Fourier sine and cosine series

"Sine and cosine series" redirects here. Not to be confused with Sine and cosine § Series definitions.

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 $$f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n\pi x}{L}}\right)$$ 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.

See also

• Fourier series
• Fourier analysis
• Least-squares spectral analysis

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.