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Fourier Series--Power Series

For $f(x)=x^k$ on the Interval $[-L, L)$ and periodic with period $2L$, the Fourier Series is given by

$\displaystyle a_n$ $\textstyle =$ $\displaystyle {1\over L}\int_{-L}^L x^k\cos\left({n\pi x\over L}\right)\,dx$  
  $\textstyle =$ $\displaystyle {2L^k\over 1+k}{}_1F_2\left({\begin{array}{c}1+{\textstyle{1\over...
...\textstyle{1\over 2}}(3+k)\end{array}; -{\textstyle{1\over 4}}\pi^2 n^2}\right)$  
$\displaystyle b_n$ $\textstyle =$ $\displaystyle {1\over L}\int_{-L}^L x^k\sin\left({n\pi x\over L}\right)\,dx$  
  $\textstyle =$ $\displaystyle {2n\pi L^k\over 2+k}{}_1F_2\left({\begin{array}{c}1+{\textstyle{1...
...2+{\textstyle{1\over 2}}k\end{array}; -{\textstyle{1\over 4}}\pi^2 n^2}\right),$  

where ${}_1F_2(a;b,c;x)$ is a generalized Hypergeometric Function.

© 1996-9 Eric W. Weisstein