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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

Chapter 3

Complex numbers and hyperbolic functions - all with Video Answers

Educators

Chapter Questions

02:31

Problem 1

Two complex numbers $z$ and $w$ are given by $z=3+4 i$ and $w=2-i .$ On an Argand diagram plot
(a) $z+w$, (b) $w-z$, (c) $w z$, (d) $z / w$,
(e) $z^{*} w+w^{2} z,\left(\right.$ f) $w^{2},(\mathrm{~g}) \ln z$, (h) $(1+z+w)^{1 / 2}$.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:41

Problem 2

By considering the real and imaginary parts of the product $e^{i \theta} e^{i \phi}$ prove the standard formulae for $\cos (\theta+\phi)$ and $\sin (\theta+\phi)$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
09:09

Problem 3

By writing $\pi / 12=(\pi / 3)-(\pi / 4)$ and considering $e^{i \pi / 12}$, evaluate $\cot (\pi / 12)$.

Timothy James
Timothy James
Numerade Educator
08:37

Problem 4

Find the locus in the complex $z$-plane of points that satisfy the following equations.
(a) $z-c=\rho\left(\frac{1+i t}{1-i t}\right)$, where $c$ is complex, $\rho$ is real and $t$ is a real parameter that varies in the range $-\infty<t<\infty$
(b) $z=a+b t+c t^{2}$, in which $t$ is a real parameter and $a, b$, and $c$ are complex numbers with $b / c$ real.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
15:29

Problem 5

Evaluate
(a) $\operatorname{Re}(\exp 2 i z)$, (b) $\operatorname{Im}\left(\cosh ^{2} z\right)$, (c) $(-1+\sqrt{3} i)^{1 / 2}$
(d) $\left|\exp \left(i^{1 / 2}\right)\right|$, (e) $\exp \left(i^{3}\right)$, (f) $\operatorname{Im}\left(2^{i+3}\right)$, (g) $i^{i}$, (h) $\ln \left[(\sqrt{3}+i)^{3}\right]$.

Timothy James
Timothy James
Numerade Educator
09:47

Problem 6

Find the equations in terms of $x$ and $y$ of the sets of points in the Argand diagram that satisfy the following:
(a) $\operatorname{Re} z^{2}=\operatorname{Im} z^{2}$;
(b) $\left(\operatorname{Im} z^{2}\right) / z^{2}=-i$
(c) $\arg [z /(z-1)]=\pi / 2$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:09

Problem 7

Show that the locus of all points $z=x+i y$ in the complex plane that satisfy
$$
|z-i a|=\lambda|z+i a|, \quad \lambda>0
$$
is a circle of radius $\left|2 \lambda /\left(1-\lambda^{2}\right)\right| a$ centred on the point $z=i a\left[\left(1+\lambda^{2}\right) /\left(1-\lambda^{2}\right)\right]$. Sketch the circles for a few typical values of $\lambda$, including $\lambda<1, \lambda>1$ and $\lambda=1$

Carson Merrill
Carson Merrill
Numerade Educator
03:52

Problem 8

The two sets of points $z=a, z=b, z=c$, and $z=A, z=B, z=C$ are the corners of two similar triangles in the Argand diagram. Express in terms of $a, b, \ldots, C$
(a) the equalities of corresponding angles, and
(b) the constant ratio of corresponding sides,
in the two triangles.
By noting that any complex quantity can be expressed as
$$
z=|z| \exp (i \arg z)
$$
deduce that
$$
a(B-C)+b(C-A)+c(A-B)=0
$$

Uma Kumari
Uma Kumari
Numerade Educator
08:37

Problem 9

For the real constant $a$ find the loci of all points $z=x+i y$ in the complex plane that satisfy
(a) $\operatorname{Re}\left\{\ln \left(\frac{z-i a}{z+i a}\right)\right\}=c, \quad c>0$,
(b) $\operatorname{Im}\left\{\ln \left(\frac{z-i a}{z+i a}\right)\right\}=k, \quad 0 \leq k \leq \pi / 2$.
Identify the two families of curves and verify that in case (b) all curves pass through the two points $\pm i a$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:16

Problem 10

The most general type of transformation between one Argand diagram, in the $z$-plane, and another, in the $Z$-plane, that gives one and only one value of $Z$ for each value of $z$ (and conversely) is known as the general bilinear transformation and takes the form
$$
z=\frac{a Z+b}{c Z+d}
$$
(a) Confirm that the transformation from the $Z$-plane to the $z$-plane is also a general bilinear transformation.
(b) Recalling that the equation of a circle can be written in the form
$$
\left|\frac{z-z_{1}}{z-z_{2}}\right|=\lambda, \quad \lambda \neq 1
$$
show that the general bilinear transformation transforms circles into circles (or straight lines). What is the condition that $z_{1}, z_{2}$ and $\lambda$ must satisfy if the transformed circle is to be a straight line?

