# NATIONAL UNIVERSITY OF SINGAPORE. Department of Mathematics. MA4247 Complex Analysis II. Lecture Notes Part IV

Save this PDF as:

Size: px
Start display at page:

Download "NATIONAL UNIVERSITY OF SINGAPORE. Department of Mathematics. MA4247 Complex Analysis II. Lecture Notes Part IV"

## Transcription

1 NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part IV Chapter 2. Further properties of analytic/holomorphic functions (continued) 2.4. The Open mapping theorem Theorem (Open mapping theorem) Suppose f is a non-constant analytic function on a domain D. If U is an open subset of D, then f(u) is open. f(z) Proof. We want to show that f(u) is open, i.e., take any point w 0 f(u), then we need to show that there exists some r > 0 such that the open ball B(w 0, r) := {w C w w 0 < r} is contained in f(u). In other words, for any w B(w 0, r), we need to show that there exists z U such that f(z) = w. f(z) Step 1. Since w 0 f(u), there exists z 0 U such that f(z 0 ) = w 0. Let g(z) = f(z) w 0 so that g(z 0 ) = f(z 0 ) w 0 = 0. Since f(z) is a non-constant analytic function, so is g(z). Thus z 0 is an isolated zero of g(z) of some order k, k 1 (see Proposition and Tutorial 1 Question 4). Then we can choose δ > 0 such that (i) B(z 0, δ) U (since U is open); 1

2 2 (ii) g(z) 0 for all z B(z 0, δ) {z 0 } (since zeroes of g are isolated), where B(z 0, δ) = {z C : z z 0 δ} ; (iii) g (z) 0 for all z B(z 0, δ) {z 0 } (since g is non-constant, g is analytic and not identically zero). Consider the function g(z) restricted to the circle γ : z z 0 = δ. By the Extreme Value Theorem, g(z) attains its minimum value on the circle γ. From (ii), the minimum value of g(z) on the circle γ is 0, and we denote it by r (so that g(z) r on γ). Now consider the open ball B(w 0, r) and let w 1 B(w 0, r) so that w 1 w 0 < r. Let h(z) = w 0 w 1 (constant function). Then on the circle γ : z z 0 = δ, g(z) r > h(z) since h(z) = w 0 w 1 < r, so by Rouché s theorem, g(z) and g(z)+h(z) has the same number of zeroes inside the circle γ. Since g(z) = f(z) w 0 has k zeroes inside γ (counting multiplicity), so does g(z) + h(z) = f(z) w 1. Now since by (iii), f (z) = g (z) 0 in the punctured ball B(z 0, δ) {z 0 }, all the zeroes of f(z) w 1 are distinct and of order 1, so there are k distinct solutions of the equation f(z) = w 1 inside the open ball B(z 0, δ). Hence, B(w 0, r) f(b(z 0, δ)) f(u). Note that we have in fact proven a stronger result, namely the following theorem from which the open mapping theorem is a corollary: Theorem Suppose that f is a non-constant function on a domain D and z 0 D with f(z 0 ) = w 0 f(d). If z 0 is a zero of order k of f(z) w 0, then we can find open balls B(z 0, δ) and B(w 0, r) with δ, r > 0 such that every w B(w 0, r) \ {w 0 } has exactly k distinct preimages in B(z 0, δ). f(z)

