MT201 - Engineering Mathematics 2
Fourier Series Tutorial 1

University of Zimbabwe, Block A, 2022

Due date: 8 September, 2022

1. For \( n, m\geq 1\) show that \[ \int_{-\pi}^{\pi} \sin(nx)\sin(mx)\,dx = \begin{cases} \pi, & \text{if } n=m\\ 0, & \text{if }n\neq m. \end{cases}\]
2. For \( n, m\geq 0\) show that \[ \int_{-\pi}^{\pi} \sin(nx)\cos(mx)\,dx = 0.\]
3. Find the Fourier Series of the following functions. Determine if the function is even, odd, or neither and choose an appropriate series.
   1. $$ f(x) = |\sin(x)|\,\,\,\text{on } [-\pi,\pi] $$
   2. $$ f(x) = \begin{cases} e^x, & \text{if } 0\leq x \leq 2\\ 0, & \text{if }-2 < x < 0. \end{cases} $$
   3. $$ f(x) = x+\pi\,\,\,\text{on }[-\pi, \pi]$$
   4. $$ f(x) = \begin{cases} x^2, & \text{if }0 \leq x < \pi,\\ 0, & \text{if } -\pi \leq x < 0 \end{cases} $$
4. Use the results of Problems 3(c), 3(d) to show convergence of the following series: $$ \frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots $$ $$ \frac{\pi^2}{12} = 1 - \frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\cdots$$