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C ONFORMAL M APS A ND M OBIUS T RANSFORMATIONS By Mariya Boyko

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O VERVIEW Introduction And Basic Definitions Basic Topology Complex Analysis Understanding Riemanns Theorem Mobius Transformations How To Find Mobius Transformations Example

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I NTROUCTION A ND B ASIC D EFINITIONS 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.

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B ASIC T OPOLOGY 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.

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B ASIC T OPOLOGY 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)

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C OMPLEX A NALYSIS 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

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U NDERSTANDING R IEMANN ' S T HEOREM Let F,G be two simply connected domains which are not the entire complex plane. By Riemanns 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.

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U NDERSTANDING R IEMANN S T HEOREM Thus by Riemanns 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.

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U NDERSTANDING R IEMANN S T HEOREM 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.

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M OBIUS T RANSFORMATIONS 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.

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M OBIUS T RANSFORMATIONS 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

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H OW T O F IND M OBIUS T RANSFORMATIONS 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 z 1, z 2 and z 3 are three distinct points in the complex plane. Suppose also that f(z 1 ) = w 1, f(z 2 ) = w 2 and f(z 3 ) = w 3. There are two possibilities, either w 1, w 2 and w 3 are non-collinear and they determine a unique circle or else they determine a unique line. (if w i = then the image is always a line)

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E XAMPLE Let C 1 and C 2 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 C 1, say a, b and c. We will first find the Möbius transformation T(z) which maps C 1 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.

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E XAMPLE (C ONTINUED ) 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 C 2, say d, e and f and in the same way find a Möbius transformation S(w) which maps C 2 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.

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E XAMPLE (C ONTINUED ) 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).

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