3 Example Consider the function f(z) = z 3 + z 4 2z 5. 3

4 4 Remark The open mapping theorem implies that if f is analytic and non-constant on a domain D, then f(d) is open. Exercise: Show that f(d) is connected, so that f(d) is also a domain. Hint. The proof that f(d) is connected is similar to Theorem A sketch of the proof is outlined below, and it will be left to the student as an exercise to fill in the details. To show that f(d) is connected, it means that for any two distinct points w 1, w 2 f(d), we have to find a polygonal line L f(d) joining w 1 to w 2. f(z) Note that w 1, w 2 f(d) implies that there exist z 1, z 2 D such that f(z 1 ) = w 1 and f(z 2 ) = w 2. Since D is connected, there exists a polygonal line l = l l k D joining z 1 to z 2 (unfortunately f(l) needs not be a polygonal line in f(d)). Similar to Theorem 2.1.2, we shall first show that each point of f(l 1 ) is connected to w 1 by a polygonal line in f(d), and then the argument is repeated to show that each point of f(l) (and in particular, the point w 2 ) is connected to w 1 by a polygonal line in f(d). Thus it remains to show that each point of f(l 1 ) is connected to w 1 be a polygonal line in f(d). Let γ : [0, 1] D be a parametrization of the line segment l 1, so that γ(a) = w 1. Consider the interval I = { s [0, 1] : f(γ(t)) is connected to w 1 by some polygonal line in f(d) for all 0 t s}. Let s o := sup I. As in Theorem 2.1.2, we need to show that (i) s o > 0 (basically because f(d) is open); (ii) for any 0 s s o, f(γ(s)) is connected to w 1 by some polygonal line in f(d); and (iii) s o = 1 (again basically because f(d) is open). f(z)

5 The proof of the above statements is similar to that of Theorem 2.1.2, and it will be left as an exercise. An alternative proof is to use the fact that a connected open set cannot be decomposed into two non-trivial open subsets. Corollary If f is analytic and non-constant on a domain D and z 0 D such that f (z 0 ) = 0, then f is never one-to-one on any open set U containing z 0. Proof. Let g(z) = f(z) f(z 0 ), so that g(z 0 ) = 0. If f (z 0 ) = 0, then g(z 0 ) = g (z 0 ) = 0, and thus g(z) has a zero of order k 2 at z 0. By the proof of the Open Mapping Theorem (or Theorem 2.4.2), there exists open balls B(z 0, δ) and B(f(z 0 ), r) with δ, r > 0 (and we can choose δ so that B(z 0, δ) U such that every w B(f(z 0 ), r) \ {f(z 0 )} has at least k inverse images of f in B(z 0, δ), so f cannot be one-to-one. Corollary If f is analytic and one-to-one on a domain D, then f (z) 0 for all z D. Proof. Corollary follows immediately from Corollary Remark: The converse to Cor is not true, give an example. Example Suppose that f is analytic on a domain D and f(z) R for all z D. Then f is constant. Proof. f(d) is not an open set, hence f must be constant. Example Suppose f(z) is analytic and non-constant on B(0, 1) and f(b(0, 1)) B(0, 1). Then f(b(0, 1)) B(0, 1). (Note: This also follows from the MMP). 5

6 6 Theorem (Inverse function theorem) If f is one-to-one and analytic on a domain D, then each of the following holds: (i) The inverse function f 1 is continuous on f(d). (ii) f 1 is analytic in f(d). (iii)if w f(d), then (f 1 ) (w) = 1 f (z) where z = f 1 (w). (Note that f (z) 0 from Corollary since f is one-to-one. Also, (iii) = (ii) = (i)) Proof. (i) First we note that since f is one-to-one on D, its inverse f 1 is a well-defined map from f(d) to D. Fix w 0 f(d) and let z 0 D such that w 0 = f(z 0 ), i.e., let z 0 = f 1 (w 0 ). To prove (i), we need to show that for any ɛ > 0, there exists δ > 0 such that w w 0 < δ = f 1 (w) z 0 < ɛ. f(z) Since D is open, shrinking ɛ if necessary, we may assume that B(z 0, ɛ) D. By the open mapping theorem, U := f(b(z 0, ɛ)) is open in f(d) and since w 0 B(w 0, δ) U. Hence U, there exists a δ > 0 such that or w B(w 0, δ) = w U = f 1 (w) B(z 0, ɛ) w w 0 < δ = f 1 (w) z 0 < ɛ. Hence, f 1 is continuous at w 0 and since w 0 is an arbitrary point of f(d), f 1 is continuous on f(d). (ii) and (iii). Suppose w 0 f(d). Then for each w f(d) such that w w 0, we have f 1 (w) f 1 (w 0 ) z z 0 = where w = f 1 (z) w w 0 f(z) f(z 0 ) 1 = where z z f(z) f(z 0 ) 0 since w w 0. z z 0 (*)

