Toronto Math Forum
APM3462012 => APM346 Math => Home Assignment 5 => Topic started by: Ian Kivlichan on October 31, 2012, 09:32:02 PM

Hopeful solutions for 4.c)! :)
edit: Note that sketch is for m=1.

Additional solution for 4.a) (essentially the same as Aida's post here http://forum.math.toronto.edu/index.php?topic=108.msg552#msg552 , but showing more of the sketch, as well as with details on the odd continuation used for sin Fourier series).

Hopeful solution for 4.b) attached!

Part (e):
We use an odd continuation for this function. Integration was done using the anglesum identity as in Problem 3 (http://forum.math.toronto.edu/index.php?topic=110.0).
\begin{equation*}
a_n = 0, a_0 = 0.
\end{equation*}
\begin{equation*}
b_n = \frac{2}{\pi} \int_{0}^{\pi} \sin{(m\frac{1}{2})x}\sin{nx} dx\\
= \frac{1}{\pi}\left[\frac{\sin{m + n  \frac{1}{2}} x }{m + n  \frac{1}{2}} + \frac{\sin{m  n  \frac{1}{2}} x }{m  n  \frac{1}{2}} \right]_{0}^{\pi}\\
= \frac{1}{\pi}\left(\frac{(1)^{m+n}}{m + n  \frac{1}{2}} + \frac{(1)^{mn}}{m  n  \frac{1}{2}} \right).
\end{equation*}
\begin{equation*}
\sin{((m\frac{1}{2})x)} = \frac{1}{\pi}(1)^m \sum_{n=1}^{\infty} (1)^n \left(\frac{1}{m + n  \frac{1}{2}} + \frac{1}{m  n  \frac{1}{2}} \right) \sin{n x}.
\end{equation*}

includes solution to part (d) and what I think is the graph to part (e).

Obviously in (d) only terms with odd $nm$ do not vanish