POWER SERIES AND ANALYTIC CONTINUATION

Size: px
Start display at page:

Download "POWER SERIES AND ANALYTIC CONTINUATION"

Transcription

1 POWER SERIES AND ANALYTIC CONTINUATION 1. Analytic functions Definition 1.1. A function f : Ω C C is complex-analytic if for each z 0 Ω there exists a power series f z0 (z) := a n (z z 0 ) n which converges to f(z) for all z in some disc D r (z 0 ) Ω. Real-analytic functions are defined analogously, except the domain Ω R is a subset of the reals and the two-dimensional disc D r (z 0 ) is replaced by 1-dimensional open interval in Ω. Unless stated otherwise, we consider functions of a complex variable and use the term analytic to mean complex-analytic. Theorem 1.1. Holomorphic functions are analytic. More precisely, if f : Ω C C is holomorphic (complex-differentiable) and D r (z 0 ) Ω is a closed disc contained in Ω, then f has a power series expansion f (n) (z 0 ) (z z 0 ) n for all z D r (z 0 ). In particular, if f : C C is entire, its power series expansion converges to f for all z C, uniformly on compact subsets of C. Remark. The theorem is false for real differentiable functions; in HW6 you encounter an example of a smooth (infinitely-differentiable) function of a real variable which is not real analytic. Proof. By Cauchy s integral formula for the closed disc D r (z 0 ), for each z D r (z 0 ) we have 1 2πi D r(z 0) ζ z dζ = 1 2πi D r(z 0) (ζ z 0 )(1 z z0 ζ z 0 ) dζ = 1 ( z z0 ) n dζ. 2πi ζ z 0 ζ z 0 D r(z 0) Since z z 0 < ζ z 0 = r, the geometric series converges uniformly in ζ D r (z 0 ), so we may interchange the sum and integral to obtain (z z 0 ) n[ 1 ] 2πi (ζ z 0 ) n+1 dζ f (n) (z 0 ) = (z z 0 ) n, D r(z 0) where we used the CIF for derivatives in the last step. 1

2 2 POWER SERIES AND ANALYTIC CONTINUATION Remark. Note that in the above theorem, the same power series f (n) (z 0 ) (z z 0 ) n converges to f(z) on every disc D r (z 0 ) Ω. In particular, its radius of convergence is at least sup{r > 0 : D r (z 0 ) Ω}. However it could conceivably converge on a larger disc D R (z 0 ) not entirely contained in Ω. In this case, however, there is no guarantee that it will converge to f(z) for all z D R (z 0 ) Ω. In the next section we shall characterize more precisely its radius of convergence. Definition 1.2. We say f has a zero of order N at z 0 if f (k) (z 0 ) = for all 0 k n 1 but f (n) (z 0 ) 0. Lemma 1.2. Assume f : Ω C is analytic. If z 0 Ω is a zero of order N, then there exists a open disc D r (z 0 ) Ω and a nonvanishing analytic function g : D r (z 0 ) C such that (z z 0 ) N g(z) for all z D r (z 0 ). Corollary 1.3. If f has a zero of order N at z 0, then there exists a disc D r (z 0 ) which contains no other zeros of f. Proof. Indeed, with D r (z 0 ) as in the lemma above, both factors in the product are nonzero when z D r (z 0 ) \ {z 0 }. Theorem 1.4. Let Ω C be open and connected, and let f : Ω C be analytic. If f 1 (0) has a limit point in Ω, then 0 for all z Ω. Proof. We begin with a Lemma. Let U C be any open set and f : U C be analytic. Then the set is both open and closed relative to U. K := {z U : f (n) (z) = 0 for all n}. Proof of lemma. Since f is locally represented by a power series, for each z 0 K there exists a disc D r (z 0 ) U such that f (n) (z 0 ) (z z 0 ) n = 0 forall z D r (z 0 ). Therefore f (n) (z) = 0 for all z D r (z 0 ), so D r (z 0 ) K and K is open. On the other hand, if z 0 U \ K, by definition there exists some integer n such that f (n) (z 0 ) 0. Since f (n) is holomorphic and in particular continuous, there exists a small ball D ε (z 0 ) U such that that f (n) (z) 0 for all z D ε (z 0 ). In other words D ε (z 0 ) U \ K so U \ K is open, and K is closed relative to U. Now if we could show that K is nonempty, the connectedness assumption would imply that K = Ω so that f is identically zero. Let z 0 be a limit point of f 1 ({0}); we claim that z 0 K. For if not, there exists an integer N such that z 0 is a zero of order N. But the above Corollary then implies that there exists a disc centered at z 0 which contains no other zeros of f, contradicting the choice of z 0 as a limit point of f 1 ({0}). This completes the proof of the theorem.

