InverseMellinTransform

InverseMellinTransform[expr,s,x]

gives the inverse Mellin transform of expr.

Details and Options

Examples

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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:

TraditionalForm formatting:

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:

MeijerG:

Other elementary and special functions:

Inverse transforms of expressions involving Bessel functions:

Inverse transform of an expression involving Zeta:

PolyGamma:

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 int_0^infty(TemplateBox[{0, t}, BesselJ] TemplateBox[{1, {x, /, t}}, BesselJ])/tdt, 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 TemplateBox[{{F, (, s, )}, s, x}, InverseMellinTransform1]=L_s^(-1)[F(s)](-log(x)):

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:

Introduced in 2016
 (11.0)