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Problems to Sections 4.3, 4.4, 4.5

• Problem 1
• Problem 3
• Problem 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 $[-l,l]$ and sketch the graph of the sum of such Fourier series:

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. $x^2$.
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 $[0,\pi]$ 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 $m\in \mathbb{N}$.

$\Uparrow$  $\uparrow$  $\Rightarrow$