3 POWER SERIES AND ANALYTIC CONTINUATION 3 Corollary 1.5. Let Ω C be open and connected. Suppose f 1, f 2 : Ω C are analytic and that {z Ω : f 1 (z) = f 2 (z)} has a limit point in Ω. Then f 1 (z) = f 2 (z) for all z Ω. Proof. Apply the previous theorem with f = f 1 f Analytic Continuation Definition 2.1. Let Ω 0 C be any set and f : Ω 0 C a function. We say a function f : Ω C C is an analytic continuation or analytic extension of f if Ω 0 Ω, f is analytic, and f Ω0 = f (that is, f(z) for all z Ω 0 ). Lemma 2.1. If Ω is open and connected, Ω 0 Ω, and Ω 0 has a limit point in Ω, then f : Ω 0 C has at most one analytic continuation to Ω. Proof. If f, g : Ω C are both analytic extensions of the same f, then by continuity they agree on the closure Ω 0 Ω which contains a limit point, so the claim follows from the last Corollary in the previous section. Examples. The complex exponential exp : C C exp(x + iy) := e x (cos(y) + i sin(y)) is the unique analytic extension of the real exponential function x e x, x R to the complex plane. On the other hand, if Ω C is an open set containing R which is not connected, then there exist multiple analytic continuations of the real exponential function x e x to Ω. Let f(x) = a n (x x 0 ) n be a real power series with radius of convergence R = (lim sup n a n 1/n ) 1 > 0; thus it converges to a smooth function f : (x 0 R, x 0 + R) R R. Then, replacing x with the complex variable z yields a complex power series which converges to a holomorphic function on the complex open disc D R (z 0 ) = {z C : z x 0 < R}. Thus F : D R (x 0 ) C, F (z) = is an analytic extension of f, and is the unique analytic extension of f to the disc D R (x 0 ) as D R (x 0 ) is connected. Using the technique of the last example, we can quickly write down power series expansions of some analytic functions. For instance, we know that the real exponential function x e x has power series expansion e x = x n, for all x R.

4 4 POWER SERIES AND ANALYTIC CONTINUATION Replacing the left side with the complex exponential exp(z) and the right side with the complex power series zn, we conclude that z n exp(z) = for all z C, since both sides are analytic on C and agree on the real line. Definition 2.2. Let Ω 0 C be a set and f : Ω 0 C a function. For z 0 Ω 0, we say an analytic function f : Ω C is a (local) analytic extension/continuation of f at z 0 if there exists an open disc D r (z 0 ) Ω 0 Ω such that f Dr(z 0) = f Dr(z 0); in other words, f is an analytic continuation of the restriction of f Dr(z 0) to some neighborhood D r (z 0 ) of z 0. Remarks. Note that Ω is not required to contain all of Ω 0 ; we merely require that Ω 0 Ω contain an open disc. Note that by the uniqueness theorem for analytic continuations, we know that f(z) for all z in the connected component of Ω 0 Ω containing z 0. However, there is no guarantee that f agrees with f on other components of Ω 0 Ω. Example 1. Define f : D 1 (0) (C \ D 1 (0)) C by { 0, z < 1 z, z > 1. Then f is analytic. The function 0, z C is an analytic continuation of f at z = 0; indeed f(z) for all z < 1. However f(z) f(z) for z > 1. Example 2. Let Log : C \ (, 0] C be the principal logarithm Log(z) := log z + i arg ( π,π) (z), and let log (0,2π) : C \ [0, ) C be the branch of log defined by log (0,2π) (z) := log z + i arg (0,2π) (z). then log (0,2π) is an analytic continuation of Log at any point z 0 with Im(z 0 ) > 0. Indeed log (0,2π) (z) = Log(z) for all such z. However log (0,2π) (z) Log(z) for any z in the lower half plane. We return to the radius of convergence of power series for analytic functions. Lemma 2.2. Let f : Ω C be analytic and for z 0 Ω, let f f(z) (n) (z 0 ) := (z z 0 ) n be the power series expansion of f at z 0. Let R z0 denote the radius of convergence of this power series. Then f : D Rz0 (z 0 ) C is an analytic continuation of f at z 0 (but need not be a (global) analytic continuation of f : Ω C). Moreover, R z0 = sup{r > 0 : F : D R (z 0 ) C an analytic continuation of f at z 0 }.