7 7 By (i), f 1 is continuous. Hence, as w w 0, we have f 1 (w) f 1 (w 0 ) or equivalently, z z 0, which, in turn, implies that 1 1 f (z 0 ). z z 0 f(z) f(z 0 ) Together with (*), we thus have f 1 (w) f 1 (w 0 ) lim = 1 w w 0 w w 0 f (z 0 ). Since w 0 is an arbitrary point of f(d), the result follows. Example. Let w = f(z) = e z on the domain D = {z : π < Im z < π}. Then f is one-to-one on D and f(d) is the cut-plane C \ (, 0], and f 1 (w) = log w, the principal log function. Furthermore, by (iii), f (w) = 1/f (z) = 1/w.

8 8 Definition A one-to-one analytic function from a domain U onto the domain V = f(u) is called an analytic isomorphism from U to V. (If U = V, f is called an analytic automorphism of U.) U and V are said to be analytically isomorphic if there exists an analytic isomorphism f : U V. Example.(i) f(z) = e z on the domain D = {z : π < Im z < π}. (ii) f(z) = z i z+i on the upper half plane H := {z : Im z > 0}.

9 Proposition Let S denote the set of domains in C. (For example, the entire complex plane C is an element of S, the upper half plane H is an element of S, all balls B(z 0, r) where r > 0 are elements of S). Define the following binary relation on S: For any U, V S, we say that U V if U is analytically isomorphic to V. Then is an equivalence relation on S, i.e., (i) (reflexivity) U U for any U S; (ii) (symmetry) If U V, then V U; (iii) (transitivity) If U V and V W, then U W. Proof. (i) For any domain U S, clearly the identity map f(z) = z on U is an one-to-one analytic function from U onto U, i.e., it is an analytic isomorphism from U to U; in other words, U U, which gives (i). To prove (ii), suppose U V. Then there exists an analytic isomorphism f from U to V, i.e., f is a one-to-one analytic function from U onto V Then the inverse function implies that if f 1 : V U is also an analytic isomorphism (Exercise), so that V U. Similarly, to prove (iii), it is an easy exercise (for you) to show that if f : U V is an analytic isomorphism and g : V W is also an analytic isomorphism, then using the Chain Rule, one can show that g f : U W is also an analytic isomorphism. Remark. From general theory of equivalence relations, one deduces from Proposition that the equivalence relation partitions S into equivalence classes of domains such that the domains in each equivalence class are analytically isomorphic to each other. Thus, it is natural to attempt to solve the classification problem, i.e., one wants to know how many equivalence classes of domains are there, and what are those domains in each equivalence class. In this generality, this is a very difficult problem and is still unsolved, you may want to read up about the Koebe conjecture. To concentrate on the complex analytic problem, we will try to classify domains with trivial topology. We recall the following definition: Definition A domain D C is simply connected if every simple closed contour in D encloses only points in D. In particular, a domain in γ is simply connected if it has no punctures or holes. Questions.(i) How many equivalence classes of analytically isomorphic simply connected domains are there in C? (ii) Which simply connected 9

10 10 domains in C are analytically isomorphic? (Equivalently, what are the simply connected domains in each equivalence class of simply connected domains?) For example, is the interior of the unit square analytically isomorphic to the interior of the unit disk? Proposition The unit ball B(0, 1) := {z C z < 1} is not analytically isomorphic to C. Proof. Suppose not, so suppose there exists an analytic isomorphism f : C B(0, 1). Apply Liouville s theorem to get a contradiction (Exercise: fill in the details). Theorem (Riemann mapping theorem) Any simply connected domain Ω which is not the whole complex plane C is analytically isomorphic to the unit ball B(0, 1). Furthermore, if z 0 Ω, there is a unique analytic isomorphism f : Ω B(0, 1) satisfying f(z 0 ) = 0 and f (z 0 ) > 0. (*) Remarks: 1. The theorem was formulated by Riemann but the first successful proof was given by Koebe. This is a remarkable and very important theorem as it allows us to have existence and uniqueness theorems which allow us to define important analytic functions without actually using analytic expressions. 2. The conditions f(z 0 ) = 0 and f (z 0 ) > 0 are often called normalising conditions to pin down the uniqueness of f. Note that f (z 0 ) > 0 means it is a positive real number.

