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Fourier Series--Power

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For a power function f(x)=x^k with k>=0 on the interval [0,2L] and periodic with period 2L, the coefficients of the Fourier series are given by

a_0=(2^(k+1)L^k)/(k+1)
(1)
a_n=(2^(k+1)L^k)/(k+1)_1F_2(1/2(k+1);1/2,1/2(k+3);-pi^2n^2)
(2)
b_n=(2^(k+2)L^knpi)/(k+2)_1F_2(1+1/2k;3/2,2+1/2k;-pi^2n^2),
(3)

where _1F_2(a;b,c;x) is a generalized hypergeometric function.

See also

Fourier Series

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Cite this as:

Weisstein, Eric W. "Fourier Series--Power." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FourierSeriesPower.html

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