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 .

Sine series[edit]

The Fourier sine series of f is defined to be

where

.

If f is continuous and , then the Fourier sine series of f is equal to f on , odd, and periodic with period .

Cosine series[edit]

The Fourier cosine series is defined to be

where

.

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

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.