Given a semicircular hump
 |  |  |
(1)
|
 |  |  |
(2)
|
the Fourier coefficients are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
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where 
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)].](/images/equations/FourierSeriesSemicircle/NumberedEquation1.svg) |
(6)
|
Fourier Series--Semicircle
See also
Fourier Series, SemicircleExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Fourier Series--Semicircle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FourierSeriesSemicircle.html