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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.

Contents

Notation

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

Sine series

The Fourier sine series of f is defined to be

∑ n = 1 ∞ c n sin ⁡ n π x L

where

c n = 2 L ∫ 0 L f ( x ) sin ⁡ n π x L d x , n ∈ 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 2 L .

Cosine series

The Fourier cosine series is defined to be

c 0 2 + ∑ n = 1 ∞ c n cos ⁡ n π x L

where

c n = 2 L ∫ 0 L f ( x ) cos ⁡ n π x L d x , n ∈ 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 2 L .

Remarks

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

References

Fourier sine and cosine series Wikipedia
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