# HankelTransform

HankelTransform[expr,r,s]

gives the Hankel transform of order 0 for expr.

HankelTransform[expr,r,s,ν]

gives the Hankel transform of order ν for expr.

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

# Examples

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## Basic Examples(2)

Compute the Hankel transform of a function:

Hankel transform for a product of functions:

## Scope(16)

### Basic Uses(5)

Compute the Hankel transform of order ν for a function:

Use the default value 0 for the parameter ν:

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

Use an exact value for s:

Use a numerical value for s:

Obtain the conditions for the convergence:

Specify assumptions:

### Elementary Functions(4)

Hankel transforms of rational functions:

Exponential and logarithmic functions:

Trigonometric functions:

Algebraic functions:

### Special Functions(5)

Hankel transforms of Bessel functions:

Airy functions:

Elliptic functions:

Error functions:

Integral functions:

### Piecewise Functions and Distributions(2)

Hankel transform of a piecewise function:

Hankel transforms of distributions:

## Options(2)

### GenerateConditions(1)

Obtain conditions for validity of the result:

### Assumptions(1)

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

Obtain a simpler result by specifying assumptions on the parameter:

## Applications(3)

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

Plot the function:

Compute its Fourier transform:

Obtain the same result using HankelTransform:

Plot the Fourier transform:

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

Compute the Hankel transforms for these functions:

Generate the gallery of 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(6)

Use Asymptotic to compute an asymptotic approximation:

HankelTransform computes the integral :

HankelTransform is its own inverse:

HankelTransform is a linear operator:

Hankel transform of a derivative:

Derivative of a Hankel transform with respect to s:

## Neat Examples(1)

Create a table of basic Hankel transforms: