Fourier sine and cosine series

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

If f is an odd function with period ${\displaystyle 2L}$, then the Fourier Half Range sine series of f is defined to be

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

which is just a form of complete Fourier series with the only difference that ${\displaystyle a_{0}}$ and ${\displaystyle a_{n}}$ are zero, and the series is defined for half of the interval.

In the formula we have

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

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

$${\displaystyle f(x)={\frac {c_{0}}{2}}+\sum _{n=1}^{\infty }c_{n}\cos {\frac {n\pi x}{L}}}$$

where

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

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

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

• 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.