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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 12

Fourier series - all with Video Answers

Educators

Chapter Questions

00:59

Problem 1

Prove the orthogonality relations stated in section $12.1$.

Sriram Soundarrajan
Sriram Soundarrajan
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Problem 2

Derive the Fourier coefficients $b_{r}$ in a similar manner to the derivation of the $a_{r}$ in section $12.2$.

Eduard Sanchez
Eduard Sanchez
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01:30

Problem 3

Which of the following functions of $x$ could be represented by a Fourier series over the range indicated?
(a) $\tanh ^{-1}(x), \quad-\infty<x<\infty$
$-\infty<x<\infty$
(c) $|\sin x|^{-1 / 2}, \quad-\infty<x<\infty$
(d) $\cos ^{-1}(\sin 2 x), \quad-\infty<x<\infty$
(e) $x \sin (1 / x) \quad-\pi^{-1}<x \leq \pi^{-1}$, cyclically repeated.

Narayan Hari
Narayan Hari
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01:01

Problem 4

By moving the origin of $t$ to the centre of an interval in which $f(t)=+1$, i.e. by changing to a new independent variable $t^{\prime}=t-\frac{1}{4} T$, express the square-wave function in the example in section $12.2$ as a cosine series. Calculate the Fourier coefficients involved (a) directly and (b) by changing the variable in result (12.8).

Amit Srivastava
Amit Srivastava
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Problem 5

Find the Fourier series of the function $f(x)=x$ in the range $-\pi<x \leq \pi$. Hence show that
$$
1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{4}
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:26

Problem 6

For the function
$$
f(x)=1-x, \quad 0 \leq x \leq 1
$$
find (a) the Fourier sine series and (b) the Fourier cosine series. Which would be better for numerical evaluation? Relate your answer to the relevant periodic continuations.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:00

Problem 7

For the continued functions used in exercise $12.6$ and the derived corresponding series, consider (i) their derivatives and (ii) their integrals. Do they give meaningful equations? You will probably find it helpful to sketch all the functions involved.

Raj Bala
Raj Bala
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02:19

Problem 8

The function $y(x)=x \sin x$ for $0 \leq x \leq \pi$ is to be represented by a Fourier series of period $2 \pi$ that is either even or odd. By sketching the function and considering its derivative, determine which series will have the more rapid convergence. Find the full expression for the better of these two series, showing that the convergence, $\sim n^{-3}$ and that alternate terms are missing.

James Kiss
James Kiss
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Problem 9

Find the Fourier coefficients in the expansion of $f(x)=\exp x$ over the range $-1<x<1$. What value will the expansion have when $x=2$ ?

Eduard Sanchez
Eduard Sanchez
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06:55

Problem 10

By integrating term by term the Fourier series found in the previous question and using the Fourier series for $f(x)=x$ found in section $12.6$, show that $\int \exp x d x=\exp x+c .$ Why is it not possible to show that $d(\exp x) / d x=\exp x$ by differentiating the Fourier series of $f(x)=\exp x$ in a similar manner?

JH
J Hardin
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Problem 11

Consider the function $f(x)=\exp \left(-x^{2}\right)$ in the range $0 \leq x \leq 1 .$ Show how it should be continued to give as its Fourier series a series (the actual form is not wanted) (a) with only cosine terms, (b) with only sine terms, (c) with period 1 and $(\mathrm{d})$ with period $2 .$

Would there be any difference between the values of the last two series at (i) $x=0$, (ii) $x=1 ?$

Eduard Sanchez
Eduard Sanchez
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Problem 12

Find, without calculation, which terms will be present in the Fourier series for the periodic functions $f(t)$, of period $T$, that are given in the range $-T / 2$ to $T / 2$ by:
(a) $f(t)=2$ for $0 \leq|t|<T / 4, f=1$ for $T / 4 \leq|t|<T / 2$
(b) $f(t)=\exp \left[-(t-T / 4)^{2}\right]$
(c) $f(t)=-1$ for $-T / 2 \leq t<-3 T / 8$ and $3 T / 8 \leq t<T / 2, f(t)=1$ for $-T / 8 \leq t<-T / 8 ;$ the graph of $f$ is completed by two straight lines in the remaining ranges so as to form a continuous function.

