MT201 - Engineering Mathematics 2
Fourier Series Tutorial 2

Due date: 8 September, 2022

1. Let \(f(x)=x\) be a periodic function with period \(2\pi\).
   1. Plot the graph of \(f(x)\) on the interval \(-3\pi\leq x\leq 3\pi\).
   2. Find the Fourier series of \(f(x)\) on the interval \(-\pi\leq x\leq \pi\)
   3. State Parseval's identity for the Fourier coefficients derived in Part (b).
   4. Use Parseval's identity stated in Part (c) to show that
      
      \[ \frac{\pi^2}{6} = \displaystyle\sum_{n=1}^\infty\frac{1}{n^2} \]
   5. Suppose that \(f(x)\) is a periodic function on \([-L,L]\) satisfying
      $$\int_{-L}^{L} (f(x))^2\,dx < \infty.$$
   6. Prove that for every \(n=0,1,2,3,\cdots\), we have
      $$ \frac{a^2_0}{2} + \sum_{m=1}^n\left(a_m^2 + b_m^2\right)\leq \frac{1}{L}\int_{-L}^{L}(f(x))^2\,dx$$
      Hint: Consider the partial sum
      $$ S_n(x)=\frac{a_0}{2}+\sum_{m=1}^n\left[a_m\cos\left(\frac{m\pi}{L}x\right) + b_m\sin\left(\frac{m\pi}{L}x\right)\right]$$
      and expand the error term as
      $$\int_{-L}^{L}\left(f(x)-S_n(x)\right)^2\,dx $$
2. 1. Compute the Fourier Transform of the function
      $$f(x)=\begin{cases} 1, & \text{if } |x|\leq 1\\ 0, & \text{if } |x| > 1. \end{cases} $$
   2. Use the method of Fourier Transforms to solve the heat equation below, leaving your solution without the imaginary unit \(i\).
      \[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2},\,\,\,-\infty < x < \infty,\,\,t> 0\]
      \[ u(x,0) = f(x) = \begin{cases}1, & \text{if } |x|\leq 1\\ 0, & \text{if } |x| > 1. \end{cases}\]
   3. Compute the Fourier sine and cosine transforms of the functions below.
      1. \[f(x) = e^{-ax},\,\,\,a > 0\]
      2. \[ f(x) = \begin{cases}1, & \text{if } |x|\leq 1\\ 0, & \text{if } |x| > 1. \end{cases}\]
   4. Use a Fourier sine transform to solve the heat equation
      \[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2},\,\,\,0 < x < \infty,\,\,t> 0\]
      \[ u(x,0) = f(x) = e^{-x},\]
      \[u(0,t)=0. \]