Fourier series

Fourier series express a piecewise continous, 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

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

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 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.