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Fourier Sine Series

If f(x) is an odd function, then a_n=0 and the Fourier series collapses to

f(x)=sum_(n=1)^inftyb_nsin(nx),
(1)

where

b_n=1/piint_(-pi)^pif(x)sin(nx)dx
(2)
=2/piint_0^pif(x)sin(nx)dx
(3)

for n=1, 2, 3, .... The last equality is true because

f(x)sin(nx)=[-f(-x)][-sin(-nx)]
(4)
=f(-x)sin(-nx).
(5)

Letting the range go to L,

b_n=2/Lint_0^Lf(x)sin((npix)/L)dx.
(6)

See also

Fourier Cosine Series, Fourier Series, Fourier Sine Transform

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Cite this as:

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

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