Problem Set 1

3. Formulae for Fourier coefficients

Construct a Fourier sine series for the odd periodic function that has period $2\pi$ and is equal to $\cos x$ for $x \in (0, \pi]$.

Spoiler

4. Fourier series computation

Consider the $2\pi$ periodic functions $f$ and $g$ defined by \[ f(x) = x \quad \text{for $x \in (-\pi, \pi]$}, \]

and \[ g(x) = \begin{cases} 0 & -\pi < x \leq 0 \\ \sin x & 0 < x \leq \pi. \end{cases} \]

Evaluate the Fourier series.

Spoiler

5. Fourier series computation

a) Sketch the graph of the periodic function $f$ of period $2\pi$ defined by \[ f(x) = e^x, \qquad x \in (-\pi, \pi]. \]

At the points of discontinuity of $f(x)$ (or rather, of the Fourier series approximation of $f$), where does the Fourier series converge?

b) Calculate the Fourier series of $f$

c) Deduce that \begin{align*} \pi \coth(\pi) &= 1 + 2 \sum_{n=1}^\infty \frac{1}{1+n^2} \\ \pi \textrm{cosech}(\pi) &= 1 + 2 \sum_{n=1}^\infty \frac{(-1)^n}{1+n^2} \end{align*}

Spoiler