# 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 \$Assumptions assumptions to make about parameters FourierParameters {0,1} parameters to define the Fourier transform GenerateConditions False whether 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

open allclose all

## Scope(6)

Elementary functions:

Special functions:

Piecewise functions and distributions:

Periodic functions:

Multivariate functions:

## 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 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_2024_inversefouriertransform, author="Wolfram Research", title="{InverseFourierTransform}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/InverseFourierTransform.html}", note=[Accessed: 23-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_inversefouriertransform, organization={Wolfram Research}, title={InverseFourierTransform}, year={1999}, url={https://reference.wolfram.com/language/ref/InverseFourierTransform.html}, note=[Accessed: 23-April-2024 ]}