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

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:

Display in TraditionalForm:

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 int_0^inftyr f(r) TemplateBox[{nu, {r, , s}}, BesselJ]dr:

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:

Wolfram Research (2017), HankelTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/HankelTransform.html.

Text

Wolfram Research (2017), HankelTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/HankelTransform.html.

CMS

Wolfram Language. 2017. "HankelTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HankelTransform.html.

APA

Wolfram Language. (2017). HankelTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HankelTransform.html

BibTeX

@misc{reference.wolfram_2024_hankeltransform, author="Wolfram Research", title="{HankelTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/HankelTransform.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_hankeltransform, organization={Wolfram Research}, title={HankelTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/HankelTransform.html}, note=[Accessed: 21-December-2024 ]}