InverseFourierTransform

InverseFourierTransform[expr,ω,t]

gives the symbolic inverse Fourier transform of expr.

InverseFourierTransform[expr,{ω1,ω2,…},{t1,t2,…}]

gives the multidimensional inverse Fourier transform of expr.

Details and Options

  • The inverse Fourier transform of a function is by default defined as .
  • The multidimensional inverse Fourier transform of a function is by default defined to be .
  • Other definitions are used in some scientific and technical fields.
  • Different choices of definitions can be specified using the option FourierParameters.
  • With the setting FourierParameters->{a,b} the inverse Fourier transform computed by InverseFourierTransform is .
  • Some common choices for {a,b} are {0,1} (default; modern physics), {1,-1} (pure mathematics; systems engineering), {-1,1} (classical physics), and {0,-2Pi} (signal processing).
  • The following options can be given: »
  • Assumptions $Assumptionsassumptions to make about parameters
    FourierParameters {0,1}parameters to define the Fourier transform
    GenerateConditions Falsewhether to generate answers that involve conditions on parameters
  • InverseFourierTransform[expr,ω,t] yields an expression depending on the continuous variable t that represents the symbolic inverse Fourier transform of expr with respect to the continuous variable ω. InverseFourier[list] takes a finite list of numbers as input, and yields as output a list representing the discrete inverse Fourier transform of the input.
  • In TraditionalForm, InverseFourierTransform is output using .

Examples

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Basic Examples  (2)

Scope  (6)

Elementary functions:

Special functions:

Piecewise functions and distributions:

Periodic functions:

Multivariate functions:

TraditionalForm formatting:

Options  (3)

Assumptions  (1)

The inverse Fourier transform of BesselJ is a piecewise function:

FourierParameters  (1)

Default modern physics convention:

Convention for pure mathematics and systems engineering:

Convention for classical physics:

Convention for signal processing:

GenerateConditions  (1)

Use GenerateConditions->True to get parameter conditions for when a result is valid:

Applications  (2)

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:

Properties & Relations  (4)

Use Asymptotic to compute an asymptotic approximation:

InverseFourierTransform and FourierTransform are mutual inverses:

InverseFourierTransform and InverseFourierCosTransform are equal for even functions:

InverseFourierTransform and InverseFourierSinTransform differ by for odd functions:

Possible Issues  (1)

The result from an inverse Fourier transform may not have the same form as the original:

Neat Examples  (1)

The InverseFourierTransform of is a convolution of box functions:

Wolfram Research (1999), InverseFourierTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierTransform.html.

Text

Wolfram Research (1999), InverseFourierTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierTransform.html.

CMS

Wolfram Language. 1999. "InverseFourierTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseFourierTransform.html.

APA

Wolfram Language. (1999). InverseFourierTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseFourierTransform.html

BibTeX

@misc{reference.wolfram_2022_inversefouriertransform, author="Wolfram Research", title="{InverseFourierTransform}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/InverseFourierTransform.html}", note=[Accessed: 22-March-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_inversefouriertransform, organization={Wolfram Research}, title={InverseFourierTransform}, year={1999}, url={https://reference.wolfram.com/language/ref/InverseFourierTransform.html}, note=[Accessed: 22-March-2023 ]}