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.

Notation[edit]

In this article, f denotes a real valued function on [0,L].

Sine series[edit]

The Fourier sine series of f is defined to be

 \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}.

If f is continuous and f(0)=f(L)=0, then the Fourier sine series of f is equal to f on [0,L], odd, and periodic with period 2L.

Cosine series[edit]

The Fourier cosine series is defined to be

 \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.

If f is continuous, then the Fourier cosine series of f is equal to f on [0,L], even, and periodic with period 2L.

Remarks[edit]

This notion can be generalized to functions which are not continuous.

See also[edit]

References[edit]

Haberman, Richard. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (4th ed.). Pearson. pp. 97–113. ISBN 978-0130652430.