Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

Chapter 20 Complex variables - all with Video Answers

Educators

Chapter Questions

03:28

Problem 1

Find an analytic function of $z=x+i y$ whose imaginary part is
$$
(y \cos y+x \sin y) \exp x
$$

Sam Stansfield

Numerade Educator

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Problem 2

Find a function $f(z)$, analytic in a suitable part of the Argand diagram, for which
$$
\operatorname{Re} f=\frac{\sin 2 x}{\cosh 2 y-\cos 2 x}
$$
Where are the singularities of $f(z)$ ?

Eduard Sanchez

Numerade Educator

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Find the radii of convergence of the following Taylor series:
(a) $\sum_{n=2}^{\infty} \frac{z^{n}}{\ln n}$,
(b) $\sum_{n=1}^{\infty} \frac{n ! z^{n}}{n^{n}}$,
(c) $\sum_{n=1}^{\infty} z^{n} n^{\ln n}$,
(d) $\sum_{n=1}^{\infty}\left(\frac{n+p}{n}\right)^{n^{2}} z^{n}$, with $p$ real.

Eduard Sanchez

Numerade Educator

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Problem 4

Find the Taylor series expansion about the origin of the function $f(z)$ defined by
$$
f(z)=\sum_{r=1}^{\infty}(-1)^{r+1} \sin \left(\frac{p z}{r}\right)
$$
where $p$ is a constant. Hence verify that $f(z)$ is a convergent series for all $z$.

Eduard Sanchez

Numerade Educator

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Problem 5

Determine the types of singularities (if any) possessed by the following functions at $z=0$ and $z=\infty$
(a) $(z-2)^{-1}$
(b) $\left(1+z^{3}\right) / z^{2}$
(c) $\sinh (1 / z)$,
(d) $e^{z} / z^{3}$
(e) $z^{1 / 2} /\left(1+z^{2}\right)^{1 / 2}$

Eduard Sanchez

Numerade Educator

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Problem 6

Identify the zeroes, poles and essential singularities of the following functions:
(a) $\tan z$
(b) $\left[(z-2) / z^{2}\right] \sin [1 /(1-z)]$,
(c) $\exp (1 / z)$,
(d) $\tan (1 / z)$,
(e) $z^{2 / 3}$

Eduard Sanchez

Numerade Educator

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Problem 7

Find the real and imaginary parts of the functions (i) $z^{2}$, (ii) $e^{2}$, and (iii) $\cosh \pi z$. By considering the values taken by these parts on the boundaries of the region $0 \leq x, y \leq 1$, determine the solution of Laplace's equation in that region that satisfies the boundary conditions
$$
\begin{array}{ll}
\phi(x, 0)=0, & \phi(0, y)=0 \\
\phi(x, 1)=x, & \phi(1, y)=y+\sin \pi y
\end{array}
$$

Eduard Sanchez

Numerade Educator

01:05

Problem 8

For the function
$$
f(z)=\ln \left(\frac{z+c}{z-c}\right)
$$
where $c$ is real, show that the real part $u$ of $f$ is constant on a circle of radius $c$ cosech $u$ centred on the point $z=c \operatorname{coth} u$. Use this result to show that the electrical capacitance per unit length of two parallel cylinders of radii $a$, placed with their axes $2 d$ apart, is proportional to $\left[\cosh ^{-1}(d / a)\right]^{-1}$.

Hunza Gilgit

Numerade Educator

00:59

Problem 9

Find a complex potential in the $z$-plane appropriate to a physical situation in which the half-plane $x>0, y=0$ has zero potential and the half-plane $x<0$ $y=0$ has potential $V .$

By making the transformation $w=a\left(z+z^{-1}\right) / 2$, with $a$ real and positive, find the electrostatic potential associated with the half-plane $r>a, s=0$ and the half-plane $r<-a, s=0$ at potentials 0 and $V$ respectively.

