Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

(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(iargz),

deduce that

a(B−C)+b(C−A)+c(A−B)=0.

3.9 For the real constantafind the loci of all pointsz=x+iyin the complex plane
that satisfy

(a) Re

{

ln

(

z−ia

z+ia

)}

=c, c>0,

(b) Im

{

ln

(

z−ia

z+ia

)}

=k,0≤k≤π/2.

Identify the two families of curves and verify that in case (b) all curves pass
through the two points±ia.
3.10 The most general type of transformation between one Argand diagram, in the
z-plane, and another, in theZ-plane, that gives one and only one value ofZfor
each value ofz(and conversely) is known as thegeneral bilinear transformation
and takes the form

z=

aZ+b

cZ+d

.

(a) Confirm that the transformation from theZ-plane to thez-plane is also a

general bilinear transformation.

(b) Recalling that the equation of a circle can be written in the form

∣

∣∣

∣

z−z 1

z−z 2

∣

∣∣

∣=λ, λ=1,

show that the general bilinear transformation transforms circles into circles

(or straight lines). What is the condition thatz 1 ,z 2 andλmust satisfy if the

transformed circle is to be a straight line?

3.11 Sketch the parts of the Argand diagram in which

(a) Rez^2 <0,|z^1 /^2 |≤2;

(b) 0≤argz∗≤π/2;

(c) |expz^3 |→0as|z|→∞.

What is the area of the region in which all three sets of conditions are satisfied?
3.12 Denote thenth roots of unity by 1,ωn,ω^2 n,...,ωnn−^1.

(a) Prove that

(i)

∑n−^1

r=0

ωrn=0, (ii)

∏n−^1

r=0

ωrn=(−1)n+1.

(b) Expressx^2 +y^2 +z^2 −yz−zx−xyas the product of two factors, each linear

inx,yandz, with coefficients dependent on the third roots of unity (and

those of thexterms arbitrarily taken as real).