Fourier sine and cosine series

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In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.

In this article f denotes a real valued function on \mathbb{R} which is periodic with period 2L.

Sine series[edit]

If f(x) is an odd function, then the Fourier sine series of f is defined to be

 f(x) = \sum_{n=1}^\infty c_n \sin \frac{n\pi x}{L}

where

c_n=\frac{2}{L}\int_0^L f(x) \sin \frac{n\pi x}{L} \, dx, n\in \mathbb{N}.

Cosine series[edit]

If f(x) is an even function, then the Fourier cosine series is defined to be

 f(x)=\frac{c_0}{2} + \sum_{n=1}^\infty c_n \cos \frac{n\pi x}{L}

where

c_n=\frac{2}{L}\int_0^L f(x) \cos \frac{n\pi x}{L} \, dx, n\in \mathbb{N}_0.

Remarks[edit]

This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.

See also[edit]