Answered: Find the Fourier series for the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the Fourier series for the triangle wave

f(x) = {π2+x,if−π≤x≤0π2−x,if0<x≤π,

defined on the interval [−π,π].

Expert Solution

Step 1

Given:

$f (x) = \{\begin{cases} π^{2} + x & - π \leq x \leq 0 \\ π^{2} - x & 0 < x \leq π \end{cases}$

We have to find the Fourier series.

Step 2

The Fourier series of a function $f (x)$ in the interval $[- L, L]$ is given by,

$f (x) = a_{0} + \sum_{n = 1}^{\infty} a_{n} \cos (\frac{n πx}{L}) + \sum_{n = 1}^{\infty} b_{n} \sin (\frac{n πx}{L})$
Where,
$a_{0} = \frac{1}{2 L} \int_{- L}^{L} f (x) d x a_{n} = \frac{1}{L} \int_{- L}^{L} f (x) \cos (\frac{n πx}{L}) d x b_{n} = \frac{1}{L} \int_{- L}^{L} f (x) \sin (\frac{n πx}{L}) d x$

Now,
$\begin{array}{rcl} a_{0} & = & \frac{1}{2 π} \int_{- π}^{π} f (x) d x \\ = & \frac{1}{2 π} [\int_{- π}^{0} f (x) d x + \int_{0}^{π} f (x) d x] \\ = & \frac{1}{2 π} [\int_{- π}^{0} (π^{2} + x) d x + \int_{0}^{π} (π^{2} - x) d x] \\ = & \frac{1}{2 π} [{[π^{2} x + \frac{x^{2}}{2}]}_{- π}^{0} + {[π^{2} x - \frac{x^{2}}{2}]}_{0}^{π}] \\ = & \frac{1}{2 π} [0 - (π^{2} (- π) + \frac{π^{2}}{2}) + (π^{2} (π) - \frac{π^{2}}{2}) - 0] \\ = & \frac{1}{2 π} [π^{3} - \frac{π^{2}}{2} + π^{3} - \frac{π^{2}}{2}] \\ = & \frac{1}{2 π} [2 π^{3} - π^{2}] \\ a_{0} & = & π^{2} - \frac{π}{2} \end{array}$

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