11 3. The proof (of existence) is beyond the scope of the module and uses the theory of normal families but is standard for first year graduate courses. The uniqueness part is relatively simple. 4. (Some terminnology) A analytic function on a domain Ω is also said to be univalent or schlicht if it is one-to-one. To help to prove the uniqueness, we have the following result: Proposition (Consequence of Schwarz s lemma) Suppose that f is an analytic isomorphism of the unit ball B(0, 1) to itself satisfying f(0) = 0 and f (0) > 0. Then f(z) = z for all z in B(0, 1). Proof. Since f(b(0, 1)) B(0, 1) (i.e., f(z) < 1 for all z < 1), it follows from Schwarz s lemma that f(z) z for all z < 1. Similarly, since f 1 (B(0, 1)) B(0, 1), it follows from Schwarz s lemma that f 1 (w) w for all w < 1. Equivalently, writing w = f(z), we have z f(z) for all z < Hence we have f(z) = z for all z < 1. Again by Schwarz s lemma, this implies f(z) = Cz, where C = 1. Now f (0) = C > 0 = C = 1 = f(z) = z. Exercise. Deduce the uniqueness part of the statement of the Riemann mapping theorem from the above result, i.e., show that if f and g are two analytic isomorphisms from Ω to B(0, 1) satisfying the conditions in (*) of the Riemann Mapping Theorem, then f(z) g(z) on B(0, 1). [ Hint: Consider g f 1, and apply Proposition ] From the Riemann Mapping Theorem and Proposition , one knows that there are only two equivalence classes of analytically isomorphic simply connected domains. One equivalence class consists only

12 12 of the complex plane C, and the other equivalence class consists of all other domains Ω C (and all such Ω are analytically isomorphic to the unit ball B(0, 1)). We remark that the classification problem is much more complicated for non-simply connected domains. In fact, one knows that there are uncountably infinite number of equivalence classes of non-simply connected domains. For example, for any R > 1, consider the annulus A(1, R) := {z C, 1 < z < R}. Proposition For R, R > 1, A(1, R) is analytically isomorphic to A(1, R ) iff R = R. Proof. (Skipped) See W. Rudin, Real and Complex Analysis, page 312, Theorem for a proof. (Project: Write up a proof and submit to the MA4247 IVLE forum page). Example.(Nov 2004 exam, question 1b) Consider the domain D = {z = x + iy C : y < x 2 and z i}. Does there exist a non-constant entire function F such that F (C) D? Justify your answer.

### NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;

### Complex Analysis Problems

Complex Analysis Problems transcribed from the originals by William J. DeMeo October 2, 2008 Contents 99 November 2 2 2 200 November 26 4 3 2006 November 3 6 4 2007 April 6 7 5 2007 November 6 8 99 NOVEMBER

### Complex Analysis Math 220C Spring 2008

Complex Analysis Math 220C Spring 2008 Bernard Russo June 2, 2008 Contents 1 Monday March 31, 2008 class cancelled due to the Master s travel plans 1 2 Wednesday April 2, 2008 Course information; Riemann

### Solutions to Complex Analysis Prelims Ben Strasser

Solutions to Complex Analysis Prelims Ben Strasser In preparation for the complex analysis prelim, I typed up solutions to some old exams. This document includes complete solutions to both exams in 23,

### Complex Analysis - Final exam - Answers

Complex Analysis - Final exam - Answers Exercise : (0 %) Let r, s R >0. Let f be an analytic function defined on D(0, r) and g be an analytic function defined on D(0, s). Prove that f +g is analytic on

### The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative,

Module 16 : Line Integrals, Conservative fields Green's Theorem and applications Lecture 47 : Fundamental Theorems of Calculus for Line integrals [Section 47.1] Objectives In this section you will learn

### INTRODUCTION TO COMPLEX ANALYSIS W W L CHEN

INTRODUTION TO OMPLEX ANALYSIS W W L HEN c W W L hen, 986, 2008. This chapter originates from material used by the author at Imperial ollege, University of London, between 98 and 990. It is available free

### THE RESIDUE THEOREM. f(z) dz = 2πi res z=z0 f(z). C

THE RESIDUE THEOREM ontents 1. The Residue Formula 1 2. Applications and corollaries of the residue formula 2 3. ontour integration over more general curves 5 4. Defining the logarithm 7 Now that we have

### NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part I

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Comple Analsis II Lecture Notes Part I Chapter 1 Preliminar results/review of Comple Analsis I These are more detailed notes for the results

### PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

### Fuchsian groups. 2.1 Definitions and discreteness

2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

### Complex Variables. Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit.

Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit. 1. The TI-89 calculator says, reasonably enough, that x 1) 1/3 1 ) 3 = 8. lim

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### (x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.

1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of

### Taylor and Laurent Series

Chapter 4 Taylor and Laurent Series 4.. Taylor Series 4... Taylor Series for Holomorphic Functions. In Real Analysis, the Taylor series of a given function f : R R is given by: f (x + f (x (x x + f (x

### FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

### RUTGERS UNIVERSITY GRADUATE PROGRAM IN MATHEMATICS Written Qualifying Examination August, Session 1. Algebra

RUTGERS UNIVERSITY GRADUATE PROGRAM IN MATHEMATICS Written Qualifying Examination August, 2014 Session 1. Algebra The Qualifying Examination consists of three two-hour sessions. This is the first session.

### Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

### Chapter 19 Clifford and the Number of Holes

Chapter 19 Clifford and the Number of Holes We saw that Riemann denoted the genus by p, a notation which is still frequently used today, in particular for the generalizations of this notion in higher dimensions.

### ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS

ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements

### Green s Theorem in the Plane

hapter 6 Green s Theorem in the Plane Introduction Recall the following special case of a general fact proved in the previous chapter. Let be a piecewise 1 plane curve, i.e., a curve in R defined by a

### From the definition of a surface, each point has a neighbourhood U and a homeomorphism. U : ϕ U(U U ) ϕ U (U U )

3 Riemann surfaces 3.1 Definitions and examples From the definition of a surface, each point has a neighbourhood U and a homeomorphism ϕ U from U to an open set V in R 2. If two such neighbourhoods U,

### Continued fractions for complex numbers and values of binary quadratic forms

arxiv:110.3754v1 [math.nt] 18 Feb 011 Continued fractions for complex numbers and values of binary quadratic forms S.G. Dani and Arnaldo Nogueira February 1, 011 Abstract We describe various properties

### f(x) f(z) c x z > 0 1

INVERSE AND IMPLICIT FUNCTION THEOREMS I use df x for the linear transformation that is the differential of f at x.. INVERSE FUNCTION THEOREM Definition. Suppose S R n is open, a S, and f : S R n is a

### Riemann sphere and rational maps

Chapter 3 Riemann sphere and rational maps 3.1 Riemann sphere It is sometimes convenient, and fruitful, to work with holomorphic (or in general continuous) functions on a compact space. However, we wish

### 3. Prove or disprove: If a space X is second countable, then every open covering of X contains a countable subcollection covering X.

Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM January 24, 2015 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least

### We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

### NOTES ON THE FUNDAMENTAL GROUP

NOTES ON THE FUNDAMENTAL GROUP AARON LANDESMAN CONTENTS 1. Introduction to the fundamental group 2 2. Preliminaries: spaces and homotopies 3 2.1. Spaces 3 2.2. Maps of spaces 3 2.3. Homotopies and Loops

### Math 361: Homework 1 Solutions

January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x

### Introduction to Topology

Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

### Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert. f(x) = f + (x) + f (x).

References: Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert Evans, Partial Differential Equations, Appendix 3 Reed and Simon, Functional Analysis,

### Solutions for Problem Set #5 due October 17, 2003 Dustin Cartwright and Dylan Thurston

Solutions for Problem Set #5 due October 17, 23 Dustin Cartwright and Dylan Thurston 1 (B&N 6.5) Suppose an analytic function f agrees with tan x, x 1. Show that f(z) = i has no solution. Could f be entire?

