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Fourier Series--Semicircle


FourierSeriesSemicircle

Given a semicircular hump

f(x)=sqrt(L^2-(x-L)^2)
(1)
=sqrt((2L-x)x),
(2)

the Fourier coefficients are

a_0=1/2piL
(3)
a_n=((-1)^nLJ_1(npi))/n
(4)
b_n=0,
(5)

where J_1(z) is a Bessel function of the first kind, so the Fourier series is therefore

 f(x)=L[1/4pi+sum_(n=1)^infty((-1)^nJ_1(npi))/ncos((npix)/L)].
(6)

See also

Fourier Series, Semicircle

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Cite this as:

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

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