Fourier sine and cosine series

Notation

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

Sine series

If f is an odd function with period 2L, then the Fourier Half Range sine series of f is defined to be f(x) = \sum_{n=1}^\infty b_n \sin \left(\frac{n\pi x}{L}\right) which is just a form of complete Fourier series with the only difference that a_0 and a_n are zero, and the series is defined for half of the interval.

In the formula we have b_n = \frac{2}{L} \int_0^L f(x) \sin \left(\frac{n\pi x}{L}\right) , dx, \quad n \in \mathbb{N} .

Cosine series

If f is an even function with a period 2L, then the Fourier cosine series is defined to be f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos \left(\frac{n \pi x}{L}\right)
where a_n = \frac{2}{L} \int_0^L f(x) \cos \left(\frac{n\pi x}{L}\right) , dx, \quad n \in \mathbb{N}_0 .

Remarks

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

Bibliography

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