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

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 $f(x)=\sum _{n=1}^{\infty }b_{n}\sin {\frac {n\pi x}{L}}$ 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 $b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin {\frac {n\pi x}{L}}\,dx,\quad n\in \mathbb {N} .$

Cosine series

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 $c_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos {\frac {n\pi x}{L}}\,dx,\quad n\in \mathbb {N} _{0}.$

Remarks

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

Bibliography

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.