$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###Problems to Sections 4.3, 4.4, 4.5 > 1. [Problem 1](#problem-4.5.P.1) > 2. [Problem 2](#problem-4.5.P.2) > 3. [Problem 3](#problem-4.5.P.3) > 4. [Problem 4](#problem-4.5.P.4) > 5. [Problem 5](#problem-4.5.P.5) > 6. [Problem 6](#problem-4.5.P.6) 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](./S4.4.html) **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 $m\in \mathbb{N}$. ____________ [$\Uparrow$](../contents.html) [$\uparrow$](./S4.5.html) [$\Rightarrow$](./S4.A.html)