Raj Bala

Numerade Educator

05:32

Problem 10

By considering in turn the transformations
$$
z=\frac{1}{2} c\left(w+w^{-1}\right), \quad w=\exp \zeta
$$
where $z=x+i y, w=r \exp i \theta, \zeta=\xi+i \eta$ and $c$ is a real positive constant, show that $z=c \cosh \zeta$ maps the strip $\xi \geq 0,0 \leq \eta \leq 2 \pi$, onto the whole $z$-plane. Which curves in the $z$-plane correspond to the lines $\xi=$ constant and $\eta=$ constant? Identify those corresponding respectively to $\xi=0, \eta=0$ and $\eta=2 \pi$

The electric potential $\phi$ of a charged conducting strip $-c \leq x \leq c, y=0$ satisfies
$$
\phi \sim-k \ln \left(x^{2}+y^{2}\right)^{1 / 2} \quad \text { for large }\left(x^{2}+y^{2}\right)^{1 / 2}
$$
with $\phi$ constant on the strip. Show that $\phi=\operatorname{Re}\left[-k \cosh ^{-1}(z / c)\right]$ and that the

Sriram Soundarrajan

Numerade Educator

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Problem 11

Show that the transformation
$$
w=\int_{0}^{z} \frac{1}{\left(\zeta^{3}-\zeta\right)^{1 / 2}} d \zeta
$$
transforms the upper half-plane into the interior of a square that has one corner at the origin of the $w$-plane and sides of length $L$, where
$$
L=\int_{0}^{\pi / 2} \operatorname{cosec}^{1 / 2} \theta d \theta
$$

Eduard Sanchez

Numerade Educator

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Problem 12

The fundamental theorem of algebra states that a complex polynomial $p_{n}(z)$ of degree $n$ has precisely $n$ complex roots. By applying Liouville's theorem (see the end of section $20.12$ ) to $f(z)=1 / p_{n}(z)$ prove that $p_{n}(z)$ has at least one complex root. Factor out that root to obtain $p_{n-1}(z)$ and, by repeating the process, prove the above theorem.

Eduard Sanchez

Numerade Educator

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Problem 13

Show that, if $a$ is a positive real constant, the function $\exp \left(i a z^{2}\right)$ is analytic and $\rightarrow 0$ as $|z| \rightarrow \infty$ for $0<\arg z \leq \pi / 4$. By applying Cauchy's theorem to a suitable contour prove that
$$
\int_{0}^{\infty} \cos \left(a x^{2}\right) d x=\sqrt{\frac{\pi}{8 a}}
$$

Eduard Sanchez

Numerade Educator

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Problem 14

For the equation $8 z^{3}+z+1=0$ :
(a) show that all three roots lie between the circles $|z|=3 / 8$ and $|z|=5 / 8$
(b) find the approximate location of the real root, and hence deduce that the complex ones lie in the first and fourth quadrants and have moduli greater than $0.5$.

Victor Salazar

Numerade Educator

01:01

Problem 15

(a) Prove that $z^{8}+3 z^{3}+7 z+5$ has two zeroes in the first quadrant.
(b) Find in which quadrants the zeroes of $2 z^{3}+7 z^{2}+10 z+6$ lie. Try to locate them.

Raj Bala

Numerade Educator

01:52

Problem 16

The following is a method of determining the number of zeroes of an $n$ th-degree polynomial $f(z)$ inside the contour $C$ given by $|z|=R$ :
(a) put $z=R(1+i t) /(1-i t)$ with $t=\tan (\theta / 2)$ in $-\infty \leq t \leq \infty$;
(b) obtain $f(z)$ as
$$
\frac{A(t)+i B(t)}{(1-i t)^{n}} \frac{(1+i t)^{n}}{(1+i t)^{n}}
$$
(c) show that $\arg f(z)=\tan ^{-1}(B / A)+n \tan ^{-1} t$;
(d) show that $\Delta_{C}[\arg f(z)]=\Delta_{C}\left[\tan ^{-1}(B / A)\right]+n \pi$;
(e) using inspection or a sketch graph, determine $\Delta_{C}\left[\tan ^{-1}(B / A)\right]$ by finding the discontinuities in $B / A$ and evaluating $\tan ^{-1}(B / A)$ at $t=\pm \infty$
Use this method, together with the results of the worked example in section $20.15$, to show that the zeroes of $z^{4}+z+1$ in the second and third quadrants have $|z|<1$

