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 $[-l,l]$:

$e^{z x}$ where $z\in \mathbb{C}$; find "exceptional" values of $z$;

$\cos(\omega x)$, $\sin (\omega x)$ where $0<\omega\in \mathbb{R}$; fins "exceptional" values of $\omega$;

$\cosh (\eta x)$, $\sinh (\eta x)$ 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:

$x$;

$|x|$;

$x^2$.

Problem 3

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

$|\sin(x)|$;

$|\cos(x)|$.

Problem 4

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

$1$;

$x$;

$\sin (m x)$ with $m\in \mathbb{N}$;

$\cos (m x)$ with $m\in \mathbb{N}$;

$\sin ((m-\frac{1}{2}) x)$ 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:

$1$;

$x$;

$\sin (m x)$ with $m\in \mathbb{N}$;

$\cos (m x)$ with $m\in \mathbb{N}$;

$\sin ((m-\frac{1}{2}) x)$ 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:

$1$;

$x$;

$\sin (m x)$ with $m\in \mathbb{N}$;

$\cos (m x)$ with $m\in \mathbb{N}$;

$\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.