CONFORMAL MAPS AND MOBIUS TRANSFORMATIONSBy Mariya Boyko

OVERVIEW

Introduction And Basic Definitions Basic Topology Complex Analysis Understanding Riemann’s Theorem Mobius Transformations How To Find Mobius Transformations Example

INTROUCTION AND BASIC DEFINITIONS

Complex functions of a complex variable require 4 dimensions to graph.

Instead, we use two Cartesian planes, one for the domain, and one for the range.

In 1851 Riemann in his PhD thesis found a theorem now known as the Riemann Mapping Theorem

Let D be any simply connected domain in the complex plane which is not the whole plane itself. Then there exists a one-to-one conformal map which takes D onto the open unit disc |z|<1.

BASIC TOPOLOGY

A Neighbourhood of a point z is a disc centered around z.

Interior Point: An interior point of a set S is a point for which there exists a neighbourhood entirely in S.

Boundary Point: A boundary point of a set S is a point for which every neighbourhood has points in S and in S’

Open Set: A set is called open if all its points are interior points. An equivalent definition is if it does not contain its boundary.

BASIC TOPOLOGY

Connected Set: A set is called connected if it cannot be expressed as the union of two disjoint, non-empty open sets.

Domain: A connected open set. Simply connected: Every loop in the set can

be continuously deformed to a point. (ie. it does not have any holes)

COMPLEX ANALYSIS

A complex function which is differentiable is called a holomorphic or analytic function.

Holomorphic is a very strong condition which has many implications.

A function (map) from an open set U is called conformal if it is holomorphic on U and whose derivative is nowhere zero on U.

Conformal maps preserve angles. The composition of conformal maps is

conformal and the inverse of a conformal map is again conformal

UNDERSTANDING RIEMANN'S THEOREM

Let F,G be two simply connected domains which are not the entire complex plane. By Riemann’s theorem, there exist two one-to-one conformal maps f,g mapping F,G respectively onto the unit disc |z|<1.

Consider the map g-1∘f. It is a one-to-one conformal map taking F to G.

Theorem: Let D by any simply connected domain in the complex plane which is not the whole plane itself. Then there exists a one-to-one conformal map which takes D onto the open unit disc |z|<1.

UNDERSTANDING RIEMANN’S THEOREM

Thus by Riemann’s Theorem there always exists a one-to-one conformal map between any two simply connected domains in the complex plane.

Even though the theorem tells us that such a map exists, it does not tell us how to find it.

Theorem: Let D by any simply connected domain in the complex plane which is not the whole plane itself. Then there exists a one-to-one conformal map which takes D onto the open unit disc |z|<1.

UNDERSTANDING RIEMANN’S THEOREM

In fact, even simple mappings such as the one from the open unit square to the open unit disc cannot be expressed in terms of elementary functions.

Theorem: Let D by any simply connected domain in the complex plane which is not the whole plane itself. Then there exists a one-to-one conformal map which takes D onto the open unit disc |z|<1.

MOBIUS TRANSFORMATIONS

The extended complex plane is the complex plane together with a point called “infinity”

The map defined by f(z) =(az + b)/(cz + d) from the extended complex plane to the extended complex plane where a,b,c and d are complex numbers and ad ≠ bc is called a Möbius transformation

It is a composition of translations, rotations, magnifications and inversions.

MOBIUS TRANSFORMATIONS

These simple one-to-one conformal functions map circles and lines to circles and lines and therefore the same is true of all Möbius transformations.

The pole of a Möbius transformation is the point where the denominator is zero.

We define f(-d/c) = ∞ and f(∞) = a/c If a line or circle passes through the pole of

the Möbius transformation then it always gets mapped to a line, otherwise it is mapped to a circle

HOW TO FIND MOBIUS TRANSFORMATIONS

To completely specify a Möbius transformation, all we need is to define the image of three distinct points.

Let f(z) be a Möbius transformation and suppose that z1, z2 and z3 are three distinct points in the complex plane. Suppose also that f(z1) = w1, f(z2) = w2 and f(z3) = w3. There are two possibilities, either w1, w2 and w3 are non-collinear and they determine a unique circle or else they determine a unique line. (if wi = ∞ then the image is always a line)

EXAMPLE

Let C1 and C2 be two circles in the complex plane. How can we find a Möbius transformation from one to the other?

Pick any three distinct points on C1, say a, b and c. We will first find the Möbius transformation T(z) which maps C1 to the real axis. To do this, let T(a) = 0, T(b) = 1 and T(c) = ∞. It is not hard to verify that T(z) =[(z – a)(b – c)]/[(z - c)(b - a)] is the required Möbius transformation.

EXAMPLE (CONTINUED)

The function [(z – a)(b – c)]/[(z - c)(b - a)] is called the cross-ratio of z, a, b and c and is denoted by (z,a,b,c)

Now take three distinct points on C2, say d, e and f and in the same way find a Möbius transformation S(w) which maps C2 to the real axis. We let S(d) = 0, S(e) = 1 and S(f) = ∞ and we obtain S(w) = [(w - d)(e - f)]/[(w - f)(e - d)] . Then, since Möbius transformations are closed under composition and inverses (as can be directly verified with a calculation), g(z) = S-1(T(z)) is the required Möbius transformation.

EXAMPLE (CONTINUED)

To find g(z) explicitly, notice that g(a)= S-

1(T(a))= S-1(0) = d. In the same way, g(b) = e and g(c) = f. Thus in general, w = g(z) = S-

1(T(z)). This implies that S(w) = T(z). Therefore we just equate the two cross-ratios (z, a, b, c) and (w, d, e, f) and solve for w in terms of z.

To map a line to circle, line to line or circle to line, we follow the same process with only a minor modification to the Möbius transformations S(w) and T(z).