### On the Class of Functions Starlike with Respect to a Boundary Point

Journal of Mathematical Analysis and Applications 261, 649 664 (2001) doi:10.1006/jmaa.2001.7564, available online at http://www.idealibrary.com on On the Class of Functions Starlike with Respect to a

### REVIEW OF ESSENTIAL MATH 346 TOPICS

REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations

### MATH 215A NOTES MOOR XU NOTES FROM A COURSE BY KANNAN SOUNDARARAJAN

MATH 25A NOTES MOOR XU NOTES FROM A COURSE BY KANNAN SOUNDARARAJAN Abstract. These notes were taken during Math 25A (Complex Analysis) taught by Kannan Soundararajan in Fall 2 at Stanford University. They

### Some Basic Notations Of Set Theory

Some Basic Notations Of Set Theory References There are some good books about set theory; we write them down. We wish the reader can get more. 1. Set Theory and Related Topics by Seymour Lipschutz. 2.

### COMPLEX ANALYSIS Spring 2014

COMPLEX ANALYSIS Spring 24 Homework 4 Solutions Exercise Do and hand in exercise, Chapter 3, p. 4. Solution. The exercise states: Show that if a

### IV.3. Zeros of an Analytic Function

IV.3. Zeros of an Analytic Function 1 IV.3. Zeros of an Analytic Function Note. We now explore factoring series in a way analogous to factoring a polynomial. Recall that if p is a polynomial with a zero

### An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010

An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 John P. D Angelo, Univ. of Illinois, Urbana IL 61801.

### Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016

Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 1. Please write your 1- or 2-digit exam number on

### ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.

ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n

### Riemann surfaces. Ian Short. Thursday 29 November 2012

Riemann surfaces Ian Short Thursday 29 November 2012 Complex analysis and geometry in the plane Complex differentiability Complex differentiability Complex differentiability Complex differentiability Complex

### Filters in Analysis and Topology

Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

### MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### M4P52 Manifolds, 2016 Problem Sheet 1

Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete

### Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting

More Books At www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com

### Math 425 Fall All About Zero

Math 425 Fall 2005 All About Zero These notes supplement the discussion of zeros of analytic functions presented in 2.4 of our text, pp. 127 128. Throughout: Unless stated otherwise, f is a function analytic

### Continuity. Chapter 4

Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

### Weak Subordination for Convex Univalent Harmonic Functions

Weak Subordination for Convex Univalent Harmonic Functions Stacey Muir Abstract For two complex-valued harmonic functions f and F defined in the open unit disk with f() = F () =, we say f is weakly subordinate

### A Non-Topological View of Dcpos as Convergence Spaces

A Non-Topological View of Dcpos as Convergence Spaces Reinhold Heckmann AbsInt Angewandte Informatik GmbH, Stuhlsatzenhausweg 69, D-66123 Saarbrücken, Germany e-mail: heckmann@absint.com Abstract The category

### FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM SAMUEL BLOOM Abstract. In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen

### Some Maximum Modulus Polynomial Rings and Constant Modulus Spaces

Pure Mathematical Sciences, Vol. 3, 2014, no. 4, 157-175 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/pms.2014.4821 Some Maximum Modulus Polynomial Rings and Constant Modulus Spaces Abtin Daghighi

### 16 1 Basic Facts from Functional Analysis and Banach Lattices

16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness

### b 0 + b 1 z b d z d

I. Introduction Definition 1. For z C, a rational function of degree d is any with a d, b d not both equal to 0. R(z) = P (z) Q(z) = a 0 + a 1 z +... + a d z d b 0 + b 1 z +... + b d z d It is exactly

### Lecture 6: April 25, 2006

Computational Game Theory Spring Semester, 2005/06 Lecture 6: April 25, 2006 Lecturer: Yishay Mansour Scribe: Lior Gavish, Andrey Stolyarenko, Asaph Arnon Partially based on scribe by Nataly Sharkov and

### Functional Analysis Exercise Class

Functional Analysis Exercise Class Wee November 30 Dec 4: Deadline to hand in the homewor: your exercise class on wee December 7 11 Exercises with solutions Recall that every normed space X can be isometrically

### AP CALCULUS AB 2011 SCORING GUIDELINES

AP CALCULUS AB 2011 SCORING GUIDELINES Question 4 The continuous function f is defined on the interval 4 x 3. The graph of f consists of two quarter circles and one line segment, as shown in the figure

### M311 Functions of Several Variables. CHAPTER 1. Continuity CHAPTER 2. The Bolzano Weierstrass Theorem and Compact Sets CHAPTER 3.

