# 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 .

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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.

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LaplaceTransform[expr,{t_{1},t_{2},...},{s_{1},s_{2},...}] | |

the multidimensional Laplace transform of expr | |

InverseLaplaceTransform[expr,{s_{1},s_{2},...},{t_{1},t_{2},...}] | |

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.

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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 | |||

pure mathematics | |||

classical physics | |||

modern physics | |||

systems engineering | |||

signal processing | {0, -2Pi} | ||

general case |

Typical settings for FourierParameters with various conventions.

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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 .

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FourierTransform[expr,{t_{1},t_{2},...},{_{1},_{2},...}] | |

the multidimensional Fourier transform of expr | |

InverseFourierTransform[expr,{_{1},_{2},...},{t_{1},t_{2},...}] | |

the multidimensional inverse Fourier transform of expr | |

FourierSinTransform[expr,{t_{1},t_{2},...},{_{1},_{2},...}],
FourierCosTransform[expr,{t_{1},t_{2},...},{_{1},_{2},...}] | |

the multidimensional sine and cosine Fourier transforms of expr | |

InverseFourierSinTransform[expr,{_{1},_{2},...},{t_{1},t_{2},...}],
InverseFourierCosTransform[expr,{_{1},_{2},...},{t_{1},t_{2},...}] | |

the multidimensional inverse Fourier sine and cosine transforms of expr |

Multidimensional Fourier transforms.

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## Z Transforms

ZTransform[expr,n,z] | Z transform of expr |

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

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.

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