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 $$f(x)=\sum _{n=1}^{\mathrm{\infty }}{b}_n\sin \left(\frac{n\pi x}{L}\right)$$ 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 $$b_n=\frac{2}{L}\underset{0}{\overset{L}{\int }}f(x)\sin \left(\frac{n\pi x}{L}\right)dx,n\in \mathbb{\mathbb{N}}.$$

Cosine series

If f is an even function with a period $2L$, then the Fourier cosine series is defined to be $$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}{a}_n\cos \left(\frac{n\pi x}{L}\right)$$ where $$a_n=\frac{2}{L}\underset{0}{\overset{L}{\int }}f(x)\cos \left(\frac{n\pi x}{L}\right)dx,n\in {\mathbb{\mathbb{N}}}_0.$$

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.