ON THE RIEMANN MAPPING THEOREM AND SOME AP- PLICATIONS M.SC THESIS BY TESFAYE NEGA OCT, 2017 ARBA MINCH, ETHIOPIA A THESIS ON THE RIEMANN MAPPING THEOREM AND SOME APPLICATIONS M.SC THESIS BY TESFAYE NEGA A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCE SCHOOL OF GRADUATE STUDIES ARBA MINCH UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS WITH SPECIALIZATION IN DIFFERENTIAL EQUATIONS OCT, 2017 ARBA MINCH ,ETHIOPIA APPROVAL SHEET-1 This is to certify that the Thesis entitled " On the Riemann Mapping Theorem and some applications" submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics with Spe- cialization in differential equation of the graduate program of the School of science in Mathematics,Arba Minch University, and is a record of original research carried out by Tesfaye Nega, under my supervision. Therefore, I recommend that it would be accepted fulfilling the thesis requirements and hence hereby can submit the thesis to the department for defense.. Dr.ABRAHAM HAILU ___________ _____________ Name of the major advisor Signature Date BEYENE CHANYALEW ___________ _____________ Name of co-advisor Signature Date APPROVAL SHEET -2 We, the undersigned, members of the Board of Examiners of the final open defense by Tesfaye Nega, have read and evaluated his thesis entitled "On the Riemann Mapping Theorem And Some Applications" and examined the candidate. This is, there, to certify that thesis has been accepted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics (Specialization: Differential Equations). Name of Advisor Signature Date Name of Co- Advisor Signature Date Name of External Examiner Signature Date Name of Internal Examiner Signature Date Chair Person Signature Date Name of Department Head Signature Date DECLARATION I, the undersigned, declare that the thesis is my original work, has not been presented for degrees in any other university and all sources of mate- rial used for the thesis have been duly acknowledged. Name: Tesfaye Nega Signature: Date: ACKNOWLEDGMENTS I would like to thank Dr.Abraham Hailu for all his support and his ideas,that helped to improve this thesis. I also would like to thank my co-advisor Beyene Chanyalew for his guidance in the preparation of this thesis. Furthermore, I would to thank every one who supported me,when I was talking about my work. i LIST OF ABBREVIATIONS Abbreviation It stands for C Complex plane C1 The extended complex plane R Real numbers <(z),=(z) Real, imaginary of z respectively ∆u Laplace of u rT Gradient of Temperature ii List of Figures 2.1 z-plane,w-plane . .7 4.1 Picturical representation of conformal mapping . 21 5.1 Graphical Representation for Example 5.2 . 31 5.2 Graphical Representation for Example 5.3 . 33 iii ABSTRACT In this thesis,we studied the Riemann Mapping Theorem.This theorem states that any non-empty simply connected domain D ⊆ C with D 6= C is conformally equivalent to the unit disk D = fz 2 C : jzj < 1g.We also studied how conformal mapping is applied in solving the Dirichlet problem over some bounded and simply connected domains. Keywords: Conformal Mapping, Mo"bius Transformations, Holomorphic Functions, Cauchy's Theorem, simply connected domain, Riemann Map- ping Theorem. iv Contents Acknowledgments . .i List of Abbreviations . ii List of Figures . iii Abstract . iv 1 INTRODUCTION 1 1.1 Back ground of the study . .1 1.2 Statement of the problem . .2 1.3 Objectives of the study . .2 1.3.1 General Objective . .2 1.3.2 Specific Objectives . .2 1.4 Operational Definitions of terms . .2 1.4.1 sets in the complex plane . .3 1.4.2 Functions on the complex plane . .4 2 CONFORMAL MAPPINGS 7 2.1 Conformal Mapping . .7 2.2 Mo"bius Transformations . .8 3 SOME IMPORTANT THEOREMS AND SEQUENCES OF HOLOMORPHIC FUNCTIONS 10 3.1 Cauchy's Theorem And it's Application . 10 3.2 Sequences of Holomorphic Functions . 18 3.3 Schwarz's Lemma . 20 4 PROOF OF THE RIEMANN MAPPING THEOREM 21 4.1 The Proof of Uniqueness . 22 4.2 The Proof of Existence . 22 5 SOME APPLICATIONS OF THE RIEMANN MAPPING v THEOREM 25 5.1 Harmonic Functions . 25 5.1.1 Definition and Basic Properties of Harmonic Function 25 5.1.2 Mean-Value Theorem . 26 5.1.3 Harmonic Functions on a Disk . 26 5.2 The Dirichlet Problem . 28 5.3 Application of the Riemann Mapping Theorem . 30 5.3.1 Physical Application of Conformal Mapping . 32 6 CONCLUSION 34 REFERENCE . 