Download presentation

Presentation is loading. Please wait.

Published byJaylin Hudman Modified over 3 years ago

1
C ONFORMAL M APS A ND M OBIUS T RANSFORMATIONS By Mariya Boyko

2
O VERVIEW Introduction And Basic Definitions Basic Topology Complex Analysis Understanding Riemanns Theorem Mobius Transformations How To Find Mobius Transformations Example

3
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.

4
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.

5
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)

6
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

7
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.

8
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.

9
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.

10
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.

11
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

12
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)

13
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.

14
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.

15
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).

Similar presentations

OK

Chapter 9 Efficiency of Algorithms. 9.1 Real Valued Functions.

Chapter 9 Efficiency of Algorithms. 9.1 Real Valued Functions.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on forex market Ppt on review writing lesson Ppt on media research jobs Ppt on surface water treatment Ppt on 3d printing technology Ppt on diode circuits on breadboard Ppt on conjunctions for grade 5 Ppt on role of computer in different fields Ppt on sustainable development for class 8 Ppt on building information modeling jobs