Integral Transforms and Related Operations

Laplace Transforms

One-dimensional Laplace transforms.

The Laplace transform of a function f(t) is given by (∫_0^∞)f(t)(e^(-st))dt. The inverse Laplace transform of F(s) is given for suitable γ by (1/(2πi))(∫_(γ-i∞)^(γ+i∞))F(s)(e^(st))ds.

Here is a simple Laplace transform:

LaplaceTransform[t^4 Sin[t],t,s]

Here is the inverse:

InverseLaplaceTransform[%,s,t]

Even simple transforms often involve special functions:

LaplaceTransform[1/(1+t^2),t,s]

Here the result involves a Meijer G function:

LaplaceTransform[1/(1+t^3),t,s]

(RefLink[InverseLaplaceTransform, paclet:ref/InverseLaplaceTransform]) returns the original function:

InverseLaplaceTransform[%,s,t]

The Laplace transform of this Bessel function just involves elementary functions:

LaplaceTransform[BesselJ[n,t],t,s]

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 1/s when one does a Laplace transform:

LaplaceTransform[Integrate[f[u],{u,0,t}],t,s]

Multidimensional Laplace transforms.

Fourier Transforms

One-dimensional Fourier transforms.

Integral transforms can produce results that involve "generalized functions" such as (RefLink[HeavisideTheta, paclet:ref/HeavisideTheta]):

FourierTransform[1/(1+t^4),t,ω]

This finds the inverse:

InverseFourierTransform[%,ω,t]

In the Wolfram Language the Fourier transform of a function f(t) is by default defined to be (1/(Sqrt[2π]))(∫_(-∞)^∞)f(t) (e^(iωt))dt. The inverse Fourier transform of F(ω) is similarly defined as (1/(Sqrt[2π]))(∫_(-∞)^∞)F(ω) (e^(-iωt))dω.

In different scientific and technical fields different conventions are often used for defining Fourier transforms. The option (RefLink[FourierParameters, paclet:ref/FourierParameters]) in the Wolfram Language allows you to choose any of these conventions you want.

Typical settings for (RefLink[FourierParameters, paclet:ref/FourierParameters]) with various conventions.

Here is a Fourier transform with the default choice of parameters:

FourierTransform[Exp[-t^2],t,ω]

Here is the same Fourier transform with the choice of parameters typically used in signal processing:

FourierTransform[Exp[-t^2],t,ω,FourierParameters->{0,-2 Pi}]

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 sin(ωt) and cos(ωt) instead of exp(iωt), and using limits 0 and ∞ rather than -∞ and ∞.

Here are the Fourier sine and cosine transforms of (e^(-t)):

{FourierSinTransform[Exp[-t],t,ω],FourierCosTransform[Exp[-t],t,ω]}

Multidimensional Fourier transforms.

This evaluates a two-dimensional Fourier transform:

FourierTransform[(u v)^2 Exp[-u^2-v^2],{u,v},{a,b}]

This inverts the transform:

InverseFourierTransform[%,{a,b},{u,v}]

Z Transforms

Z transforms.

The Z transform of a function f(n) is given by (∑_(n=0)^∞)f(n)(z^(-n)). The inverse Z transform of F(z) is given by the contour integral (1/(2πi))∮F(z)(z^(n-1))dz. 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 (2^(-n)):

ZTransform[2^-n,n,z]

Here is the inverse Z transform:

InverseZTransform[%,z,n]

The generating function for 1/n! is an exponential function:

ZTransform[1/n!,n,z]