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Fourier Series--Triangle Wave

\begin{figure}\begin{center}\BoxedEPSF{FourierSeriesTriangleWave.epsf scaled 700}\end{center}\end{figure}

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

$\displaystyle b_n$ $\textstyle =$ $\displaystyle {2\over L}\left\{{\int_0^{L/2} {x\over L/2}\sin\left({n\pi x\over...
...textstyle{1\over 2}}L)}\right]\sin\left({n\pi x\over L}\right)\,dx}\right\}\,dx$  
  $\textstyle =$ $\displaystyle {32\over\pi^2 n^2} \cos({\textstyle{1\over 4}}n\pi)\sin^3({\textstyle{1\over 4}}n\pi)$  
  $\textstyle =$ $\displaystyle {32\over\pi^2 n^2}\left\{\begin{array}{ll} 0 & \mbox{$n=0$, 4, \d...
... 6, \dots}\\  -{\textstyle{1\over 4}}& \mbox{$n=3$, 7, \dots}\end{array}\right.$  
  $\textstyle =$ $\displaystyle {8\over \pi^2 n^2}\left\{\begin{array}{ll} (-1)^{(n-1)/2} & \mbox{for $n$\ odd}\\  0 & \mbox{for $n$\ even.}\end{array}\right.$  

The Fourier series is therefore

f(x)={8\over \pi^2}\sum_{n=1,3,5,\ldots}^\infty {(-1)^{(n-1)/2}\over n^2}\sin\left({n\pi x\over L}\right).

See also Fourier Series

© 1996-9 Eric W. Weisstein