Aamir Mithaiwala

Numerade Educator

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Problem 17

By considering the real part of
$$
\int \frac{-i z^{n-1} d z}{1-a\left(z+z^{-1}\right)+a^{2}}
$$
where $z=\exp i \theta$ and $n$ is a non-negative integer, evaluate
$$
\int_{0}^{\pi} \frac{\cos n \theta}{1-2 a \cos \theta+a^{2}} d \theta
$$
for $a$ real and $>1$

Eduard Sanchez

Numerade Educator

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Problem 18

Prove that if $f(z)$ has a simple pole at $z_{0}$ then $1 / f(z)$ has residue $1 / f^{\prime}\left(z_{0}\right)$ there. Hence evaluate
$$
\int_{-\pi}^{\pi} \frac{\sin \theta}{a-\sin \theta} d \theta
$$
where $a$ is real and $>1$.

Eduard Sanchez

Numerade Educator

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Problem 19

The equation of an ellipse in plane polar coordinates $r, \theta$, with one of its foci at the origin, is
$$
\frac{l}{r}=1-\epsilon \cos \theta
$$
where $l$ is a length (that of the latus rectum) and $\epsilon(0<\epsilon<1)$ is the eccentricity of the ellipse. Express the area of the ellipse as an integral around the unit circle in the complex plane, and show that the only singularity of the integrand inside the circle is a double pole at $z_{0}=\epsilon^{-1}-\left(\epsilon^{-2}-1\right)^{1 / 2}$
By setting $z=z_{0}+\xi$ and expanding the integrand in powers of $\xi$, find the residue at $z_{0}$ and hence show that the area is equal to $\pi l^{2}\left(1-\epsilon^{2}\right)^{-3 / 2}$. (In terms of the semi-axes $a$ and $b$ of the ellipse, $l=b^{2} / a$ and $\left.\epsilon^{2}=\left(a^{2}-b^{2}\right) / a^{2} .\right)$

Eduard Sanchez

Numerade Educator

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Problem 20

Prove that, for $\alpha>0$, the integral
$$
\int_{0}^{\infty} \frac{t \sin \alpha t}{1+t^{2}} d t
$$
has the value $(\pi / 2) \exp (-\alpha)$.

Eduard Sanchez

Numerade Educator

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Problem 21

Prove that
$$
\int_{0}^{\infty} \frac{\cos m x}{4 x^{4}+5 x^{2}+1} d x=\frac{\pi}{6}\left(4 e^{-m / 2}-e^{-m}\right) \quad \text { for } m>0
$$

Eduard Sanchez

Numerade Educator

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Problem 22

Show that the principal value of the integral
$$
\int_{-\infty}^{\infty} \frac{\cos (x / a)}{x^{2}-a^{2}} d x
$$
is $-(\pi / a) \sin 1 .$

Eduard Sanchez

Numerade Educator

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Problem 23

(a) Prove that the integral of $\left[\exp \left(i \pi z^{2}\right)\right] \operatorname{cosec} \pi z$ around the parallelogram with corners $\pm 1 / 2 \pm R \exp (i \pi / 4)$ has the value $2 i$
(b) Show that the parts of the contour parallel to the real axis give no contribution when $R \rightarrow \infty$.
(c) Evaluate the integrals along the other two sides by putting $z^{\prime}=r \exp (i \pi / 4)$ and working in terms of $z^{\prime}+\frac{1}{2}$ and $z^{\prime}-\frac{1}{2} .$ Hence by letting $R \rightarrow \infty$ show that
$$
\int_{-\infty}^{\infty} e^{-\pi r^{2}} d r=1
$$

Eduard Sanchez

Numerade Educator

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Problem 24

By applying the residue theorem around a wedge-shaped contour of angle $2 \pi / n$, with one side along the real axis, prove that the integral
$$
\int_{0}^{\infty} \frac{d x}{1+x^{n}}
$$
where $n$ is real and $\geq 2$, has the value $(\pi / n) \operatorname{cosec}(\pi / n)$.

Eduard Sanchez

Numerade Educator

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Problem 25

Using a suitable cut plane, prove that if $\alpha$ is real and $0<\alpha<1$ then
$$
\int_{0}^{\infty} \frac{x^{-\alpha}}{1+x} d x
$$
has the value $\pi \operatorname{cosec} \pi \alpha$.

