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Fourier Series--Square Wave

\begin{figure}\begin{center}\BoxedEPSF{FourierSeriesSquare.epsf scaled 680}\end{center}\end{figure}

Consider a square wave of length $2L$. Since the function is Odd, $a_0=a_n=0$, and

$\displaystyle b_n$ $\textstyle =$ $\displaystyle {2\over L}\int_0^L\sin\left({n\pi x\over L}\right)\,dx$  
  $\textstyle =$ $\displaystyle {4\over n\pi} \sin^2({\textstyle{1\over 2}}n\pi) = {4\over n\pi} ...
...begin{array}{ll} 0 & \mbox{$n$\ even}\\  1 & \mbox{$n$\ odd.}\end{array}\right.$  

The Fourier series is therefore

f(x)={4\over\pi}\sum_{n=1,3,5,\ldots}^\infty {1\over n}\sin\left({n\pi x\over L}\right).

See also Fourier Series, Square Wave

© 1996-9 Eric W. Weisstein