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(x) is an odd function, then the Fourier sine series of f is defined to be

f(x)=\sum _{n=1}^{\infty }c_{n}\sin {\frac {n\pi x}{L}}

where

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

.

Cosine series

If f(x) is an even function, 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,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

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.

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