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{\bf Exam Question

Topic: Fourier Series}

Find the Fourier Series for the function given by $$ f(x)=\left\{
\begin{array}{ll} 0 & \mbox{if\ \ $-\pi< x<0$};\\ x & \mbox{if\ \ $0\le
x\le\pi$}.\end{array} \right. $$

{\bf Solution}

$$a_0=\frac{1}{\pi}\int_{-\pi}{\pi}f(x)\,
dx=\frac{1}{\pi}\mathrm{area\ of\
triangle}=\frac{1}{\pi}\frac{\pi^2}{2}=\frac{\pi}{2}.$$

\begin{eqnarray*}
a_n&=&\frac{1}{\pi}\int_0^{\pi} x\cos nx\, dx
=\frac{1}{\pi}\left[x\frac{\sin
nx}{n}\right]_0^{\pi}-\frac{1}{\pi}\int_0^{\pi}\frac{\sin nx}{n}\,
dx\\ &=& \frac{1}{\pi}\left[\frac{\cos
nx}{n^2}\right]_0^{\pi}=\frac{1}{n^2\pi}\left[(-1)^n-1\right]
\end{eqnarray*}

\begin{eqnarray*}
b_n&=&\frac{1}{\pi}\int_0^{\pi} x\sin nx\, dx
=\frac{1}{\pi}\left[-x\frac{\cos
nx}{n}\right]_0^{\pi}+\frac{1}{\pi}\int_0^{\pi}\frac{\cos nx}{n}\,
dx\\ &=&
\frac{1}{\pi}\left[\frac{-\pi(-1)^n}{n}\right]+\frac{1}{\pi}\left[\frac{\sin
nx}{n^2} \right]_0^{\pi}=\frac{(-1)^{n+1}}{n}
\end{eqnarray*}

So the Fourier Series is

$$Frac{\pi}{4}+\sum_{n=1}^{\infty}\left[\frac{1}{\pi
n^2}\left((-1)^n-1\right)\cos nx+\frac{(-1)^{n+1}}{n}\sin
nx\right]$$

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