3.3 Conformal Maps

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There was an exercise in chapter two which invited you to notice that if

you took any of the functions you had been working with at the time, all of

which were analytic almost everywhere, then the image by such a function of

a rectangle gave something which had corners. Moreover, although the edges

of the rectangle were sent to curves, the curves intersected at right angles.

The only exception was the case when f(z) = z2 and the corner was at the

origin.

The question was asked, why is this happening and why is there an exception

in the one case?

If you are really smart you will have seen the answer: if you take a corner

where the edges are lines intersecting at right angles, then if the map f is analytic

at the corner, it may be approximated by its derivative there. And this

means that in a su_ciently small neighbourhood, the map is approximable

as an a_ne map, multiplication by a complex number together with a shift.

And multiplication by a complex number is just a rotation and a similarity.

None of these will stop a right angle being a right angle. The only exception

is when the derivative is zero, when all bets are o_.

It is clear that not just right angles are preserved by analytic functions;

any angle is preserved. This is rather a striking restriction, forced by the

3.3. CONFORMAL MAPS 103

properties of complex numbers and derivatives.

This property of a complex function is called isogonality2 or conformality,

with the latter sometimes being restricted to the case where the sense of the

angle is preserved. For our purposes, the term conformal means that angles

are preserved everywhere, which is guaranteed if the map is analytic and has

derivative non-zero everywhere.

Exercise 3.3.1 For which complex numbers w is multiplication by w going

to preserve the sense of two intersecting lines?

Exercise 3.3.2 Give an example of a conformal map in this sense which is

not analytic.

There are a lot of applications of Complex Function Theory which depend

on this property; I do not, alas, have time to do more than warn you of what

your lecturers in Engineering may exploit at some later time.

It is very commonly desired to transform some one shape in the plane into

some other shape, by a conformal map. Some very remarkable such transforms

are known; see [11] for a dictionary of very unlikely looking conformal

maps. See [9] for the Schwartz-Christo_el transformations, which take the

half plane to any polygon, and are conformal on the interior.

It is a remarkable fact that

Theorem 3.1 ( The Riemann Mapping Theorem)

If U is some connected and simply connected region of the complex plane (i.e.

it is in one piece and has no holes in it), and if it is open (i.e. every point in

the U has a disk centred on it also contained in U) then providing U is not

the whole plane, there is a 1-1 conformal mapping of U onto the interior of

the unit disk. 2

3

2From the greek isos meaning equal and agon an angle, as in pentagon and polygon.

3Malcolm suggested that I point out that the selection of the interior of the unit disk is

for ease of stating the theorem. It works for a much larger range of regions; it is particularly

useful on occasion to take a half plane as the `universal' region onto which all manner of

unlikely regions can be taken by conformal maps.

104 CHAPTER 3. C - DIFFERENTIABLE FUNCTIONS

It follows that for any two open regions of C which are connected and simply

connected, there is an invertible conformal map which takes one to the other.

This may seem somewhat unlikely, but it has been proved. See [10] for

details.

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