$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
Some of the problems in this assignment could be solved based on the other problems and such solutions are much shorter than from the scratch; seeing and exploiting connections is a plus.
Here $\mathbb{N}=\{1,2,3,\ldots\}$,
Problem 1. Decompose into full Fourier series on interval $[-l,l]$:
a. $e^{z x}$ where $z\in \mathbb{C}$; find "exceptional" values of $z$; b. $\cos(\omega x)$, $\sin (\omega x)$ where $0<\omega\in \mathbb{R}$; fins "exceptional" values of $\omega$; c. $\cosh (\eta x)$, $\sinh (\eta x)$ where $0<\eta\in \mathbb{R}$;
Problem 2. Decompose into full Fourier series on interval $[-l,l]$ and sketch the graph of the sum of such Fourier series:
Problem 3. Decompose into full Fourier series on interval $[-\pi,\pi]$ and sketch the graph of the sum of such Fourier series:
Problem 4. Decompose into $\sin$ Fourier series on interval $[0,\pi]$ and sketch the graph of the sum of such Fourier series:
Problem 5. Decompose into $\cos$ Fourier series on interval $[0,\pi]$ and sketch the graph of the sum of such Fourier series:
Problem 6. Decompose into Fourier series with respect to $\sin ((n+\frac{1}{2})x)$ ($n=0,1,\ldots$) on interval $[0,2\pi]$ and sketch the graph of the sum of such Fourier series: