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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) = {\textstyle{1\over 2}}a_0 + \sum_{n=1}^\infty a_n\cos(nx),
\end{displaymath} (1)

$\displaystyle a_0$ $\textstyle =$ $\displaystyle {1\over\pi} \int_{-\pi}^\pi f(x)\,dx = {2\over\pi} \int^\pi_0 f(x)\,dx$ (2)
$\displaystyle a_n$ $\textstyle =$ $\displaystyle {1\over\pi} \int_{-\pi}^\pi f(x)\cos(nx)\,dx$  
  $\textstyle =$ $\displaystyle {2\over\pi} \int^\pi_0 f(x)\cos(nx)\,dx,$ (3)

where the last equality is true because
f(x)\cos(nx) = f(-x)\cos(-nx).
\end{displaymath} (4)

Letting the range go to $L$,
$\displaystyle a_0$ $\textstyle =$ $\displaystyle {2\over L} \int^L_0 f(x)\,dx$ (5)
$\displaystyle a_n$ $\textstyle =$ $\displaystyle {2\over L} \int^L_0 f(x)\cos\left({n\pi x\over L}\right)\,dx.$ (6)

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

© 1996-9 Eric W. Weisstein