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]] '' | In [[mathematics]], the '''Fourier series''', named after [[Joseph Fourier]] (1768—1830), refers to an infinite series representation of a [[complex number|complex]]-valued [[periodic function]] ''ƒ'' of a [[real number|real]] variable ξ, of period ''P'': | ||
:<math>f(\xi+P)=f(\xi) \ | :<math>f(\xi+P)=f(\xi) \ . </math> | ||
is equivalent (in some sense) to | ''Fourier's theorem'' states that an [[infinite series]], known as a Fourier series, is equivalent (in some sense) to such a function: | ||
:<math>f(\xi) =\sum_{n=-\infty}^\infty c_n e^{2\pi in\xi/P}</math> | :<math>f(\xi) =\sum_{n=-\infty}^\infty c_n e^{2\pi in\xi/P}</math> | ||
defined by | where the coefficients {''c<sub>n</sub>''} are defined by | ||
:<math> c_n = \frac{1}{P} \int_0^P f(\xi) \exp\left(\frac{-2\pi in\xi}{P}\right)\,d\xi \ . </math> | :<math> c_n = \frac{1}{P} \int_0^P f(\xi) \exp\left(\frac{-2\pi in\xi}{P}\right)\,d\xi \ . </math> | ||
In what sense it may be said that this series converges to '' | In what sense it may be said that this series converges to ''ƒ''(ξ) is a complicated question.<ref name=Hardy/><ref name=Jahnke/> 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> | :<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 '' | and usually do not worry too much about the conditions to be imposed on the arbitrary function ''ƒ''(ξ) of period ''P'' in order that this expansion converge to the function. | ||
==References== | ==References== | ||
Revision as of 11:52, 4 June 2012
In mathematics, the Fourier series, named after Joseph Fourier (1768—1830), refers to an infinite series representation of a complex-valued periodic function ƒ of a real variable ξ, of period P:
Fourier's theorem states that an infinite series, known as a Fourier series, is equivalent (in some sense) to such a function:
where the coefficients {cn} are defined by
In what sense it may be said that this series converges to ƒ(ξ) is a complicated question.[1][2] 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 ƒ(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.
- ↑ For an historical account, see Hans Niels Jahnke (2003). “§6.5 Convergence of Fourier series”, A History of Analysis. American Mathematical Society, pp. 178 ff. ISBN 0821826239.