Eduard Sanchez
Eduard Sanchez
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00:47

Problem 13

Consider the representation as a Fourier series of the displacement of a string lying in the interval $0 \leq x \leq L$ and fixed at its ends, when it is pulled aside by $y_{0}$ at the point $x=L / 4$. Sketch the continuations for the region outside the interval that will
(a) produce a series of period $L$,
(b) produce a series that is antisymmetric about $x=0$, and
(c) produce a series that will contain only cosine terms.
(d) What are (i) the periods of the series in (b) and (c) and (ii) the value of the ${ }^{4} a_{0}$-term' in (c)?
(e) Show that a typical term of the series obtained in (b) is
$$
\frac{32 y_{0}}{3 n^{2} \pi^{2}} \sin \frac{n \pi}{4} \sin \frac{n \pi x}{L}
$$

Manik Pulyani
Manik Pulyani
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Problem 14

Show that the Fourier series for the function $y(x)=|x|$ in the range $-\pi \leq x<\pi$ is
$$
y(x)=\frac{\pi}{2}-\frac{4}{\pi} \sum_{m=0}^{\infty} \frac{\cos (2 m+1) x}{(2 m+1)^{2}}
$$
By integrating this equation term by term from 0 to $x$, find the function $g(x)$ whose Fourier series is
$$
\frac{4}{\pi} \sum_{m=0}^{\infty} \frac{\sin (2 m+1) x}{(2 m+1)^{3}}
$$
Deduce the value of the sum $S$ of the series
$$
1-\frac{1}{3^{3}}+\frac{1}{5^{3}}-\frac{1}{7^{3}}+\cdots
$$

Eduard Sanchez
Eduard Sanchez
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01:58

Problem 15

Using the result of exercise $12.14$, determine, as far as possible by inspection, the form of the functions of which the following are the Fourier series:
(a)
$$
\cos \theta+\frac{1}{9} \cos 3 \theta+\frac{1}{25} \cos 5 \theta+\cdots
$$
(b)
$$
\sin \theta+\frac{1}{27} \sin 3 \theta+\frac{1}{125} \sin 5 \theta+\cdots
$$
(c)
$$
\frac{L^{2}}{3}-\frac{4 L^{2}}{\pi^{2}}\left[\cos \frac{\pi x}{L}-\frac{1}{4} \cos \frac{2 \pi x}{L}+\frac{1}{9} \cos \frac{3 \pi x}{L}-\cdots\right]
$$
(You may find it helpful to first set $x=0$ in the quoted result and so obtain values for $S_{0}=\sum(2 m+1)^{-2}$ and other sums derivable from it.)

Kajal Gautam
Kajal Gautam
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Problem 16

By finding a cosine Fourier series of period 2 for the function $f(t)$ that takes the form $f(t)=\cosh (t-1)$ in the range $0 \leq t \leq 1$, prove that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{2} \pi^{2}+1}=\frac{1}{e^{2}-1}
$$

Eduard Sanchez
Eduard Sanchez
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Problem 17

Find the (real) Fourier series of period 2 for $f(x)=\cosh x$ and $g(x)=x^{2}$ in the range $-1 \leq x \leq 1 .$ By integrating the series for $f(x)$ twice, prove that
$$
\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2} \pi^{2}\left(n^{2} \pi^{2}+1\right)}=\frac{1}{2}\left(\frac{1}{\sinh 1}-\frac{5}{6}\right)
$$

Eduard Sanchez
Eduard Sanchez
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Problem 18

Express the function $f(x)=x^{2}$ as a Fourier sine series in the range $0<x \leq 2$ and show that it converges to zero at $x=\pm 2$

Eduard Sanchez
Eduard Sanchez
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Problem 19

Demonstrate explicitly for the square-wave function discussed in section $12.2$ that Parseval's theorem (12.13) is valid. You will need to use the relationship
$$
\sum_{m=0}^{\infty} \frac{1}{(2 m+1)^{2}}=\frac{\pi^{2}}{8}
$$
Show that a filter that transmits frequencies only up to $8 \pi / T$ will still transmit

Eduard Sanchez
Eduard Sanchez
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01:31