35 vi Chapter 1 INTRODUCTION 1.1 Back ground of the study The study of complex variable is important for physical sciences and is a central subject in mathematics. Complex number is an expression of the form z = x + iy, x is the real part of z, <(z) and y is the imaginary part of z, =(z). If y = 0, we say that z is real and if x = 0, we say that z is pure imaginary. We often denote z, an element of the complex numbers as z 2 C; where x, an element is the p real numbers denoted by x 2 R and i = −1. Complex conjugate of z = x + iy 2 C is given by z = x − iy and modulus of z is defined by jzj = px2 + y2, for z 6= 0. Geometrically, we represent equation z = x + iy in a two dimensional coordinate system called the complex plane. Hence the complex number z can be written in the alternative polar form : z = x + iy = reiθ = r(cos θ + i sin θ). Conformal mapping is an important technique used in complex analysis and has many applications in different physical situations. If the function is harmonic (i.e it satisfies Laplace's equation) then the transformation of such functions via conformal mapping is also harmonic. So equations pertaining to any field that can be represented by a potential function (all conservative fields) can be solved via conformal mapping. Holomor- phic functions are to complex function theory, much as real differentiable functions are one of the main subjects of calculus. We call a function biholomorphic if it is a holomorphic bijection. This thesis gives a brief introduction to conformal mappings such that the uniqueness and the ex- istence of the Riemann Mapping Theorem . 1 Chapter 1 INTRODUCTION 1.2 Statement of the problem One of the most important theorems in geometric function theory is the Riemann mapping theorem. This theorem gives the existence of a confor- mal mapping of a simply connected proper domain of the complex plane and the unit disk. The main purpose of conformal mapping : more important to the engineer and physicist is the use of conformal mapping in connection with potential problems. It yields a standard method for solving boundary value problem in two dimensional potential theory by transforming a complicated region into simpler one. Therefore, the paper deals with the following basic questions • What are the conditions that the Riemann Mapping Theorem holds ? • What are some of the applications of the Riemann Mapping Theorem ? 1.3 Objectives of the study 1.3.1 General Objective The overall objectives of this thesis are to understand the basic ideas, con- cepts, definition, theorem and properties about the existence of conformal mapping function of a simply connected domain which is proper subset of the complex plane to the unit disk. 1.3.2 Specific Objectives The specific objectives of this thesis are the following :- • To study the conditions that the Riemann Mapping Theorem holds. • To study the proof of the Riemann mapping theorem. • To discuss some applications of the Riemann Mapping Theorem. 1.4 Operational Definitions of terms In this subsection there are sets in the complex plane and functions on the complex plane which are essential for the study of the Riemann Mapping 2 Chapter 1 INTRODUCTION Theorem. 1.4.1 sets in the complex plane Definition 1.1. The open disk of radius r and center zo is the set fz 2 C : jz − zoj < rg. Definition 1.2. The unit disk is the disk centered at 0 of radius 1. Symbolically D = fz 2 C : jzj < 1g. Definition 1.3. A neighborhood of point zo 2 C is the set of the points z such that jz − zoj < , where > 0. Definition 1.4. A point zo of a set S is called an interior point of S if S contains a neighborhood of zo. Definition 1.5. A region is an open subset of the complex plane ,plus some ,all or none of the boundary points. Definition 1.6. Region R is said to be open if every point of R has neigh- borhood entirely contained in R. Definition 1.7. A boundary point of a set S is a point every neighborhood of which contains both points that belong to S and points that do not belong to S. Definition 1.8. A set is closed if it contains all of its boundary points. Definition 1.9. A set S is called connected if any two of its can be joined by a chain of finitely many straight line segments all of whose points belong to S . Example 1.1. The set A is (0; 1)[(1; 2) is disconnected. choose u = (0; 1) and v = (1; 2) then u \ v = ; ,u [ v = A, A \ u 6= ; and A \ v 6= ;, where A is the set of real number. Definition 1.10. A region R is said to be bounded if 9M > 0 such that jzj ≤ M for all z 2 R and else unbounded.