## 3.5.12 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

One-dimensional 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.
 In[1]:=  LaplaceTransform[t^4 Sin[t], t, s]
 Out[1]=
Here is the inverse.
 In[2]:=  InverseLaplaceTransform[%, s, t]
 Out[2]=
Even simple transforms often involve special functions.
 In[3]:=  LaplaceTransform[1/(1 + t^2), t, s]
 Out[3]=
Here the result involves a Meijer G function.
 In[4]:=  LaplaceTransform[1/(1 + t^3), t, s]
 Out[4]=
The Laplace transform of this Bessel function just involves elementary functions.
 In[5]:=  LaplaceTransform[BesselJ[n, t], t, s]
 Out[5]=

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.
 In[6]:=  LaplaceTransform[Integrate[f[u], {u, 0, t}], t, s]
 Out[6]=

 LaplaceTransform[expr, {, , ... }, {, , ... }] the multidimensional Laplace transform of expr InverseLaplaceTransform[expr, {, , ... }, {, , ... }] the multidimensional inverse Laplace transform of expr

Multidimensional Laplace transforms.

### Fourier Transforms

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

One-dimensional Fourier transforms.
Here is a Fourier transform.
 In[1]:=  FourierTransform[1/(1 + t^4), t, ]
 Out[1]=
This finds the inverse.
 In[2]:=  InverseFourierTransform[%, , t]
 Out[2]=

In Mathematica 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 Mathematica allows you to choose any of these conventions you want.

 common convention setting Fourier transform inverse Fourier transform Mathematica default {0, 1} pure mathematics {1, -1} classical physics {-1, 1} modern physics {0, 1} systems engineering {1, -1} signal processing {0, -2 Pi} general case {a, b}

Typical settings for FourierParameters with various conventions.
Here is a Fourier transform with the default choice of parameters.
 In[3]:=  FourierTransform[Exp[-t^2], t, ]
 Out[3]=
Here is the same Fourier transform with the choice of parameters typically used in signal processing.
 In[4]:=  FourierTransform[Exp[-t^2], t, , FourierParameters->{0, -2 Pi}]
 Out[4]=

 FourierSinTransform[expr, t, ] Fourier sine transform FourierCosTransform[expr, t, ] Fourier cosine transform InverseFourierSinTransform[expr, , t] inverse Fourier sine transform InverseFourierCosTransform[expr, , 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 .
 In[5]:=  {FourierSinTransform[Exp[-t], t, ], FourierCosTransform[Exp[-t], t, ]}
 Out[5]=

 FourierTransform[expr, {, , ... }, {, , ... }] the multidimensional Fourier transform of expr InverseFourierTransform[expr, {, , ... }, {, , ... }] the multidimensional inverse Fourier transform of expr FourierSinTransform[expr, {, , ... }, {, , ... }], FourierCosTransform[expr, {, , ... }, {, , ... }] the multidimensional sine and cosine Fourier transforms of expr InverseFourierSinTransform[expr, {, , ... }, {, , ... }], InverseFourierCosTransform[expr, {, , ... }, {, , ... }] the multidimensional inverse Fourier sine and cosine transforms of expr

Multidimensional Fourier transforms.
This evaluates a two-dimensional Fourier transform.
 In[6]:=  FourierTransform[(u v)^2 Exp[-u^2-v^2], {u, v}, {a, b}]
 Out[6]=
This inverts the transform.
 In[7]:=  InverseFourierTransform[%, {a, b}, {u, v}]
 Out[7]=

### 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 .
 In[1]:=  ZTransform[2^-n, n, z]
 Out[1]=
Here is the inverse Z transform.
 In[2]:=  InverseZTransform[%, z, n]
 Out[2]=
The generating function for is an exponential function.
 In[3]:=  ZTransform[1/n!, n, z]
 Out[3]=

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