5 POWER SERIES AND ANALYTIC CONTINUATION 5 Proof. The first bullet point was already proved in the Theorem at the beginning of these notes. To provve the second bullet point, note that follows immediately from the first bullet point. To prove, let F : D R (z 0 ) C be an analytic continuation of f at z 0. By definition there exists some disc D r (z 0 ) Ω such that F Dr(z 0) = f Dr(z 0). In particular, F (n) (z 0 ) = f (n) (z 0 ) for all n. Now as F is itself analytic, by the Theorem at the beginning of these notes we have F (z) = F (n) (z 0 ) (z z 0 ) n = f (n) (z 0 ) (z z 0 ) n for all z D R (z 0 ). Therefore the power series for f has radius of convergence at least R. Example 3. Let 1 z 2 +1 for z Ω := C \ {±i}. Then its power series at z 0 = 0 has radius of convergence 1. To see this, note that f : Ω C is a local analytic continuation of itself at 0, and D 1 (0) Ω. On the other hand, f admits no analytic extension to a larger disc D r (0) for r > 1. Indeed, if f : D r (0) C were such an extension, then f would agree with f on D r (0) \ {±i} by the uniqueness of analytic extensions on connected domains. But f would be bounded near z = ±i, while lim z ±i f(z) =.

MORE CONSEQUENCES OF CAUCHY S THEOREM

MORE CONSEQUENCES OF CAUCHY S THEOREM MOE CONSEQUENCES OF CAUCHY S THEOEM Contents. The Mean Value Property and the Maximum-Modulus Principle 2. Morera s Theorem and some applications 3 3. The Schwarz eflection Principle 6 We have stated Cauchy

More information

Complex Variables. Cathal Ormond

Complex Variables. Cathal Ormond Complex Variables Cathal Ormond Contents 1 Introduction 3 1.1 Definition: Polar Form.............................. 3 1.2 Definition: Length................................ 3 1.3 Definitions.....................................

More information

Hartogs Theorem: separate analyticity implies joint Paul Garrett garrett/

Hartogs Theorem: separate analyticity implies joint Paul Garrett  garrett/ (February 9, 25) Hartogs Theorem: separate analyticity implies joint Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ (The present proof of this old result roughly follows the proof

More information

Considering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.

Considering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial. Lecture 3 Usual complex functions MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and

More information

RIEMANN MAPPING THEOREM

RIEMANN MAPPING THEOREM RIEMANN MAPPING THEOREM VED V. DATAR Recall that two domains are called conformally equivalent if there exists a holomorphic bijection from one to the other. This automatically implies that there is an

More information

Complex Analysis Homework 9: Solutions

Complex Analysis Homework 9: Solutions Complex Analysis Fall 2007 Homework 9: Solutions 3..4 (a) Let z C \ {ni : n Z}. Then /(n 2 + z 2 ) n /n 2 n 2 n n 2 + z 2. According to the it comparison test from calculus, the series n 2 + z 2 converges

More information

MA3111S COMPLEX ANALYSIS I

MA3111S COMPLEX ANALYSIS I MA3111S COMPLEX ANALYSIS I 1. The Algebra of Complex Numbers A complex number is an expression of the form a + ib, where a and b are real numbers. a is called the real part of a + ib and b the imaginary

More information

1 Holomorphic functions

1 Holomorphic functions Robert Oeckl CA NOTES 1 15/09/2009 1 1 Holomorphic functions 11 The complex derivative The basic objects of complex analysis are the holomorphic functions These are functions that posses a complex derivative

More information

Course 214 Section 2: Infinite Series Second Semester 2008

Course 214 Section 2: Infinite Series Second Semester 2008 Course 214 Section 2: Infinite Series Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 2 Infinite Series 25 2.1 The Comparison Test and Ratio Test.............. 26

More information

13 Maximum Modulus Principle

13 Maximum Modulus Principle 3 Maximum Modulus Principle Theorem 3. (maximum modulus principle). If f is non-constant and analytic on an open connected set Ω, then there is no point z 0 Ω such that f(z) f(z 0 ) for all z Ω. Remark

More information

LECTURE-15 : LOGARITHMS AND COMPLEX POWERS

LECTURE-15 : LOGARITHMS AND COMPLEX POWERS LECTURE-5 : LOGARITHMS AND COMPLEX POWERS VED V. DATAR The purpose of this lecture is twofold - first, to characterize domains on which a holomorphic logarithm can be defined, and second, to show that

More information

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5 MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have

More information

Math 411, Complex Analysis Definitions, Formulas and Theorems Winter y = sinα

Math 411, Complex Analysis Definitions, Formulas and Theorems Winter y = sinα Math 411, Complex Analysis Definitions, Formulas and Theorems Winter 014 Trigonometric Functions of Special Angles α, degrees α, radians sin α cos α tan α 0 0 0 1 0 30 π 6 45 π 4 1 3 1 3 1 y = sinα π 90,

More information

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 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.

More information

Notes on Complex Analysis

Notes on Complex Analysis Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................

More information

Lecture 9. = 1+z + 2! + z3. 1 = 0, it follows that the radius of convergence of (1) is.

