# FourierCoefficient

As of Version 7.0, FourierCoefficient is part of the built-in Wolfram Language kernel.

FourierCoefficient[expr,t,n]

gives the n coefficient in the Fourier exponential series expansion of expr, where expr is a periodic function of t with period 1.

# Details and Options

• To use FourierCoefficient, you first need to load the Fourier Series Package using Needs["FourierSeries`"].
• The n coefficient in the Fourier exponential series expansion of expr is by default defined to be Integrate[expr 2πnt,{t,-,}].
• If n is numeric, it should be an explicit integer.
• Different choices for the definition of the Fourier exponential series expansion can be specified using the option FourierParameters.
• With the setting FourierParameters->{a,b}, expr is assumed to have a period of , and the n coefficient computed by FourierCoefficient is bIntegrate[expr 2 πbnt,{t,-,}].
• In addition to the option FourierParameters, FourierCoefficient can also accept the options available to Integrate. These options are passed directly to Integrate.

# Examples

## Basic Examples(1)

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Use different definitions for calculating a coefficient in a Fourier series:

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Compare with the answer from a numerical approximation:

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