# InverseHankelTransform

InverseHankelTransform[expr,s,r]

gives the inverse Hankel transform of order 0 for expr.

InverseHankelTransform[expr,s,r,ν]

gives the inverse Hankel transform of order ν for expr.

# Details and Options • The inverse Hankel transform of order ν for a function is defined to be .
• The inverse Hankel transform is defined for and .
• 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, InverseHankelTransform is output using .

# Examples

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## Basic Examples(2)

Compute the inverse Hankel transform of a function:

Inverse Hankel transform for a product of functions:

## Scope(16)

### Basic Uses(5)

Compute the inverse Hankel transform of order ν for a function:

Use the default value 0 for the parameter ν:

Compute the inverse Hankel transform of a function for a symbolic parameter r:

Use an exact value for r:

Use a numerical value for r:

Obtain the conditions for the convergence:

Specify assumptions:

### Elementary Functions(4)

Inverse Hankel transforms of rational functions:

Exponential and logarithmic functions:

Trigonometric functions:

Algebraic functions:

### Special Functions(5)

Inverse Hankel transforms of Bessel functions:

Airy functions:

Elliptic functions:

Error functions:

Integral functions:

### Piecewise Functions and Distributions(2)

Inverse Hankel transform of a piecewise function:

Inverse Hankel transforms of distributions:

## Options(2)

### GenerateConditions(1)

Obtain conditions for validity of the result:

### Assumptions(1)

Compute the inverse Hankel transform of a function depending on a parameter a:

Obtain a simpler result by specifying assumptions on the parameter:

## Applications(3)

The inverse Fourier transform of a radially symmetric function in the plane can be expressed as an inverse Hankel transform. Verify this relation for the function defined by:

Plot the function:

Compute its inverse Fourier transform:

Obtain the same result using InverseHankelTransform:

Plot the inverse Fourier transform:

Generate a gallery of inverse Fourier transforms for a list of radially symmetric functions:

Compute the inverse Hankel transforms for these functions:

Generate the gallery of inverse Fourier transforms as required:

Obtain a particular solution for an inhomogeneous equation involving the radial Laplacian:

Apply HankelTransform to the equation:

Solve for the Hankel transform:

Apply InverseHankelTransform to obtain a particular solution:

Verify the solution:

## Properties & Relations(7)

Use Asymptotic to compute an asymptotic approximation:

InverseHankelTransform computes the integral :

InverseHankelTransform is the inverse of HankelTransform:

InverseHankelTransform is its own inverse:

InverseHankelTransform is a linear operator:

InverseHankelTransform of derivatives:

Derivative of an inverse Hankel transform with respect to r:

## Neat Examples(1)

Create a table of basic inverse Hankel transforms: