Fourier series

Fourier series express a piecewise, continuous, periodic function as a linear combination of orthogonal sine and cosine functions.

The Fourier series of a function f(t) is of the form:

$f(t)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(\omega _{n}t)+b_{n}\sin(\omega _{n}t)]$

where, n is an integer and

$\omega _{n}=n{\frac {2\pi }{T}}$

$a_{n}={\frac {2}{T}}\int _{t_{1}}^{t_{2}}f(t)\cos(\omega _{n}t)\,dt$

$b_{n}={\frac {2}{T}}\int _{t_{1}}^{t_{2}}f(t)\sin(\omega _{n}t)\,dt$

Fourier series can be generalized to Fourier transformations for non-perodic, piecewise continuous, square integrable functions.