# Fourier Series Package

The *Mathematica* kernel provides the functions FourierTransform and InverseFourierTransform for computing the symbolic Fourier exponential transform and inverse transform. It also provides the functions FourierSinTransform, InverseFourierSinTransform, FourierCosTransform, and InverseFourierCosTransform for computing the symbolic Fourier sine and cosine transforms and their inverses. As of Version 7, FourierSeries and related functions are also included in the *Mathematica* kernel.

This package provides functions for numerical approximations of Fourier transforms, Fourier series, and discrete-time Fourier transforms. The numerical approximations use *Mathematica*'s numerical integration and summation directly without first trying for an exact solution.

NFourierTransform[expr,t,] | find a numerical approximation to the Fourier transform of expr evaluated at the numerical value , where expr is treated as a function of t |

NInverseFourierTransform[expr,,t] | find a numerical approximation to the inverse Fourier transform of expr evaluated at the numerical value t, where expr is treated as a function of |

NFourierSinTransform[expr,t,] | find a numerical approximation to the Fourier sine transform of expr evaluated at the numerical value , where expr is treated as a function of t |

NInverseFourierSinTransform[expr,,t] | find a numerical approximation to the inverse Fourier sine transform of expr evaluated at the numerical value t, where expr is treated as a function of |

NFourierCosTransform[expr,t,] | find a numerical approximation to the Fourier cosine transform of expr evaluated at the numerical value , where expr is treated as a function of t |

NInverseFourierCosTransform[expr,,t] | find a numerical approximation to the inverse Fourier cosine transform of expr evaluated at the numerical value t, where expr is treated as a function of |

Finding numerical approximations to Fourier transforms.

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In addition to supporting the NIntegrate options, the numerical Fourier transform functions support the option FourierParameters. This option allows you to choose among the various conventions used for defining Fourier transforms.

Effect of FourierParameters setting on Fourier transform.

Effect of FourierParameters setting on Fourier sine transform.

Effect of FourierParameters setting on Fourier cosine transform.

To calculate a numerical approximation to a Fourier series or coefficient, you can use numerical versions of the Fourier series functions which are given below. The numerical approximation functions accept the FourierParameters option, allowing you to specify the period.

NFourierCoefficient[expr,t,n] | find a numerical approximation to the n coefficient in the exponential series expansion |

NFourierSinCoefficient[expr,t,n] | find a numerical approximation to the n coefficient in the sine series expansion |

NFourierCosCoefficient[expr,t,n] | find a numerical approximation to the n coefficient in the cosine series expansion |

NFourierSeries[expr,t,k] | find the exponential series expansion to order k using numerical approximations for the coefficients |

NFourierTrigSeries[expr,t,k] | find the trigonometric series expansion to order k using numerical approximations for the coefficients |

Finding approximate numerical values for Fourier coefficients and series.

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NFourierCoefficient[expr,t,n] | find a numerical approximation to the function |

Numerical approximation to FourierCoefficient.

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The Fourier transform from the discrete time domain into the continuous frequency domain is usually termed the discrete-time Fourier transform. Just as the Z transform is the discrete analog of the Laplace transform, the discrete-time Fourier transform is the discrete analog of the continuous-time Fourier transform. The option setting FourierParameters -> {a, b} can be used to specify the period.

Numerical approximations to these transforms are given below.

NDTFourierTransform[expr,n,omega] | find a numerical approximation to the function |

NInverseDTFourierTransform[expr,omega,n] | find a numerical approximation to the function |

Numerical approximations for discrete-time Fourier transforms.

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