M311 Functions of Several Variables 2006 CHAPTER 1. Continuity CHAPTER 2. The Bolzano Weierstrass Theorem and Compact Sets CHAPTER 3. Differentiability 1 2 CHAPTER 1. Continuity If (a, b) R 2 then we write

### CHAPTER 7. Connectedness

CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

### Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f

1 Bernstein's analytic continuation of complex powers c1995, Paul Garrett, garrettmath.umn.edu version January 27, 1998 Analytic continuation of distributions Statement of the theorems on analytic continuation

### FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM ANG LI Abstract. In this paper, we start with the definitions and properties of the fundamental group of a topological space, and then proceed to prove Van-

### THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS

THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU Abstract. This is an edited version of a

### Properties of Analytic Functions

Properties of Analytic Functions Generalizing Results to Analytic Functions In the last few sections, we completely described entire functions through the use of everywhere convergent power series. Our

### The Sphere OPTIONAL - I Vectors and three dimensional Geometry THE SPHERE

36 THE SPHERE You must have played or seen students playing football, basketball or table tennis. Football, basketball, table tennis ball are all examples of geometrical figures which we call "spheres"

### Group construction in geometric C-minimal non-trivial structures.

Group construction in geometric C-minimal non-trivial structures. Françoise Delon, Fares Maalouf January 14, 2013 Abstract We show for some geometric C-minimal structures that they define infinite C-minimal

### CHAPTER 1. Metric Spaces. 1. Definition and examples

CHAPTER Metric Spaces. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. The definitions will provide us with a useful tool for more general applications

### Modern Analysis Series Edited by Chung-Chun Yang AN INTRODUCTION TO COMPLEX ANALYSIS

Modern Analysis Series Edited by Chung-Chun Yang AN INTRODUCTION TO COMPLEX ANALYSIS Classical and Modern Approaches Wolfgang Tutschke Harkrishan L. Vasudeva ««CHAPMAN & HALL/CRC A CRC Press Company Boca

### After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.

Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric

### 3 Measurable Functions

3 Measurable Functions Notation A pair (X, F) where F is a σ-field of subsets of X is a measurable space. If µ is a measure on F then (X, F, µ) is a measure space. If µ(x) < then (X, F, µ) is a probability

### Lecture 4 Lebesgue spaces and inequalities

Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how

### L p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by

L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may

### 3 Fatou and Julia sets

3 Fatou and Julia sets The following properties follow immediately from our definitions at the end of the previous chapter: 1. F (f) is open (by definition); hence J(f) is closed and therefore compact

### Section 7.5: Cardinality

Section 7: Cardinality In this section, we shall consider extend some of the ideas we have developed to infinite sets One interesting consequence of this discussion is that we shall see there are as many

### 1 Partitions and Equivalence Relations

Today we re going to talk about partitions of sets, equivalence relations and how they are equivalent. Then we are going to talk about the size of a set and will see our first example of a diagonalisation

### Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm

Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify

### SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS

Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 36, 2011, 449 460 SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS Martin Chuaqui, Peter Duren and Brad Osgood P. Universidad Católica de Chile, Facultad

### MATH 215B. SOLUTIONS TO HOMEWORK (6 marks) Construct a path connected space X such that π 1 (X, x 0 ) = D 4, the dihedral group with 8 elements.

