Problem 1
Calculate the integral $$ \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty} \frac{1}{2^{n}} e^{i n x}\right|^{2} d x . $$
Problem 1
Let \(f(x)=|x|,-\pi \leq x \leq \pi\), and let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote the Fourier series of \(f\). (a) Determine the coefficients \(a_{n}\) and \(b_{n}\). (b) Prove that the series \(\sum_{n=1}^{\infty} n a_{n} \sin n x\) converges for every \(x\). (c) For each real \(x\), we set \(g(x)=-\sum_{n=1}^{\infty} n a_{n} \sin n x\). Sketch the graph of \(g\) on the interval \([-2 \pi, 2 \pi]\). (d) Calculate \(\sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{2}}\) and \(\sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{4}}\).
Problem 1
Set \(f(x)=1-x^{2}\) in the interval \([-\pi, \pi]\) and let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ be the Fourier series of \(f\). (a) Calculate the \(a_{n}\) and \(b_{n}\). (b) To what values does the Fourier series of \(f\) converge at the points \(x=5 \pi\) and \(x=6 \pi\) ? Explain.
Problem 1
Find the Fourier series of each of the following functions.
(a) \(f(x)=|\sin x|\)
(b) \(f(x)= \begin{cases}0, & -\pi \leq x \leq 0 \\ e^{x}, & 0
Problem 1
Let \(f(x)=x+\cos x\) and $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos \frac{n x}{2}+b_{n} \sin \frac{n x}{2}\right] $$ be the Fourier series of \(f\) on \([0,4 \pi]\). (a) Determine the \(a_{n}\) and \(b_{n}\). (b) Let \(g(x)=\frac{A_{0}}{2}+\sum_{n=1}^{5} B_{n} \sin \frac{n x}{2}\). For what values of \(A_{0}\) and \(B_{n}\), \(1 \leq n \leq 5\), is the distance between \(f\) and \(g\) minimal?
Problem 1
Let $$ f(x)= \begin{cases}A x+B, & -\pi \leq x<0, \\ \cos x, & 0 \leq x \leq \pi .\end{cases} $$ For what values \(A\) and \(B\) does the Fourier series of \(f\) converge uniformly to \(f\) on all of \([-\pi, \pi]\) ?
Problem 1
Let $$ f(x)= \begin{cases}x, & 0 \leq x \leq \frac{\pi}{2}, \\ \pi-x, & \frac{\pi}{2} \leq x \leq \pi,\end{cases} $$ and let \(\tilde{f}\) be the odd extension of \(f\) to \([-\pi, \pi]\). Find the Fourier series of \(\tilde{f}\) on \([-\pi, \pi]\).
Problem 2
For each real number \(p \neq 0\), set \(f_{p}(x)=e^{p x}\) in the interval \([-\pi, \pi]\). Let $$ f_{p}(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote the Fourier series of \(f_{p}\). (a) Calculate \(a_{n}\) and \(b_{n}\). (b) Determine \(\sum_{n=0}^{\infty} a_{n}\) and \(\sum_{n=0}^{\infty}(-1)^{n} a_{n}\).
Problem 2
Let $$ f(x)= \begin{cases}0, & -\pi \leq x<0 \\ e^{i x}, & 0 \leq x<\pi\end{cases} $$ Find the complex Fourier series of \(f\).
Problem 2
Let \(g(x)= \begin{cases}\cos x, & -\pi
