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Fourier Cosine Series

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

f(x)=1/2a_0+sum_(n=1)^inftya_ncos(nx),
(1)

where

a_0=1/piint_(-pi)^pif(x)dx
(2)
=2/piint_0^pif(x)dx
(3)
a_n=1/piint_(-pi)^pif(x)cos(nx)dx
(4)
=2/piint_0^pif(x)cos(nx)dx,
(5)

where the last equality is true because

f(x)cos(nx)=f(-x)cos(-nx).
(6)

Letting the range go to L,

a_0=2/Lint_0^Lf(x)dx
(7)
a_n=2/Lint_0^Lf(x)cos((npix)/L)dx.
(8)

See also

Even Function, Fourier Cosine Transform, Fourier Series, Fourier Sine Series

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Cite this as:

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

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