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Showing posts with label Fourier Series. Show all posts
Showing posts with label Fourier Series. Show all posts

Wednesday, December 7, 2016

Poisson integral

Evaluate the integral:

$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$

Solution

Tuesday, November 15, 2016

Not Lebesgue integrable function

Let $x \in \mathbb{R}$. Given the series:

\begin{equation} \sum_{n=2}^{\infty} \frac{\sin nx}{\ln n} \end{equation}
  1. Prove that $(1)$ converges forall $x \in \mathbb{R}$.
  2. Prove that $(1)$ is not a Fourier series of a Lebesgue integrable function.
Solution

Saturday, August 8, 2015

On a very well known identity

Let $f$ be a $2\pi$ periodical function defined as $f(x)=\cos ax , \; |x|\leq \pi, \; a \notin \mathbb{Z}$ . Expand $f$ in a Fourier Series and prove the identity:

$$\pi \cot \pi a = \lim_{N \rightarrow +\infty} \sum_{n=-N}^{N} \frac{1}{n+a}$$

Solution

Thursday, July 30, 2015

Piecewise function into Fourier Series and evaluation of Series

Let $0<\delta<\pi$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ be a $2\pi$ periodical function defined as:

$$f(x)= \left\{\begin{matrix}
1 &, &\left | x \right |\leq \delta \\
 0&, &\delta< \left | x \right |\leq \pi
\end{matrix}\right.$$

Expand $f$ in a Fourier series and in continuity show that:

$$\begin{matrix}
\displaystyle \sum_{n=1}^{\infty}\frac{\sin n \delta}{n}= \frac{\pi-\delta}{2} &, &\displaystyle \sum_{n=1}^{\infty}\frac{\sin^2 n\delta}{n^2 \delta}= \frac{\pi-\delta}{2}  &, &\displaystyle \sum_{n=1}^{\infty}\frac{1}{\left ( 2n-1 \right )^2}= \frac{\pi^2}{8}\end{matrix} $$

Solution

Friday, May 1, 2015

Fourier cosine series of $\sin ax$

Let $f(x)=\sin ax, \; x \in (0, \pi) $ and let $a \in \mathbb{Z}$ . Prove that $f$ can be expanded into Fourier cosine series and that:

$$\sin ax \sim \left\{\begin{matrix} \displaystyle \dfrac{4a}{\pi}\sum_{n=0}^{\infty}\frac{\cos (2n+1)x}{a^2-(2n+1)^2} &,\;\;\;  a {\rm\; is \; even}\\  \displaystyle \frac{4a}{\pi}\left [ \frac{1}{2a^2}+\sum_{n=1}^{\infty}\frac{\cos 2nx}{a^2-4n^2} \right ] &,\;\;\;  a {\rm\; is \; odd} \end{matrix}\right.$$

Solution:

Wednesday, April 22, 2015

Not a Fourier series of a Riemann integrable function

Prove that the series $\displaystyle \sum_{n=1}^{\infty}\frac{\cos nx}{\ln (n+1)}$ is not a Fourier series of a $2\pi$ periodical Riemann integrable function.

Solution:

Saturday, April 18, 2015

On Fourier Series

Let $f:[-\pi, \pi] \rightarrow \mathbb{R}$ be defined as $f(x)=|x|$. Expand the function in a Fourier series and then evaluate the series:

$$a) \sum_{n={\rm odd} \geq 1}^{\infty}\frac{1}{n^2}, \; \; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \beta)\sum_{n=1}^{\infty}\frac{1}{n^2}$$

Solution: