Conformal Maps And Mobius Transformations

1. Conformal Maps And MobiusTransformations By MariyaBoyko

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

3. IntrouctionAnd 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.

4. 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.

5. 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)

6. 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

7. 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.

8. 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.

9. 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.

10. 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.

11. 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

12. 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)

13. 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.

14. 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.

15. 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).