InverseFourierTransform
✖
InverseFourierTransform
gives the multidimensional inverse Fourier transform of expr.
Details and Options



- The Fourier transform and its inverse are a way to transform between the time domain and the frequency domain.
- Fourier transforms are typically used to reduce ordinary and partial differential equations to algebraic or ordinary differential equations, respectively. They are also used extensively in control theory and signal processing. Finally, they have applications in studying quantum mechanical phenomena, noise filtering, etc.
- The inverse Fourier transform of the frequency domain function
is the time domain function
:
- The inverse Fourier transform of a function
is by default defined to be
.
- The multidimensional inverse Fourier transform of a function
is by default defined to be
or when using vector notation,
.
- Different choices of definitions can be specified using the option FourierParameters.
- The integral is computed using numerical methods if the third argument,
, is given a numerical value.
- The asymptotic inverse Fourier transform can be computed using Asymptotic.
- There are several related Fourier transformations:
-
FourierTransform infinite continuous-time functions (FT) FourierSequenceTransform infinite discrete-time functions (DTFT) FourierCoefficient finite continuous-time functions (FS) Fourier finite discrete-time functions (DFT) - The inverse Fourier transform is an automorphism in the Schwartz vector space of functions whose derivatives are rapidly decreasing and thus induces an automorphism in its dual: the space of tempered distributions. These include absolutely integrable functions, well-behaved functions of polynomial growth and compactly supported distributions.
- Hence, InverseFourierTransform not only works with absolutely integrable functions, but it can also handle a variety of tempered distributions such as DiracDelta to enlarge the pool of functions or generalized functions it can effectively transform.
- The following options can be given:
-
AccuracyGoal Automatic digits of absolute accuracy sought Assumptions $Assumptions assumptions to make about parameters FourierParameters {0,1} parameters to define the inverse Fourier transform GenerateConditions False whether to generate answers that involve conditions on parameters PerformanceGoal $PerformanceGoal aspects of performance to optimize PrecisionGoal Automatic digits of precision sought WorkingPrecision Automatic the precision used in internal computations - Common settings for FourierParameters include:
-
{0,1} default setting/physics {1,-1} systems engineering/mathematics {-1,1} classical physics {0,-2Pi} ordinary frequency {a,b} general setting - In TraditionalForm, InverseFourierTransform is output using
. »

Examples
open allclose allBasic Examples (5)Summary of the most common use cases
Compute the inverse Fourier transform of a function:

https://wolfram.com/xid/0bnicimo4y5hu-wz9b3s

Plot the function and its Fourier transform:

https://wolfram.com/xid/0bnicimo4y5hu-sycnet

For the systems engineering convention, change the Fourier parameters:

https://wolfram.com/xid/0bnicimo4y5hu-0xyrxq

Inverse Fourier transform of a function with parameters:

https://wolfram.com/xid/0bnicimo4y5hu-gt9aro


https://wolfram.com/xid/0bnicimo4y5hu-p2uldj

Compute a numerical inverse Fourier transform:

https://wolfram.com/xid/0bnicimo4y5hu-cgbnci

Inverse Fourier transform of a bivariate function:

https://wolfram.com/xid/0bnicimo4y5hu-3os46q

Plot the magnitude of the transform:

https://wolfram.com/xid/0bnicimo4y5hu-fz1okg

Scope (41)Survey of the scope of standard use cases
Basic Uses (3)
Compute the inverse Fourier transform of a function for a symbolic parameter :

https://wolfram.com/xid/0bnicimo4y5hu-tfqkww

Use a numerical value for the parameter:

https://wolfram.com/xid/0bnicimo4y5hu-8dz08b

TraditionalForm formatting:

https://wolfram.com/xid/0bnicimo4y5hu-8rchg2

Elementary Functions (7)

https://wolfram.com/xid/0bnicimo4y5hu-qpu8dl

Inverse Fourier transform of a Gaussian is another Gaussian:

https://wolfram.com/xid/0bnicimo4y5hu-moy0yd


https://wolfram.com/xid/0bnicimo4y5hu-ou0d0m


https://wolfram.com/xid/0bnicimo4y5hu-w80w6h

Function involving a square root:

https://wolfram.com/xid/0bnicimo4y5hu-gar47i

Plot the real and imaginary parts of the transform:

https://wolfram.com/xid/0bnicimo4y5hu-z166et

Function involving complex exponential and square root functions:

https://wolfram.com/xid/0bnicimo4y5hu-zhitf0


https://wolfram.com/xid/0bnicimo4y5hu-bl2a36


https://wolfram.com/xid/0bnicimo4y5hu-f9h7by


https://wolfram.com/xid/0bnicimo4y5hu-2z37wy

Inverse transform of a Sech function:

https://wolfram.com/xid/0bnicimo4y5hu-mfymjy


https://wolfram.com/xid/0bnicimo4y5hu-jbk3ui

Rational Functions (4)

https://wolfram.com/xid/0bnicimo4y5hu-en1m8h

Inverse Fourier transform for complex rational function:

https://wolfram.com/xid/0bnicimo4y5hu-hw2rik


https://wolfram.com/xid/0bnicimo4y5hu-lh0fuq

Reciprocal of quadratic polynomial:

https://wolfram.com/xid/0bnicimo4y5hu-xyub7r


https://wolfram.com/xid/0bnicimo4y5hu-n767cq


https://wolfram.com/xid/0bnicimo4y5hu-z7ngog

Real and imaginary part plots:

https://wolfram.com/xid/0bnicimo4y5hu-rf8ncd

Special Functions (6)
Sinc function:

https://wolfram.com/xid/0bnicimo4y5hu-z4jzdp


https://wolfram.com/xid/0bnicimo4y5hu-1ndpvd

Inverse Fourier transform leading to a BesselK function:

https://wolfram.com/xid/0bnicimo4y5hu-jkgndr


https://wolfram.com/xid/0bnicimo4y5hu-b2kot5

Expression involving BesselJ function:

https://wolfram.com/xid/0bnicimo4y5hu-sm70cm


https://wolfram.com/xid/0bnicimo4y5hu-dwnnk2

Inverse transform of a BesselY function:

https://wolfram.com/xid/0bnicimo4y5hu-kcau58


https://wolfram.com/xid/0bnicimo4y5hu-2u41pj

Inverse transform of a square root function on , leading to a BesselJ function:

https://wolfram.com/xid/0bnicimo4y5hu-0tzezf


https://wolfram.com/xid/0bnicimo4y5hu-6yquv9

Product of Gamma, cosine and absolute value functions:

https://wolfram.com/xid/0bnicimo4y5hu-iz8hot


https://wolfram.com/xid/0bnicimo4y5hu-s3t8ea

Piecewise Functions (4)
The inverse of the following function is a piecewise function:

https://wolfram.com/xid/0bnicimo4y5hu-xjyzgc


https://wolfram.com/xid/0bnicimo4y5hu-gecoi4

Inverse transform for a complex rational function leading to a piecewise exponential function:

https://wolfram.com/xid/0bnicimo4y5hu-gl5tes


https://wolfram.com/xid/0bnicimo4y5hu-qn6eaq

Inverse transform of a box function:

https://wolfram.com/xid/0bnicimo4y5hu-bv6dr1


https://wolfram.com/xid/0bnicimo4y5hu-3ss529

Inverse Fourier transform of a piecewise function:

https://wolfram.com/xid/0bnicimo4y5hu-qcm6f


https://wolfram.com/xid/0bnicimo4y5hu-8bj49i

Periodic Functions (4)
The inverse transform of the sum of sine and cosine:

https://wolfram.com/xid/0bnicimo4y5hu-8rg1nh


https://wolfram.com/xid/0bnicimo4y5hu-27s7ai


https://wolfram.com/xid/0bnicimo4y5hu-9bvljc

Functions whose inverse transforms are trigonometric functions:

https://wolfram.com/xid/0bnicimo4y5hu-106m7d


https://wolfram.com/xid/0bnicimo4y5hu-tq75gs

Generalized Functions (2)
Inverse transform of an exponential imaginary function is the DiracDelta function:

https://wolfram.com/xid/0bnicimo4y5hu-55jzxk

Inverse transform of the product of an exponential imaginary function and a power of is a derivative of the DiracDelta function:

https://wolfram.com/xid/0bnicimo4y5hu-tbyuof

Multivariate Functions (5)
Inverse Fourier transform of a bivariate rational function:

https://wolfram.com/xid/0bnicimo4y5hu-mzaoky

Inverse Fourier transform of a bivariate exponential function:

https://wolfram.com/xid/0bnicimo4y5hu-fsc0zy

Reciprocal of square root function whose inverse transform is itself:

https://wolfram.com/xid/0bnicimo4y5hu-6uyp85


https://wolfram.com/xid/0bnicimo4y5hu-y3a16o

Inverse Fourier transform of a trivariate function:

https://wolfram.com/xid/0bnicimo4y5hu-inssd9


https://wolfram.com/xid/0bnicimo4y5hu-7b7kbk

Formal Properties (3)
Inverse Fourier transform of a first-order derivative:

https://wolfram.com/xid/0bnicimo4y5hu-igf4ta

Inverse Fourier transform of a second-order derivative:

https://wolfram.com/xid/0bnicimo4y5hu-sahdax

Inverse Fourier transform threads itself over equations:

https://wolfram.com/xid/0bnicimo4y5hu-818kex

Numerical Inversion (3)
Calculate the inverse Fourier transform at a single point:

https://wolfram.com/xid/0bnicimo4y5hu-mfc7y1

Alternatively, calculate the inverse Fourier transform symbolically:

https://wolfram.com/xid/0bnicimo4y5hu-y6w8fe

Then evaluate it for a specific value of t:

https://wolfram.com/xid/0bnicimo4y5hu-qgczmv

Plot the inverse Fourier transform numerically and compare it with the exact result:

https://wolfram.com/xid/0bnicimo4y5hu-9gt6dr

Options (7)Common values & functionality for each option
AccuracyGoal (1)
The option AccuracyGoal sets the number of digits of accuracy:

https://wolfram.com/xid/0bnicimo4y5hu-5ymjb


https://wolfram.com/xid/0bnicimo4y5hu-f6u3ua


https://wolfram.com/xid/0bnicimo4y5hu-kqpox6

Assumptions (1)
The inverse Fourier transform of BesselJ is a piecewise function:

https://wolfram.com/xid/0bnicimo4y5hu-dfqdc5


https://wolfram.com/xid/0bnicimo4y5hu-c1fdjf


https://wolfram.com/xid/0bnicimo4y5hu-di59uc

FourierParameters (2)
Inverse Fourier transform for the unit box function and different parameters:

Set up your particular global choice of parameters to work with ordinary frequency once per session:

https://wolfram.com/xid/0bnicimo4y5hu-eddx0d


https://wolfram.com/xid/0bnicimo4y5hu-p50knq


https://wolfram.com/xid/0bnicimo4y5hu-euzngr

GenerateConditions (1)
Use GenerateConditionsTrue to get parameter conditions for when a result is valid:

https://wolfram.com/xid/0bnicimo4y5hu-ngoxw

PrecisionGoal (1)
The option PrecisionGoal sets the relative tolerance in the integration:

https://wolfram.com/xid/0bnicimo4y5hu-n3x5d0


https://wolfram.com/xid/0bnicimo4y5hu-9119x


https://wolfram.com/xid/0bnicimo4y5hu-b3dg8e

WorkingPrecision (1)
If a WorkingPrecision is specified, the computation is done at that working precision:

https://wolfram.com/xid/0bnicimo4y5hu-hr3ys


https://wolfram.com/xid/0bnicimo4y5hu-duvdjv


https://wolfram.com/xid/0bnicimo4y5hu-j4rvt

Applications (7)Sample problems that can be solved with this function
Signals and Systems (2)
Find the convolution of signals:

https://wolfram.com/xid/0bnicimo4y5hu-o8bg4r
The product of their Fourier transforms:

https://wolfram.com/xid/0bnicimo4y5hu-50mw7b


https://wolfram.com/xid/0bnicimo4y5hu-quvbx0

Compare with Convolve:

https://wolfram.com/xid/0bnicimo4y5hu-cui6p8

The ideal lowpass filter is defined:

https://wolfram.com/xid/0bnicimo4y5hu-sch5ft

The impulse response is the inverse Fourier transform of :

https://wolfram.com/xid/0bnicimo4y5hu-4ncymb


https://wolfram.com/xid/0bnicimo4y5hu-x3v7dk


https://wolfram.com/xid/0bnicimo4y5hu-wwq3ga


https://wolfram.com/xid/0bnicimo4y5hu-71xaqr

Ordinary Differential Equations (1)
Solve a differential equation using Fourier transforms:

https://wolfram.com/xid/0bnicimo4y5hu-i5lg1e
Apply the Fourier transform over the equation:

https://wolfram.com/xid/0bnicimo4y5hu-l1adei

Solve for the Fourier transform:

https://wolfram.com/xid/0bnicimo4y5hu-isjqfn

Find the inverse transform to get the solution:

https://wolfram.com/xid/0bnicimo4y5hu-lkgcs9

Compare with DSolveValue:

https://wolfram.com/xid/0bnicimo4y5hu-q2w10j

Partial Differential Equations (1)
Consider the heat equation: with initial condition
:

https://wolfram.com/xid/0bnicimo4y5hu-0fpofs
Fourier transform with respect to :

https://wolfram.com/xid/0bnicimo4y5hu-63ojg8


https://wolfram.com/xid/0bnicimo4y5hu-jb4llw

Compute the inverse Fourier transform:

https://wolfram.com/xid/0bnicimo4y5hu-rud1ra

And convolution to get the solution:

https://wolfram.com/xid/0bnicimo4y5hu-24j399

Consider the special case with initial condition and
:

https://wolfram.com/xid/0bnicimo4y5hu-yqg0qw

Compare with DSolveValue:

https://wolfram.com/xid/0bnicimo4y5hu-dkim4w

Plot the initial conditions and solutions for different values of :

https://wolfram.com/xid/0bnicimo4y5hu-r0y1oc

Plot the solution over the -
plane:

https://wolfram.com/xid/0bnicimo4y5hu-zeojgv

Evaluation of Integrals (1)
Calculate the following definite integral:

https://wolfram.com/xid/0bnicimo4y5hu-2qlvnt

Compute the Fourier transform with respect to and interchange the order of transform and integration:

https://wolfram.com/xid/0bnicimo4y5hu-htosvo


https://wolfram.com/xid/0bnicimo4y5hu-udwfsc

Use the inverse Fourier transform to get the result:

https://wolfram.com/xid/0bnicimo4y5hu-7c4rl1

Compare with Integrate:

https://wolfram.com/xid/0bnicimo4y5hu-h1wan9

Other 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:

https://wolfram.com/xid/0bnicimo4y5hu-bk3wkz

https://wolfram.com/xid/0bnicimo4y5hu-dn7ss

Compute its inverse Fourier transform:

https://wolfram.com/xid/0bnicimo4y5hu-nzfcoa

Obtain the same result using InverseHankelTransform:

https://wolfram.com/xid/0bnicimo4y5hu-j9fyka

Plot the inverse Fourier transform:

https://wolfram.com/xid/0bnicimo4y5hu-diqik5

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

https://wolfram.com/xid/0bnicimo4y5hu-ebri30
Compute the inverse Hankel transforms for these functions:

https://wolfram.com/xid/0bnicimo4y5hu-fp1in1

Generate the gallery of inverse Fourier transforms as required:

https://wolfram.com/xid/0bnicimo4y5hu-bbdw9w

Properties & Relations (5)Properties of the function, and connections to other functions
By default, the inverse Fourier transform of is:

https://wolfram.com/xid/0bnicimo4y5hu-f62581

For , the definite integral becomes:

https://wolfram.com/xid/0bnicimo4y5hu-hgqezg

Compare with InverseFourierTransform:

https://wolfram.com/xid/0bnicimo4y5hu-745lka

Use Asymptotic to compute an asymptotic approximation:

https://wolfram.com/xid/0bnicimo4y5hu-kc6zu

InverseFourierTransform and FourierTransform are mutual inverses:

https://wolfram.com/xid/0bnicimo4y5hu-bf3zt1


https://wolfram.com/xid/0bnicimo4y5hu-4dn9t


https://wolfram.com/xid/0bnicimo4y5hu-it7bfn


https://wolfram.com/xid/0bnicimo4y5hu-byd6ra

InverseFourierTransform and InverseFourierCosTransform are equal for even functions:

https://wolfram.com/xid/0bnicimo4y5hu-fslxt9


https://wolfram.com/xid/0bnicimo4y5hu-f33own

InverseFourierTransform and InverseFourierSinTransform differ by - for odd functions:

https://wolfram.com/xid/0bnicimo4y5hu-d7ly8o


https://wolfram.com/xid/0bnicimo4y5hu-ichu8t

Possible Issues (1)Common pitfalls and unexpected behavior
Neat Examples (2)Surprising or curious use cases
The InverseFourierTransform of is a
convolution of box functions:

https://wolfram.com/xid/0bnicimo4y5hu-c14tqx

Create a table of basic inverse Fourier transforms:

https://wolfram.com/xid/0bnicimo4y5hu-ntocdm

Wolfram Research (1999), InverseFourierTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierTransform.html (updated 2025).
Text
Wolfram Research (1999), InverseFourierTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierTransform.html (updated 2025).
Wolfram Research (1999), InverseFourierTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierTransform.html (updated 2025).
CMS
Wolfram Language. 1999. "InverseFourierTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/InverseFourierTransform.html.
Wolfram Language. 1999. "InverseFourierTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. 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
Wolfram Language. (1999). InverseFourierTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseFourierTransform.html
BibTeX
@misc{reference.wolfram_2025_inversefouriertransform, author="Wolfram Research", title="{InverseFourierTransform}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/InverseFourierTransform.html}", note=[Accessed: 10-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_inversefouriertransform, organization={Wolfram Research}, title={InverseFourierTransform}, year={2025}, url={https://reference.wolfram.com/language/ref/InverseFourierTransform.html}, note=[Accessed: 10-July-2025
]}