Eduard Sanchez

Numerade Educator

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Problem 26

Show that
$$
\int_{0}^{\infty} \frac{\ln x}{x^{3 / 4}(1+x)} d x=-\sqrt{2} \pi^{2}
$$

Eduard Sanchez

Numerade Educator

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Problem 27

By integrating a suitable function around a large semicircle in the upper half plane and a small semicircle centred on the origin, determine the value of
$$
I=\int_{0}^{\infty} \frac{(\ln x)^{2}}{1+x^{2}} d x
$$
and deduce, as a by-product of your calculation, that
$$
\int_{0}^{\infty} \frac{\ln x}{1+x^{2}} d x=0
$$

Eduard Sanchez

Numerade Educator

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Problem 28

Prove that
$$
\sum_{-\infty}^{\infty} \frac{1}{n^{2}+\frac{3}{4} n+\frac{1}{8}}=4 \pi
$$
Carry out the summation numerically, say between $-4$ and 4 , and note how much of the sum comes from values near the poles of the contour integration.

Suzanne W.

Numerade Educator

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Problem 29

(a) Determine the residues at all the poles of the function
$$
f(z)=\frac{\pi \cot \pi z}{a^{2}+z^{2}}
$$
where $a$ is a positive real constant.
(b) By evaluating, in two different ways, the integral $I$ of $f(z)$ along the straight line joining $-\infty-i a / 2$ and $+\infty-i a / 2$, show that
$$
\sum_{n=1}^{\infty} \frac{1}{a^{2}+n^{2}}=\frac{\pi \operatorname{coth} \pi a}{2 a}-\frac{1}{2 a^{2}}
$$
(c) Deduce the value of $\sum_{1}^{\infty} n^{-2}$.

Victor Salazar

Numerade Educator

03:08

Problem 30

By considering the integral of
$$
\left(\frac{\sin \alpha z}{\alpha z}\right)^{2} \frac{\pi}{\sin \pi z}, \quad \alpha<\frac{\pi}{2}
$$
around a circle of large radius, prove that
$$
\sum_{m=1}^{\infty}(-1)^{m-1} \frac{\sin ^{2} m \alpha}{(m \alpha)^{2}}=\frac{1}{2}
$$

Gaurav Kalra

Numerade Educator

11:01

Problem 31

Use the Bromwich inversion, and contours such as those shown in figure $20.23(a)$, to find the functions of which the following are the Laplace transforms:
(a) $s\left(s^{2}+b^{2}\right)^{-1}$
(b) $n !(s-a)^{-(n+1)}$, with $n$ a positive integer and $s>a ;$
(c) $a\left(s^{2}-a^{2}\right)^{-1}$, with $s>|a|$. (Change variable to $\left.t=s-|a| .\right)$
Compare your answers with those given in table $13.1 .$

Aidan Mcnabb

Numerade Educator

02:49

Problem 32

Find the function $f(t)$ whose Laplace transform is
$$
\bar{f}(s)=\frac{e^{-s}-1+s}{s^{2}}
$$

Ryan Williams

Numerade Educator

04:24

Problem 33

A function $f(t)$ has the Laplace transform
$$
F(s)=\frac{1}{2 i} \ln \left(\frac{s+i}{s-i}\right)
$$
the complex logarithm being defined by a finite branch cut running along the imaginary axis from $-i$ to $i$.
(a) Convince yourself that, for $t>0, f(t)$ can be expressed as a closed contour integral that encloses only the branch cut.
(b) Calculate $F(s)$ on either side of the branch cut, evaluate the integral and hence determine $f(t)$
(c) Confirm that the derivative with respect to $s$ of the Laplace transform integral of your answer is the same as that given by $d F / d s$.

Israel Hernandez

Numerade Educator

04:09

Problem 34

Use the contour in figure $20.23(c)$ to show that the function with Laplace transform $s^{-1 / 2}$ is $(\pi x)^{-1 / 2}$. (For an integrand of the form $r^{-1 / 2} \exp (-r x)$ change variable to $t=r^{1 / 2}$.)

Uma Kumari

Numerade Educator