FourierSinCoefficient

gives the n coefficient in the Fourier sine series expansion of expr.

FourierSinCoefficient[expr,{t₁,t₂,…},{n₁,n₂,…}]

gives a multidimensional Fourier sine coefficient.

Details and Options

The coefficient in the Fourier sine series expansion of is by default given by .
The -dimensional Fourier sine coefficient is given by .
In the form FourierSinCoefficient[expr,t,n], n can be symbolic or a positive integer.
The following options can be given:

Assumptions	$Assumptions	assumptions on parameters
FourierParameters	{1,1}	parameters to define Fourier series
GenerateConditions	False	whether to generate results that involve conditions on parameters

The function expr is assumed to be periodic in t with period , except when otherwise specified by FourierParameters.
Common settings for FourierParameters include:
{1,1} default settings

{1,2Pi} period 1

{a,b} general setting

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Find the 3 Fourier sine series coefficient:

Find the general term coefficient:

Plot the coefficient sequence:

Find the {3,5} Fourier sine coefficient:

The general term:

Plot the coefficient sequence:

Find the 5 Fourier sine coefficient for a quadratic polynomial:

Find the general coefficient for a piecewise function:

General Fourier sine coefficient for a Gaussian function:

Fourier sine coefficient for a basis function:

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