Problems to Sections 4.3, 4.4, 4.5

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

### Problems to Sections 4.3, 4.4, 4.5

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:

1. $x$;
2. $|x|$;
3. $x^2$.
4. For problem (b) with $l=5$ plot 4 first partial sums like on the figure in the end of Section 4.4

Problem 3. Decompose into full Fourier series on interval $[-\pi,\pi]$ and sketch the graph of the sum of such Fourier series:

1. $|\sin(x)|$;
2. $|\cos(x)|$.

Problem 4. Decompose into $\sin$ Fourier series on interval $[0,\pi]$ and sketch the graph of the sum of such Fourier series:

1. $1$;
2. $x$;
3. $x(\pi -x)$;
4. $\sin (m x)$ with $m\in \mathbb{N}$;
5. $\cos (m x)$ with $m\in \mathbb{N}$;
6. $\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.

Problem 5. Decompose into $\cos$ Fourier series on interval $[0,\pi]$ and sketch the graph of the sum of such Fourier series:

1. $1$;
2. $x$;
3. $x(\pi -x)$;
4. $\sin (m x)$ with $m\in \mathbb{N}$;
5. $\cos (m x)$ with $m\in \mathbb{N}$;
6. $\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.

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:

1. $1$;
2. $x$;
3. $x(\pi -x)$;
4. $\sin (m x)$ with $m\in \mathbb{N}$;
5. $\cos (m x)$ with $m\in \mathbb{N}$;
6. $\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.