Chapter 14, Fourier Series Video Solutions, Mathematical Methods for Physicists

Problem 1

A function $f(x)$ (quadratically integrable) is to be represented by a finite Fourier series. A convenient measure of the accuracy of the series is given by the integrated square of the deviation,
$$
\Delta_p=\int_0^{2 \pi}\left[f(x)-\frac{a_0}{2}-\sum_{n=1}^p\left(a_n \cos n x+b_n \sin n x\right)\right]^2 d x .
$$

Show that the requirement that $\Delta_p$ be minimized, that is,
$$
\frac{\partial \Delta_p}{\partial a_n}=0, \quad \frac{\partial \Delta_p}{\partial b_n}=0,
$$
for all $n$, leads to choosing $a_n$ and $b_n$ as given in Eqs. (14.2) and (14.3).
Note. Your coefficients $a_n$ and $b_n$ are independent of $p$. This independence is a consequence of orthogonality and would not hold for powers of $x$, fitting a curve with polynomials.

James Kiss

Numerade Educator

05:07

Problem 2

In the analysis of a complex waveform (ocean tides, earthquakes, musical tones, etc.) it might be more convenient to have the Fourier series written as
$$
f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty} \alpha_n \cos \left(n x-\theta_n\right)
$$
Show that this is equivalent to Eq. (14.1) with
$$
\begin{array}{ll}
a_n=\alpha_n \cos \theta_n, & \alpha_n^2=a_n^2+b_n^2, \\
b_n=\alpha_n \sin \theta_n, & \tan \theta_n=b_n / a_n .
\end{array}
$$

Note. The coefficients $\alpha_n^2$ as a function of $n$ define what is called the power spectrum. The importance of $\alpha_n^2$ lies in their invariance under a shift in the phase $\theta_n$.

James Kiss

Numerade Educator

01:58

Problem 3

A function $f(x)$ is expanded in an exponential Fourier series
$$
f(x)=\sum_{n=-\infty}^{\infty} c_n e^{i n x} .
$$

If $f(x)$ is real, $f(x)=f^*(x)$, what restriction is imposed on the coefficients $c_n$ ?

Sarah Gift

Numerade Educator

Problem 4

Assuming that $\int_{-\pi}^\pi[f(x)]^2 d x$ is finite, show that
$$
\lim _{m \rightarrow \infty} a_m=0, \quad \lim _{m \rightarrow \infty} b_m=0 .
$$

Check back soon!

01:54

Problem 5

Apply the summation technique of this section to show that
$$
\sum_{n=1}^{\infty} \frac{\sin n x}{n}= \begin{cases}\frac{1}{2}(\pi-x), & 0<x \leq \pi \\ -\frac{1}{2}(\pi+x), & -\pi \leq x<0\end{cases}
$$
(Fig. 14.2).

Aymara Gallardo

Numerade Educator

02:11

Problem 6

Sum the trigonometric series
$$
\sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin n x}{n}
$$
and show that it equals $x / 2$.

Darshan Maheshwari

Numerade Educator

Problem 7

Sum the trigonometric series
$$
\sum_{n=0}^{\infty} \frac{\sin (2 n+1) x}{2 n+1}
$$
and show that it equals
$$
\begin{cases}\pi / 4, & 0<x<\pi \\ -\pi / 4, & -\pi<x<0 .\end{cases}
$$

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01:06

Problem 8

Calculate the sum of the finite Fourier sine series for the sawtooth wave, $f(x)=$ $x,(-\pi, \pi)$, Eq. (14.21). Use 4-, 6-, 8-, and 10-term series and $x / \pi=0.00(0.02) 1.00$. If a plotting routine is available, plot your results and compare with Fig. 14.1.

Chai Santi

Numerade Educator

Problem 9

Let $f(z)=\ln (1+z)=\sum_{n=1}^{\infty}(-1)^{n+1} z^n / n$. (This series converges to $\ln (1+z)$ for $|z| \leq 1$, except at the point $z=-1$.)
(a) From the real parts show that
$$
\ln \left(2 \cos \frac{\theta}{2}\right)=\sum_{n=1}^{\infty}(-1)^{n+1} \frac{\cos n \theta}{n}, \quad-\pi<\theta<\pi .
$$
(b) Using a change of variable, transform part (a) into
$$
-\ln \left(2 \sin \frac{\theta}{2}\right)=\sum_{n=1}^{\infty} \frac{\cos n \theta}{n}, \quad 0<\theta<2 \pi .
$$

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