Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

y

Q R S T x

i P

w=g(z)

R′

P′

S′

Q′

T′

s

r

Figure 24.4 Transforming the upper half of thez-plane into the interior of

the unit circle in thew-plane, in such a way thatz=iis mapped ontow=0

and the pointsx=±∞are mapped ontow=1.

Further, suppose that (say) Ref(z)=φis constant over a boundaryCin the

z-plane; then ReF(w) = Φ is constant overCin thez-plane. But this is the same

as saying that ReF(w) is constant over the boundaryC′in thew-plane,C′being

the curve into whichCis transformed by the conformal transformationw=g(z).

This result is exploited extensively in the next chapter to solve Laplace’s equation

for a variety of two-dimensional geometries.

Examples of useful conformal transformations are numerous. For instance,

w=z+b,w=(expiφ)zandw=azcorrespond, respectively, to a translation by

b, a rotation through an angleφand a stretching (or contraction) in the radial

direction (forareal). These three examples can be combined into the general

linear transformationw=az+b,where,ingeneral,aandbare complex. Another

example is the inversion mappingw=1/z, which maps the interior of the unit

circle to the exterior and vice versa. Other, more complicated, examples also exist.

Show that if the pointz 0 lies in the upper half of thez-plane then the transformation

w=(expiφ)

z−z 0

z−z∗ 0

maps the upper half of thez-plane into the interior of the unit circle in thew-plane. Hence

find a similar transformation that maps the pointz=iontow=0and the pointsx=±∞

ontow=1.

Taking the modulus ofw, we have

|w|=

∣∣

∣

∣(expiφ)

z−z 0

z−z∗ 0

∣∣

∣

∣=

∣∣

∣

∣

z−z 0

z−z∗ 0

∣∣

∣

∣.

However, since the complex conjugatez 0 ∗is the reflection ofz 0 in the real axis, ifzandz 0
both lie in the upper half of thez-plane then|z−z 0 |≤|z−z∗ 0 |; thus|w|≤1, as required.
We also note that (i) the equality holds only whenzlies on the real axis, and so this axis
is mapped onto the boundary of the unit circle in thew-plane; (ii) the pointz 0 is mapped
ontow= 0, the origin of thew-plane.
By fixing the images of two points in thez-plane, the constantsz 0 andφcan also be
fixed. Since we require the pointz=ito be mapped ontow= 0, we have immediately