Harmonic Analysis

Fourier Series; Harmonic Analysis Exercise 1 : Symmetric rectangular pulse Obtain the Fourier series of the 2\pi-periodic function $$ f(x)=\begin{cases} 0, & -\pi\le x< -\tfrac{\pi}{2},\\ 1, & -\tfrac{\pi}{2}\le x< \tfrac{\pi}{2},\\ 0, & \tfrac{\pi}{2}\le x\le \pi, \end{cases} $$State whether the periodic extension is even or odd, compute the nonzero Fourier coefficients, and write the series in sine/cosine form. Exercise 2 : Odd square wave For the 2\pi-periodic function $$ f(x)=\begin{cases} 1, & -\pi\le x<0,\\ -1, & 0\le x\le\pi, \end{cases} $$Compute the Fourier series, indicate whether only sine terms appear, and derive the explicit formula for the coefficients. ...