Problems to Sections 4.3, 4.4, 4.5

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Problems to Sections 4.3, 4.4, 4.5

  1. Problem 1
  2. Problem 2
  3. Problem 3
  4. Problem 4
  5. Problem 5
  6. Problem 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 $[-l,l]$:

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

$\Uparrow$  $\uparrow$  $\Rightarrow$