Fourier Cosine Series

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

In this article, denotes a real valued function on $\mathbb$ which is periodic with period 2''L''.

Sine series

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

In the formula we have
$b_n = \frac \int_0^L f(x) \sin \frac \, dx, \quad n \in \mathbb .$

Cosine series

If is an even function with a period $2L$, then the Fourier cosine series is defined to be
$f(x) = \frac + \sum_^ c_n \cos \frac$
where
$c_n = \frac \int_0^L f(x) \cos \frac \, dx, \quad n \in \mathbb_0 .$

Remarks

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

See also

*Fourier series
*Fourier analysis
* Least-squares spectral analysis

Bibliography

*
* {{cite book
, first=Horatio Scott , last=Carslaw
, title=Introduction to the Theory of Fourier's Series and Integrals, Volume 1
, edition=2
, publisher=Macmillan and Company
, date=1921
, chapter=Chapter 7: Fourier's Series , chapter-url=https://books.google.com/books?id=JNVAAAAAIAAJ&pg=PA196
, page=196

Fourier series