apply MellinTransform to each function:

Obtain the required integral by performing an inverse Mellin transform:

Compute the integral directly using Integrate:

Obtain the same result using MellinConvolve:

Find the general solution of the Bessel equation using MellinTransform:

Apply MellinTransform to the equation:

Use RSolveValue to solve the recurrence equation:

Use InverseMellinTransform to find the required general solution:

Verify the result using DSolveValue:

Use MellinTransform to find the first two terms in the asymptotic expansion for a function that is defined by an infinite series:

Compute the Mellin transform of :

Compute the residues at and to obtain the required asymptotic expansion:

## Properties & Relations(11)

Use Asymptotic to compute an asymptotic approximation:

MellinTransform computes the integral :

Obtain the same result using Integrate:

MellinTransform and InverseMellinTransform are mutual inverses:

Verify the relationship for a specific function:

MellinTransform is a linear operator:

The Mellin transform of is given by :

The Mellin transform of is given by for positive values of a:

The Mellin transform of is given by for real values of a:

The Mellin transform of is given by :

The Mellin transform of is given by :

The Mellin transform of a Mellin convolution is the product of the individual Mellin transforms:

Verify the relationship for a specific pair of functions:

MellinTransform is related to FourierTransform by :

## Possible Issues(1)

Different expressions may have the same MellinTransform:

The transforms in these examples have different regions of convergence:

With region of convergence given, the inverse transform gives back the input:

## Neat Examples(1)

Create a table of basic Mellin transforms: