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Exercises 5.7 Exercise
1.
Expand the following functions in a cosine series and a sine series:
\(f(x) = x\) when \(0 \lt x \lt \pi\)
\(f(x) = \pi-x\) when \(0 \lt x \lt \pi\text{.}\)
Answer.
\begin{equation*}
\frac{\pi}{2}-\frac{4}{\pi}+\left(\cos x+\frac{\cos 3x}{3^{2}}+ \frac{\cos 5x}{5^{2}}+\cdots\right),
\end{equation*}
\begin{equation*}
2\left(\sin x-\frac{\sin 2x}{2}+ \frac{\sin 3x}{3}-\cdots\right).
\end{equation*}
\begin{equation*}
2\left(\sin x-\frac{\sin 2x}{2}+ \frac{\sin 3x}{3}-\cdots\right),
\end{equation*}
\begin{equation*}
\frac{\pi}{2}-\frac{4}{\pi}+\left(\cos x+\frac{\cos 3x}{3^{2}}+ \frac{\cos 5x}{5^{2}}+\cdots\right)
\end{equation*}
2.
Expand the following function \(f(x) x\sin x\) in the Fourier series in the interval \(-\pi \lt x \lt \pi\text{.}\)
Answer.
\begin{equation*}
1-2\left(-\frac{\cos x}{4}+ \frac{\cos 2x}{1\cdot 3}-\frac{\cos 3x}{2\cdot 4}+\frac{\cos 4x}{3\cdot 5}-\cdots\right).
\end{equation*}
3.
Find the Fourier series representation of the half wave rectifier given by the following function:
\begin{equation*}
f(x) = \begin{cases}
\sin \omega t, & \text{for} \quad 0 \leq \omega t \leq \pi\\
0, & \text{for} \quad -\pi \leq \omega t \leq 0
\end{cases}
\end{equation*}
Hint.
\begin{equation*}
f(t) = \begin{cases}
0, & \text{from} \quad -T/2 \leq t \leq 0\\
\sin \omega t, & \text{from} \quad 0 \leq t \leq T/2
\end{cases}
\end{equation*}
Answer.
hence,
\begin{equation*}
f(t) = \frac{1}{\pi}+\frac{\sin\omega t}{2}-\frac{2}{\pi}\left[\frac{\cos 2\omega t}{1\cdot 3}\right.
\end{equation*}
\begin{equation*}
\left. +\frac{\cos 4\omega t}{3\cdot 5}+\frac{\cos 6\omega t}{5\cdot 7}+\cdots \right]
\end{equation*}
4.
Find the Fourier series representation of a full wave rectifier given by the following function:
\begin{equation*}
f(t) = \begin{cases}
i_{o}\sin\omega t, & \text{from} \quad 0 \leq t \leq T/2\\
-i_{o}\sin\omega t, & \text{from} \quad T/2 \leq t \leq T
\end{cases}
\end{equation*}
Answer.
\begin{equation*}
f(t) = i-\frac{2 i_{o}}{\pi}-\frac{4i_{o}}{3\pi}\cos 2\omega t
\end{equation*}
\begin{equation*}
- \frac{4i_{o}}{15\pi}\cos 4\omega t -\frac{4i_{o}}{35\pi}\cos 6\omega t - \cdots
\end{equation*}
5.
Develop the Fourier expansion for a triangular wave represented by the following function:
\begin{equation*}
f(x) = \begin{cases}
-x, & \text{from} \quad -\pi \lt x \lt 0\\
x, & \text{from} \quad 0 \lt x \lt \pi
\end{cases}
\end{equation*}
Answer.
\begin{equation*}
f(x) = \frac{\pi}{2}-\frac{4}{\pi}\sum\limits_{n_{odd}=1}^{\infty}\frac{1}{n^{2}}\cos nx.
\end{equation*}
6.
Determine the Fourier series expansion of the function \(f(x) = |x|\text{,}\) for \(-\pi \leq x \leq \pi\text{.}\) Hence deduce that
\begin{equation*}
\frac{\pi^{2}}{8}=1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\frac{1}{7^{2}}+\cdots.
\end{equation*}
Python Code 1.
from sympy import fourier_series, pi, plot from sympy.abc import x f = x s = fourier_series(f, (x, 0, pi)) s1 = s.truncate(n = 5) s2 = s.truncate(n = 7) s3 = s.truncate(n = 9) p = plot(f, s1, s2, s3, (x, 0, pi), show=False, legend=True) s
p.show()
