Fourier sine and 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[edit]

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

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 {\frac {n\pi x}{L}}

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 {\frac {n\pi x}{L}}\,dx,\quad n\in \mathbb {N} .

If $f$ is an even function with a period $2L$ , then the Fourier cosine series is defined to be

f(x)={\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,\quad n\in \mathbb {N} _{0}.

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

Byerly, William Elwood (1893). "Chapter 2: Development in Trigonometric Series". An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30.
Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.