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InverseFourierTransform

InverseFourierTransform[expr,

, t]
gives the symbolic inverse Fourier transform of expr.

InverseFourierTransform[expr, {

₁,

₂, ...}, {t₁, t₂, ...}]
gives the multidimensional inverse Fourier transform of expr.

MORE INFORMATION

The inverse Fourier transform of a function is by default defined as .

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), {0, -2Pi} (signal processing).

Assumptions and other options to Integrate can also be given in InverseFourierTransform. »

InverseFourierTransform[expr, , 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 . 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 .

Elementary functions:

Special functions:

Piecewise functions and distributions:

TraditionalForm formatting:

Generalizations & Extensions (1)

Multidimensional inverse Fourier transform:

The inverse Fourier transform of BesselJ is a piecewise function:

Default modern physics convention:

Convention for pure mathematics and systems engineering:

Convention for classical physics:

Convention for signal processing:

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

Properties & Relations (3)

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:

InverseFourierSinTransform InverseFourierCosTransform InverseFourier FourierTransform InverseFourierSequenceTransform Convolve InverseLaplaceTransform Integrate

Integral Transforms and Related Operations

Fourier Analysis

Integral Transforms

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