Raj Bala
Raj Bala
Numerade Educator
03:45

Problem 11

Sketch the parts of the Argand diagram in which
(a) $\operatorname{Re} z^{2}<0,\left|z^{1 / 2}\right| \leq 2$,
(b) $0 \leq \arg z^{*} \leq \pi / 2$
(c) $\left|\exp z^{3}\right| \rightarrow 0$ as $|z| \rightarrow \infty$.
What is the area of the region in which all three conditions are satisfied?

Carlos Pinilla
Carlos Pinilla
Numerade Educator
06:08

Problem 12

Denote the $n$th roots of unity by $1, \omega_{n}, \omega_{n}^{2}, \ldots, \omega_{n}^{n-1}$.
(a) Prove that
(i) $\sum_{r=0}^{n-1} \omega_{n}^{r}=0$,
(ii) $\prod_{r=0}^{n-1} \omega_{n}^{r}=(-1)^{n+1}$.
(b) Express $x^{2}+y^{2}+z^{2}-y z-z x-x y$ as the product of two factors, each linear in $x, y$ and $z$, with coefficients dependent on the third roots of unity (and those of the $x$ terms arbitrarily taken as real).

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator
01:37

Problem 13

Prove that $x^{2 m+1}-a^{2 m+1}$, where $m$ is an integer $\geq 1$, can be written as
$$
x^{2 m+1}-a^{2 m+1}=(x-a) \prod_{r=1}^{m}\left[x^{2}-2 a x \cos \left(\frac{2 \pi r}{2 m+1}\right)+a^{2}\right]
$$

Adriano Chikande
Adriano Chikande
Numerade Educator
01:15

Problem 14

The complex position vectors of two parallel interacting equal fluid vortices moving with their axes of rotation always perpendicular to the $z$-plane are $z_{1}$ and $z_{2}$. The equations governing their motions are
$$
\frac{d z_{1}^{*}}{d t}=-\frac{i}{z_{1}-z_{2}}, \quad \frac{d z_{2}^{*}}{d t}=-\frac{i}{z_{2}-z_{1}}
$$
Deduce that (a) $z_{1}+z_{2}$, (b) $\left|z_{1}-z_{2}\right|$ and (c) $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}$ are all constant in time, and hence describe the motion geometrically.

Raj Bala
Raj Bala
Numerade Educator
03:26

Problem 15

Solve the equation
$$
z^{7}-4 z^{6}+6 z^{5}-6 z^{4}+6 z^{3}-12 z^{2}+8 z+4=0
$$
(a) by examining the effect of setting $z^{3}$ equal to 2 , and
(b) by factorising and using the binomial expansion of $(z+a)^{4}$.
Plot the seven roots of the equation on an Argand plot, exemplifying that complex roots of a polynomial equation always occur in conjugate pairs if the polynomial has real coefficients.

Chris Trentman
Chris Trentman
Numerade Educator
08:44

Problem 16

The polynomial $f(z)$ is defined by
$$
f(z)=z^{5}-6 z^{4}+15 z^{3}-34 z^{2}+36 z-48
$$
(a) Show that the equation $f(z)=0$ has roots of the form $z=\lambda i$ where $\lambda$ is real, and hence factorize $f(z)$
(b) Show further that the cubic factor of $f(z)$ can be written in the form $(z+a)^{3}+b$, where $a$ and $b$ are real, and hence solve the equation $f(z)=0$ completely.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:30

Problem 17

The binomial expansion of $(1+x)^{n}$, discussed in chapter 1 , can be written for a positive integer $n$ as
$$
(1+x)^{n}=\sum_{r=0}^{n}{ }^{n} C_{r} x^{r}
$$
where ${ }^{n} C_{r}=n ! /[r !(n-r) !]$.
(a) Use de Moivre's theorem to show that the sum
$$
S_{1}(n)={ }^{n} C_{0}-{ }^{n} C_{2}+{ }^{n} C_{4}-\cdots+(-1)^{m}{ }^{n} C_{2 m}, \quad n-1 \leq 2 m \leq n
$$
has the value $2^{n / 2} \cos (n \pi / 4)$.
(b) Derive a similar result for the sum
$$
S_{2}(n)={ }^{n} C_{1}-{ }^{n} C_{3}+{ }^{n} C_{5}-\cdots+(-1)^{m n}{ }^{n} C_{2 m+1}, \quad n-1 \leq 2 m+1 \leq n
$$
and verify it for the cases $n=6,7$ and 8 .

