Difference between revisions of "Fourier series"

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'''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]].
 
'''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:
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The Fourier [[series (mathematics)|series]] of a function <math>f(t)</math> is of the form:
:<math> f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] </math>
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where, ''n'' is an integer and
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:<math> f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \bigg[a_n \cos(\omega_n t) + b_n \sin(\omega_n t) \bigg] </math>
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 +
where, <math>n</math> is an [[integer]] and
 +
 
 
:<math> \omega_n = n\frac{2\pi}{T}</math>
 
:<math> \omega_n = n\frac{2\pi}{T}</math>
 
:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>
 
:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>
 
:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>
 
:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>
  
Fourier series can be generalized to [[Fourier transformation]]s for other classes of functions, for instance the space <math>L^2(\mathbb R^n)</math> of [[square integrable function]]s.  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.   
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Fourier series can be generalized to [[Fourier transformation]]s for other classes of functions, for instance the space <math>L^2(\mathbb R^n)</math> of square integrable functions.  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.   
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The fourier series can also be described using [[complex numbers]]. The complex Fourier series is:
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 +
<math>
 +
f(t) = \sum_{n=-\infty}^{\infty} \bigg[c_n e^{i \omega_n t} \bigg]
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</math>
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 +
where
  
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<math>c_n = \frac{1}{T} \int^{t_2}_{t_1} f(t) e^{i \omega_n t} dt</math>
 
[[Category:Calculus]]
 
[[Category:Calculus]]

Latest revision as of 16:41, 23 November 2016

This article/section deals with mathematical concepts appropriate for late high school or early college.

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 of a function is of the form:

where, is an integer and

Fourier series can be generalized to Fourier transformations for other classes of functions, for instance the space of square integrable functions. Even more generally, it is possible to carry out Fourier analysis in the setting of locally compact abelian topological groups, 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:

where