Fourier series

Fourier series express a piecewise continous, periodic function as a linear combination of orthonormal Sine and Cosine functions.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.