$\newcommand{\const}{\mathrm{const}}$ $\newcommand{\erf}{\operatorname{erf}}$

### APM 346 (2012) Home Assignment 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 $[-l,l]$:

1. $e^{z x}$ where $z\in \mathbb{C}$; find "exceptional" values of $z$;
2. $\cos(\omega x)$, $\sin (\omega x)$ where $0<\omega\in \mathbb{R}$; fins "exceptional" values of $\omega$;
3. $\cosh (\eta x)$, $\sinh (\eta x)$ 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. $x^2$.

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