Problem 2
Let \(g(x)= \begin{cases}\cos x, & -\pi
Problem 2
Find the Fourier series of \(f_{p}(x)=\cos p x\), for \(0 \leq p \leq \pi\).
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 \(f \in E\) be an even function satisfying \(\int_{-\pi}^{\pi} f(t) d t=5\). Define the function \(F\) by $$ F(x)=\int_{-\pi}^{x} f(t) d t, \quad-\pi \leq x \leq \pi . $$ 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 \(F\) and set $$ G(x)=\frac{A_{0}}{2}+\sum_{n=1}^{\infty}\left[A_{n} \cos n x+B_{n} \sin n x\right] $$ Calculate \(G(-\pi), G(\pi)\), and \(G(0)\).
Problem 2
Let \(f(x)=x^{2}\) and $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote the Fourier series of \(f\) on \([\pi, 3 \pi]\). (a) Calculate the \(a_{n}\) and \(b_{n}\). (b) Set \(g(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos \frac{n x}{2},-\pi \leq x \leq \pi\). Determine \(g\) and sketch the graph of \(g\) on \([-\pi, \pi]\). (c) Set \(h(x)=\sum_{n=1}^{\infty} b_{n} \sin \frac{n x}{2},-\pi \leq x \leq \pi\). Determine \(h\) and sketch the graph of \(h\) on \([-\pi, \pi]\).
Problem 2
For each real \(p,-\pi \leq p \leq \pi\), find the Fourier series of the
function
$$
f_{p}(x)= \begin{cases}0, & -\pi \leq x \leq p, \\ 1, & p
Problem 3
For each natural integer \(n\) we define $$ f_{n}(x)=1+\sum_{k=1}^{n}[\cos k x-\sin k x] . $$ Calculate the value of the integral \(\int_{-\pi}^{\pi}\left|f_{n}(x)\right|^{2} d x\).
Problem 3
Assume \(f\) satisfies the assumptions of Dirichlet's Theorem. Determine the following limits: (a) \(\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin n t d t\) (b) \(\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin \left(n-\frac{1}{2}\right) t d t\) (c) \(\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} \frac{f(t)}{t} \sin \left(n-\frac{1}{2}\right) t d t\)
Problem 3
Let \(f \in E\) and $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote its Fourier series. Define the two functions $$ g(x)=\frac{f(x)+f(-x)}{2}, \quad h(x)=\frac{f(x)-f(-x)}{2} . $$ Find the Fourier series of \(g\) and of \(h\).
Problem 3
Suppose \(f(x)= \begin{cases}\frac{\pi}{4}, & -\pi
