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

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:

Display in TraditionalForm:

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 int_0^inftys F(s) TemplateBox[{nu, {s, , r}}, BesselJ]ds:

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:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_inversehankeltransform, author="Wolfram Research", title="{InverseHankelTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/InverseHankelTransform.html}", note=[Accessed: 13-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_inversehankeltransform, organization={Wolfram Research}, title={InverseHankelTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/InverseHankelTransform.html}, note=[Accessed: 13-November-2024 ]}