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###[Problems to Sections 4.3, 4.4, 4.5](id:sect-4.5.P)
> 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.](id:problem-4.5.P.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.](id:problem-4.5.P.2)**
Decompose into full Fourier series on interval $[-l,l]$ and sketch the
graph of the sum of such Fourier series:
a. $x$;
b. $|x|$;
c. $x^2$.
d. 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.](id:problem-4.5.P.3)**
Decompose into full Fourier series on interval $[-\pi,\pi]$ and
sketch the graph of the sum of such Fourier series:
a. $|\sin(x)|$;
b. $|\cos(x)|$.
**[Problem 4.](id:problem-4.5.P.4)**
Decompose into $\sin$ Fourier series on interval $[0,\pi]$ and
sketch the graph of the sum of such Fourier series:
a. $1$;
b. $x$;
c. $x(\pi -x)$;
d. $\sin (m x)$ with $m\in \mathbb{N}$;
e. $\cos (m x)$ with $m\in \mathbb{N}$;
f. $\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.
**[Problem 5.](id:problem-4.5.P.5)**
Decompose into $\cos$ Fourier series on interval $[0,\pi]$ and
sketch the graph of the sum of such Fourier series:
a. $1$;
b. $x$;
c. $x(\pi -x)$;
d. $\sin (m x)$ with $m\in \mathbb{N}$;
e. $\cos (m x)$ with $m\in \mathbb{N}$;
f. $\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.
**[Problem 6.](id:problem-4.5.P.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:
a. $1$;
b. $x$;
c. $x(\pi -x)$;
d. $\sin (m x)$ with $m\in \mathbb{N}$;
e. $\cos (m x)$ with $m\in \mathbb{N}$;
f. $\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.
____________
[$\Uparrow$](../contents.html) [$\uparrow$](./S4.5.html) [$\Rightarrow$](./S4.A.html)