Problem 20

Show that the Fourier series for $|\sin \theta|$ in the range $-\pi \leq \theta \leq \pi$ is given by
$$
|\sin \theta|=\frac{2}{\pi}-\frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos 2 m \theta}{4 m^{2}-1}
$$
By setting $\theta=0$ and $\theta=\pi / 2$, deduce values for
$$
\sum_{m=1}^{\infty} \frac{1}{4 m^{2}-1} \quad \text { and } \sum_{m=1}^{\infty} \frac{1}{16 m^{2}-1}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
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Problem 21

Find the complex Fourier series for the periodic function of period $2 \pi$ defined in the range $-\pi \leq x \leq \pi$ by $y(x)=\cosh x$. By setting $t=0$ prove that
$$
\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}+1}=\frac{1}{2}\left(\frac{\pi}{\sinh \pi}-1\right)
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:33

Problem 22

The repeating output from an electronic oscillator takes the form of a sine wave $f(t)=\sin t$ for $0 \leq t \leq \pi / 2 ;$ it then drops instantaneously to zero and starts again. The output is to be represented by a complex Fourier series of the form
$$
\sum_{n=-\infty}^{\infty} c_{n} e^{4 n t i}
$$
Sketch the function and find an expression for $c_{n}$. Verify that $c_{-n}=c_{n}^{*} .$ Demonstrate that setting $t=0$ and $t=\pi / 2$ produces differing values for the sum
$$
\sum_{n=1}^{\infty} \frac{1}{16 n^{2}-1}
$$
Determine the correct value and check it using the quoted result of exercise $12.5 .$

Amit Srivastava
Amit Srivastava
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Problem 23

Apply Parseval's theorem to the series found in the previous exercise and so derive a value for the sum of the series
$$
\frac{17}{(15)^{2}}+\frac{65}{(63)^{2}}+\frac{145}{(143)^{2}}+\cdots+\frac{16 n^{2}+1}{\left(16 n^{2}-1\right)^{2}}+\cdots
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:03

Problem 24

A string, anchored at $x=\pm L / 2$, has a fundamental vibration frequency of $2 L / c$, where $c$ is the speed of transverse waves on the string. It is pulled aside at its centre point by a distance $y_{0}$ and released at time $t=0 .$ Its subsequent motion can be described by the series
$$
y(x, t)=\sum_{n=1}^{\infty} a_{n} \cos \frac{n \pi x}{L} \cos \frac{n \pi c t}{L}
$$
Find a general expression for $a_{n}$ and show that only odd harmonics of the fundamental frequency are present in the sound generated by the released string. By applying Parseval's theorem, find the sum $S$ of the series $\sum_{0}^{\infty}(2 m+1)^{-4}$.

Raj Bala
Raj Bala
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Problem 25

Show that Parseval's theorem for two functions whose Fourier expansions have. cosine and sine coefficients $a_{n}, b_{n}$ and $\alpha_{n}, \beta_{n}$ takes the form
$$
\frac{1}{L} \int_{0}^{L} f(x) g^{*}(x) d x=\frac{1}{4} a_{0} \alpha_{0}+\frac{1}{2} \sum_{n=1}^{\infty}\left(a_{n} \alpha_{n}+b_{n} \beta_{n}\right)
$$
(a) Demonstrate that for $g(x)=\sin m x$ or $\cos m x$ this reduces to the definition of the Fourier coefficients.
(b) Explicitly verify the above result for the case in which $f(x)=x$ and $g(x)$ is the square-wave function, both in the interval $-1 \leq x \leq 1$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:02

Problem 26

An odd function $f(x)$ of period $2 \pi$ is to be approximated by a Fourier sine series having only $m$ terms. The error in this approximation is measured by the square deviation
$$
E_{m}=\int_{-\pi}^{\pi}\left[f(x)-\sum_{n=1}^{m} b_{n} \sin n x\right]^{2} d x
$$
By differentiating $E_{m}$ with respect to the coefficients $b_{n}$, find the values of $b_{n}$ that minimise $E_{m-}$ Sketch the graph of the function $f(x)$, where
$$
f(x)=\left\{\begin{array}{cl}
-x(\pi+x) & \text { for }-\pi \leq x<0 \\
x(x-\pi) & \text { for } 0 \leq x<\pi
\end{array}\right.
$$
$f(x)$ is to be approximated by the first three terms of a Fourier sine series. What coefficients minimise $E_{3} ?$ What is the resulting value of $E_{3} ?$

Raj Bala
Raj Bala
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