Fourier series: Difference between revisions
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In [[mathematics]], the '''Fourier series''', named after [[Joseph Fourier]] (1768—1830), of a [[complex number|complex]]-valued [[periodic function]] ''f'' of a [[real number|real]] variable, | In [[mathematics]], the '''Fourier series''', named after [[Joseph Fourier]] (1768—1830), of a [[complex number|complex]]-valued [[periodic function]] ''f'' of a [[real number|real]] variable ξ, of period ''P'': | ||
:<math>\sum_{n=-\infty}^\infty c_n e^{2\pi | :<math>f(\xi+P)=f(\xi) \ , </math> | ||
is equivalent to an [[infinite series]] | |||
:<math>f(\xi) =\sum_{n=-\infty}^\infty c_n e^{2\pi in\xi/P}</math> | |||
defined by | defined by | ||
:<math> c_n = \frac{1}{ | :<math> c_n = \frac{1}{P} \int_0^P f(\xi) \exp\left(\frac{-2\pi inx}{P}\right)\,d\xi \ . </math> | ||
In what sense it may be said that this series converges to ''f''(''x'') is a somewhat delicate question.<ref name=Hardy/> However, physicists being less delicate than mathematicians in these matters, simply write | |||
:<math>f(\xi) = \sum_{n=-\infty}^\infty c_n e^{2\pi in\xi/P} \ ,</math> | |||
and usually do not worry too much about the conditions to be imposed on the arbitrary function ''f''(ξ) of period ''P'' in order that this expansion converge to the function. | |||
==References== | |||
{{reflist|refs= | |||
<ref name=Hardy> | |||
{{cite book |title=Fourier Series |author=G. H. Hardy, Werner Rogosinski |chapter=Chapter IV: Convergence of Fourier series |pages=pp. 37 ''ff'' |isbn= 0486406814 |year=1999 |edition=Reprint of Cambridge University Press 1956 ed|publisher=Courier Dover Publications}} | |||
</ref> | |||
}} | |||
Revision as of 10:26, 4 June 2012
In mathematics, the Fourier series, named after Joseph Fourier (1768—1830), of a complex-valued periodic function f of a real variable ξ, of period P:
is equivalent to an infinite series
defined by
In what sense it may be said that this series converges to f(x) is a somewhat delicate question.[1] However, physicists being less delicate than mathematicians in these matters, simply write
and usually do not worry too much about the conditions to be imposed on the arbitrary function f(ξ) of period P in order that this expansion converge to the function.
References
- ↑ G. H. Hardy, Werner Rogosinski (1999). “Chapter IV: Convergence of Fourier series”, Fourier Series, Reprint of Cambridge University Press 1956 ed. Courier Dover Publications, pp. 37 ff. ISBN 0486406814.