Toronto Math Forum
APM3462016F => APM346Tests => Q4 => Topic started by: Victor Ivrii on October 28, 2016, 09:02:21 AM

Decompose into full Fourier series on interval $[\pi,\pi]$ and sketch the graph of the sum of such Fourier series:
\begin{equation}
f(x)=x.
\end{equation}

Since $f(x) = x$ is an even function, $b_n = 0 \ \ \forall n$
$a_0 = \frac{1}{\pi}\int_{\pi}^\pi x\,dx = \pi$
$a_n = \frac{1}{\pi}\int_{\pi}^\pi x\cos(nx) \,dx = \int_0^\pi \frac{2x}{\pi}\cos(nx) \,dx $
Integrating by parts
$a_n =\frac{2x}{n\pi}\sin(nx) \big_{0}^{\pi} \int_0^\pi \frac{2}{n\pi} \sin(nx) \,dx = \frac{2}{n^2\pi}\cos(nx)\big_{0}^{\pi} $
$a_n = \begin{cases}\frac{4}{n^2\pi} && n \ is \ odd \\0 && n \ is \ even\end{cases}$
$f(x) = \frac{\pi}{2} +\sum_{m=0}^\infty \frac{4}{(2m+1)^2\pi}\cos((2m +1)x) $

Everybody understands that on the graph bottoms are at $2n\pi $ and picks at $(2n+1)\pi$, $n\in \mathbb{Z}$.