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Fourier series

294 bytes added, 21:41, 23 November 2016
Added complex Fourier series definition
'''Fourier series''' express a piecewise continuous, bounded, periodic function as a linear combination of [[orthogonal]] [[sine]] and [[cosine]] functions. The seeds of the modern theory were developed by [[Joseph Fourier]].
The Fourier [[series (mathematics)|series]] of a function ''$f(t)''$ is of the form: :$f(t) = \frac{1a_0}{2} a_0 + \sum_{n=1}^{\infty}\bigg[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)\bigg]$ 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 transformation]]s for other classes of functions, for instance the space $L^2(\mathbb R^n)$ of [[square integrable function]]sfunctions. Even more generally, it is possible to carry out Fourier analysis in the setting of [[compact space|locally compact]] [[abelian]] [[topology|topological]] [[group]]s, where the fundamental observation of Pontryagin duality provides the necessary theoretical underpinnings.  The fourier series can also be described using [[complex numbers]]. The complex Fourier series is: $f(t) = \sum_{n=-\infty}^{\infty} \bigg[c_n e^{i \omega_n t} \bigg]$ where
$c_n = \frac{1}{T} \int^{t_2}_{t_1} f(t) e^{i \omega_n t} dt$
[[Category:Calculus]]
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