Integral Transforms and Related Operations

Laplace Transforms

LaplaceTransform[expr,t,s]the Laplace transform of expr
InverseLaplaceTransform[expr,s,t]the inverse Laplace transform of expr

Onedimensional Laplace transforms.

The Laplace transform of a function is given by . The inverse Laplace transform of is given for suitable by .

Here is a simple Laplace transform.
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Here is the inverse.
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Even simple transforms often involve special functions.
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Here the result involves a Meijer G function.
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InverseLaplaceTransform returns the original function.
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The Laplace transform of this Bessel function just involves elementary functions.
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Laplace transforms have the property that they turn integration and differentiation into essentially algebraic operations. They are therefore commonly used in studying systems governed by differential equations.

Integration becomes multiplication by when one does a Laplace transform.
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LaplaceTransform[expr,{t1,t2,},{s1,s2,}]
the multidimensional Laplace transform of expr
InverseLaplaceTransform[expr,{s1,s2,},{t1,t2,}]
the multidimensional inverse Laplace transform of expr

Multidimensional Laplace transforms.

Fourier Transforms

FourierTransform[expr,t,ω]the Fourier transform of expr
InverseFourierTransform[expr,ω,t]the inverse Fourier transform of expr

Onedimensional Fourier transforms.

Integral transforms can produce results that involve "generalized functions" such as HeavisideTheta.
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This finds the inverse.
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In the Wolfram Language the Fourier transform of a function is by default defined to be . The inverse Fourier transform of is similarly defined as .

In different scientific and technical fields different conventions are often used for defining Fourier transforms. The option FourierParameters in the Wolfram Language allows you to choose any of these conventions you want.

common conventionsettingFourier transforminverse Fourier transform
Wolfram Language default
pure mathematics
classical physics
modern physics
systems engineering
signal processing{0,-2Pi}
general case

Typical settings for FourierParameters with various conventions.

Here is a Fourier transform with the default choice of parameters.
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Here is the same Fourier transform with the choice of parameters typically used in signal processing.
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FourierSinTransform[expr,t,ω]Fourier sine transform
FourierCosTransform[expr,t,ω]Fourier cosine transform
InverseFourierSinTransform[expr,ω,t]
inverse Fourier sine transform
InverseFourierCosTransform[expr,ω,t]
inverse Fourier cosine transform

Fourier sine and cosine transforms.

In some applications of Fourier transforms, it is convenient to avoid ever introducing complex exponentials. Fourier sine and cosine transforms correspond to integrating respectively with and instead of , and using limits 0 and rather than and .

Here are the Fourier sine and cosine transforms of .
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FourierTransform[expr,{t1,t2,},{ω1,ω2,}]
the multidimensional Fourier transform of expr
InverseFourierTransform[expr,{ω1,ω2,},{t1,t2,}]
the multidimensional inverse Fourier transform of expr
FourierSinTransform[expr,{t1,t2,},{ω1,ω2,}], FourierCosTransform[expr,{t1,t2,},{ω1,ω2,}]
the multidimensional sine and cosine Fourier transforms of expr
InverseFourierSinTransform[expr,{ω1,ω2,},{t1,t2,}], InverseFourierCosTransform[expr,{ω1,ω2,},{t1,t2,}]
the multidimensional inverse Fourier sine and cosine transforms of expr

Multidimensional Fourier transforms.

This evaluates a twodimensional Fourier transform.
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This inverts the transform.
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Z Transforms

ZTransform[expr,n,z]Z transform of expr
InverseZTransform[expr,z,n]inverse Z transform of expr

Z transforms.

The Z transform of a function is given by . The inverse Z transform of is given by the contour integral . Z transforms are effectively discrete analogs of Laplace transforms. They are widely used for solving difference equations, especially in digital signal processing and control theory. They can be thought of as producing generating functions, of the kind commonly used in combinatorics and number theory.

This computes the Z transform of .
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Here is the inverse Z transform.
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The generating function for is an exponential function.
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