MATH 215B. SOLUTIONS TO HOMEWORK 2 1. (6 marks) Construct a path connected space X such that π 1 (X, x 0 ) = D 4, the dihedral group with 8 elements. Solution A presentation of D 4 is a, b a 4 = b 2 =

### Problem Set 5. 2 n k. Then a nk (x) = 1+( 1)k

Problem Set 5 1. (Folland 2.43) For x [, 1), let 1 a n (x)2 n (a n (x) = or 1) be the base-2 expansion of x. (If x is a dyadic rational, choose the expansion such that a n (x) = for large n.) Then the

### Lecture 5 - Hausdorff and Gromov-Hausdorff Distance

Lecture 5 - Hausdorff and Gromov-Hausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is

### Complex Continued Fraction Algorithms

Complex Continued Fraction Algorithms A thesis presented in partial fulfilment of the requirements for the degree of Master of Mathematics Author: Bastiaan Cijsouw (s3022706) Supervisor: Dr W Bosma Second

### 2. The Concept of Convergence: Ultrafilters and Nets

2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two

### Problem set 1, Real Analysis I, Spring, 2015.

Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n

### Mapping problems and harmonic univalent mappings

Mapping problems and harmonic univalent mappings Antti Rasila Helsinki University of Technology antti.rasila@tkk.fi (Mainly based on P. Duren s book Harmonic mappings in the plane) Helsinki Analysis Seminar,

### Local dynamics of holomorphic maps

5 Local dynamics of holomorphic maps In this chapter we study the local dynamics of a holomorphic map near a fixed point. The setting will be the following. We have a holomorphic map defined in a neighbourhood

### SUPPLEMENT ON THE SYMMETRIC GROUP

SUPPLEMENT ON THE SYMMETRIC GROUP RUSS WOODROOFE I presented a couple of aspects of the theory of the symmetric group S n differently than what is in Herstein. These notes will sketch this material. You

### Real Analysis Problems

Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.

### 2.2 Annihilators, complemented subspaces

34CHAPTER 2. WEAK TOPOLOGIES, REFLEXIVITY, ADJOINT OPERATORS 2.2 Annihilators, complemented subspaces Definition 2.2.1. [Annihilators, Pre-Annihilators] Assume X is a Banach space. Let M X and N X. We

### u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)

M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation

### Indeed, if we want m to be compatible with taking limits, it should be countably additive, meaning that ( )

Lebesgue Measure The idea of the Lebesgue integral is to first define a measure on subsets of R. That is, we wish to assign a number m(s to each subset S of R, representing the total length that S takes

### Two sets X, Y have the same cardinality (cardinal number, cardinal),

3. Cardinal Numbers Cardinality Two sets X, Y have the same cardinality (cardinal number, cardinal), (3.1) X = Y, if there exists a one-to-one mapping of X onto Y. The relation (3.1) is an equivalence

### Lecture Notes 3 Convergence (Chapter 5)

Lecture Notes 3 Convergence (Chapter 5) 1 Convergence of Random Variables Let X 1, X 2,... be a sequence of random variables and let X be another random variable. Let F n denote the cdf of X n and let

### Complex Analysis Math 185A, Winter 2010 Final: Solutions

Complex Analysis Math 85A, Winter 200 Final: Solutions. [25 pts] The Jacobian of two real-valued functions u(x, y), v(x, y) of (x, y) is defined by the determinant (u, v) J = (x, y) = u x u y v x v y.

### MAT 3271: Selected solutions to problem set 7

MT 3271: Selected solutions to problem set 7 Chapter 3, Exercises: 16. Consider the Real ffine Plane (that is what the text means by the usual Euclidean model ), which is a model of incidence geometry.

### Vector Calculus. Lecture Notes

Vector Calculus Lecture Notes Adolfo J. Rumbos c Draft date November 23, 211 2 Contents 1 Motivation for the course 5 2 Euclidean Space 7 2.1 Definition of n Dimensional Euclidean Space........... 7 2.2

### Functional Analysis F3/F4/NVP (2005) Homework assignment 2

Functional Analysis F3/F4/NVP (25) Homework assignment 2 All students should solve the following problems:. Section 2.7: Problem 8. 2. Let x (t) = t 2 e t/2, x 2 (t) = te t/2 and x 3 (t) = e t/2. Orthonormalize

### 3 COUNTABILITY AND CONNECTEDNESS AXIOMS

3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first