John Vanschoick
John Vanschoick
Numerade Educator
06:06

Problem 18

By considering $(1+\exp i \theta)^{n}$, prove that
$$
\begin{aligned}
&\sum_{r=0}^{n}{ }^{n} C_{r} \cos n \theta=2^{n} \cos ^{n}(\theta / 2) \cos (n \theta / 2) \\
&\sum_{r=0}^{n}{ }^{n} C_{r} \sin n \theta=2^{n} \cos ^{n}(\theta / 2) \sin (n \theta / 2)
\end{aligned}
$$
where ${ }^{n} C_{r}=n ! /[r !(n-r) !]$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:27

Problem 19

Use de Moivre's theorem with $n=4$ to prove that
$$
\cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1
$$
and deduce that
$$
\cos \frac{\pi}{8}=\left(\frac{2+\sqrt{2}}{4}\right)^{1 / 2}
$$

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator
04:05

Problem 20

Express $\sin ^{4} \theta$ entirely in terms of the trigonometric functions of multiple angles and deduce that its average value over a complete cycle is $\frac{3}{8}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:53

Problem 21

Use de Moivre's theorem to prove that
$$
\tan 5 \theta=\frac{t^{5}-10 t^{3}+5 t}{5 t^{4}-10 t^{2}+1}
$$
where $t=\tan \theta$. Deduce the values of $\tan (n \pi / 10)$ for $n=1,2,3,4$

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator
04:22

Problem 22

(a) Prove that
$$
\cosh x-\cosh y=2 \sinh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)
$$
(b) Prove that, if $y=\sinh ^{-1} x$,
$$
\left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=0
$$

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator
01:57

Problem 23

Determine the conditions under which the equation
$$
a \cosh x+b \sinh x=c, \quad c>0
$$
has zero, one, or two real solutions for $x$. What is the solution if $a^{2}=c^{2}+b^{2}$ ?

Nick Johnson
Nick Johnson
Numerade Educator
01:56

Problem 24

(a) Solve $\cosh x=\sinh x+2 \operatorname{sech} x$.
(b) Show that the real solution $x$ of $\tanh x=\operatorname{cosech} x$ can be written in the form $x=\ln (u+\sqrt{u}) .$ Find an explicit value for $u$.
(c) Evaluate $\tanh x$ when $x$ is the real solution of $\cosh 2 x=2 \cosh x$.

M Hassan Anwar
M Hassan Anwar
Numerade Educator
03:32

Problem 25

Express $\sinh ^{4} x$ in terms of hyperbolic cosines of multiples of $x$, and hence solve
$$
2 \cosh 4 x-8 \cosh 2 x+5=0
$$

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator
04:24

Problem 26

In the theory of special relativity, the relationship between the position and time coordinates of an event as measured in two frames of reference that have parallel $x$-axes can be expressed in terms of hyperbolic functions. If the coordinates are $x$ and $t$ in one frame and $x^{\prime}$ and $t^{\prime}$ in the other then the relationship take the form
$$
\begin{aligned}
x^{\prime} &=x \cosh \phi-c t \sinh \phi \\
c t^{\prime} &=-x \sinh \phi+c t \cosh \phi
\end{aligned}
$$
Express $x$ and $c t$ in terms of $x^{\prime}, c t^{\prime}$ and $\phi$ and show that
$$
x^{2}-(c t)^{2}=\left(x^{\prime}\right)^{2}-\left(c t^{\prime}\right)^{2}
$$

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator
10:53

Problem 27

A closed barrel has as its curved surface that obtained by rotating about the $x$-axis the part of the curve
$$
y=a[2-\cosh (x / a)]
$$
lying in the range $-b \leq x \leq b .$ Show that the total surface area $A$ of the barrel is given by
$$
A=\pi a[9 a-8 a \exp (-b / a)+a \exp (-2 b / a)-2 b]
$$

Carlos Pinilla
Carlos Pinilla
Numerade Educator
05:38

Problem 28

The principal value of the logarithmic function of a complex variable is defined to have its argument in the range $-\pi<\arg z \leq \pi$. By writing $z=\tan w$ in terms of exponentials show that
$$
\tan ^{-1} z=\frac{1}{2 i} \ln \left(\frac{1+i z}{1-i z}\right)
$$
Use this result to evaluate
$$
\tan ^{-1}\left(\frac{2 \sqrt{3}-3 i}{7}\right)
$$

Jarrett Lancaster
Jarrett Lancaster
Numerade Educator