Trigonometric series

In mathematics, a trigonometric series is an infinite series of the form

${\displaystyle A_{0}+\sum _{n=1}^{\infty }A_{n}\cos {nx}+B_{n}\sin {nx},}$

where ${\displaystyle x}$ is the variable and ${\displaystyle \{A_{n}\}}$ and ${\displaystyle \{B_{n}\}}$ are coefficients. It is an infinite version of a trigonometric polynomial.

A trigonometric series is called the Fourier series of the integrable function ${\textstyle f}$ if the coefficients have the form:

${\displaystyle A_{n}={\frac {1}{\pi }}\int _{0}^{2\pi }\!f(x)\cos {nx}\,dx}$

${\displaystyle B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }\!f(x)\sin {nx}\,dx}$