# InverseMellinTransform

InverseMellinTransform[expr,s,x]

gives the inverse Mellin transform of expr.

# Details and Options

• The inverse Mellin transform of a function is defined to be , where the integration is along a vertical line , lying in a strip in which the function is holomorphic. In some cases, the strip of holomorphy may extend to a half-plane.
• ConditionalExpression[expr,α<Re[s]<β] can be used to indicate the strip of holomorphy. »
• 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
• GenerateConditions can be used to obtain the strip of holomorphy. »
• Assumptions can be used to specify a strip of holomorphy. »
• If both Assumptions and ConditionalExpression are used to constrain the strip of holomorphy, then the intersection of the strips is used.
• In TraditionalForm, InverseMellinTransform is output using .

# Examples

open allclose all

## Basic Examples(2)

Compute the InverseMellinTransform of a function:

InverseMellinTransform for a product of functions:

## Scope(10)

### Basic Uses(4)

Compute the inverse Mellin transform of a function for a symbolic parameter x:

Use an exact value for the parameter:

Use an inexact value for the parameter:

Obtain the strip of holomorphy assumed by InverseMellinTransform:

Specify assumptions:

### Elementary Functions(3)

Inverse Mellin transforms of rational functions:

Exponential function:

Rational-exponential function:

Trigonometric functions:

### Special Functions(3)

Inverse transforms of Gamma expressed in terms of exponential functions:

Bessel functions:

Other elementary and special functions:

Inverse transforms of expressions involving Bessel functions:

Inverse transform of an expression involving Zeta:

## Options(2)

### Assumptions(1)

The answer returned by InverseMellinTransform in this example is valid for Re[s]>0:

Use Assumptions to specify a different strip of holomorphy:

Use ConditionalExpression to specify the same assumptions:

### GenerateConditions(1)

Obtain conditions for validity of the result in InverseMellinTransform:

GenerateConditions is set to False by default in this case:

## Applications(2)

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 a particular solution for a differential equation using a Mellin transform technique:

Apply MellinTransform to the equation:

Solve for the Mellin transform:

Compute the required solution using InverseMellinTransform:

Obtain the solution using DSolveValue with a boundary condition at infinity:

Verify that the two solutions are the same for t>0:

## Properties & Relations(4)

Use Asymptotic to compute an asymptotic approximation:

InverseMellinTransform and MellinTransform are mutual inverses:

Verify the relationship for a specific function:

InverseMellinTransform is a linear operator:

The inverse Mellin and Laplace transforms are related by :

## Possible Issues(2)

InverseMellinTransform may return different results depending on the assumptions:

In this example, the default answer is valid in a right half-plane:

The inverse Mellin transform may only exist for a certain range of values for x:

## Neat Examples(1)

Create a table of basic inverse Mellin transforms:

Wolfram Research (2016), InverseMellinTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseMellinTransform.html.

#### Text

Wolfram Research (2016), InverseMellinTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseMellinTransform.html.

#### CMS

Wolfram Language. 2016. "InverseMellinTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseMellinTransform.html.

#### APA

Wolfram Language. (2016). InverseMellinTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseMellinTransform.html

#### BibTeX

@misc{reference.wolfram_2023_inversemellintransform, author="Wolfram Research", title="{InverseMellinTransform}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/InverseMellinTransform.html}", note=[Accessed: 17-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_inversemellintransform, organization={Wolfram Research}, title={InverseMellinTransform}, year={2016}, url={https://reference.wolfram.com/language/ref/InverseMellinTransform.html}, note=[Accessed: 17-April-2024 ]}