Lecture 9. = 1+z + 2! + z3. 1 = 0, it follows that the radius of convergence of (1) is. The Exponential Function Lecture 9 The exponential function 1 plays a central role in analysis, more so in the case of complex analysis and is going to be our first example using the power series method.

More information

4 Uniform convergence

4 Uniform convergence 4 Uniform convergence In the last few sections we have seen several functions which have been defined via series or integrals. We now want to develop tools that will allow us to show that these functions

More information

Fundamental Properties of Holomorphic Functions

Fundamental Properties of Holomorphic Functions Complex Analysis Contents Chapter 1. Fundamental Properties of Holomorphic Functions 5 1. Basic definitions 5 2. Integration and Integral formulas 6 3. Some consequences of the integral formulas 8 Chapter

More information

Complex Analysis Qualifying Exam Solutions

Complex Analysis Qualifying Exam Solutions Complex Analysis Qualifying Exam Solutions May, 04 Part.. Let log z be the principal branch of the logarithm defined on G = {z C z (, 0]}. Show that if t > 0, then the equation log z = t has exactly one

More information

Theorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r

Theorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r 2. A harmonic conjugate always exists locally: if u is a harmonic function in an open set U, then for any disk D(z 0, r) U, there is f, which is analytic in D(z 0, r) and satisfies that Re f u. Since such

More information

INTRODUCTION TO REAL ANALYTIC GEOMETRY

INTRODUCTION TO REAL ANALYTIC GEOMETRY INTRODUCTION TO REAL ANALYTIC GEOMETRY KRZYSZTOF KURDYKA 1. Analytic functions in several variables 1.1. Summable families. Let (E, ) be a normed space over the field R or C, dim E

More information

Analysis Comprehensive Exam, January 2011 Instructions: Do as many problems as you can. You should attempt to answer completely some questions in both

Analysis Comprehensive Exam, January 2011 Instructions: Do as many problems as you can. You should attempt to answer completely some questions in both Analysis Comprehensive Exam, January 2011 Instructions: Do as many problems as you can. You should attempt to answer completely some questions in both real and complex analysis. You have 3 hours. Real

More information

9. Series representation for analytic functions

9. Series representation for analytic functions 9. Series representation for analytic functions 9.. Power series. Definition: A power series is the formal expression S(z) := c n (z a) n, a, c i, i =,,, fixed, z C. () The n.th partial sum S n (z) is

More information

F (z) =f(z). f(z) = a n (z z 0 ) n. F (z) = a n (z z 0 ) n

F (z) =f(z). f(z) = a n (z z 0 ) n. F (z) = a n (z z 0 ) n 6 Chapter 2. CAUCHY S THEOREM AND ITS APPLICATIONS Theorem 5.6 (Schwarz reflection principle) Suppose that f is a holomorphic function in Ω + that extends continuously to I and such that f is real-valued

More information

Math 220A - Fall Final Exam Solutions

Math 220A - Fall Final Exam Solutions Math 22A - Fall 216 - Final Exam Solutions Problem 1. Let f be an entire function and let n 2. Show that there exists an entire function g with g n = f if and only if the orders of all zeroes of f are

More information

4.6 Montel's Theorem. Robert Oeckl CA NOTES 7 17/11/2009 1

4.6 Montel's Theorem. Robert Oeckl CA NOTES 7 17/11/2009 1 Robert Oeckl CA NOTES 7 17/11/2009 1 4.6 Montel's Theorem Let X be a topological space. We denote by C(X) the set of complex valued continuous functions on X. Denition 4.26. A topological space is called

More information

Math Homework 2

Math Homework 2 Math 73 Homework Due: September 8, 6 Suppose that f is holomorphic in a region Ω, ie an open connected set Prove that in any of the following cases (a) R(f) is constant; (b) I(f) is constant; (c) f is

More information

Qualifying Exam Complex Analysis (Math 530) January 2019

Qualifying Exam Complex Analysis (Math 530) January 2019 Qualifying Exam Complex Analysis (Math 53) January 219 1. Let D be a domain. A function f : D C is antiholomorphic if for every z D the limit f(z + h) f(z) lim h h exists. Write f(z) = f(x + iy) = u(x,

More information

VII.5. The Weierstrass Factorization Theorem

VII.5. The Weierstrass Factorization Theorem VII.5. The Weierstrass Factorization Theorem 1 VII.5. The Weierstrass Factorization Theorem Note. Conway motivates this section with the following question: Given a sequence {a k } in G which has no limit

More information

LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 15: COMPLETENESS AND CONVEXITY LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other

More information

CONSEQUENCES OF POWER SERIES REPRESENTATION

CONSEQUENCES OF POWER SERIES REPRESENTATION CONSEQUENCES OF POWER SERIES REPRESENTATION 1. The Uniqueness Theorem Theorem 1.1 (Uniqueness). Let Ω C be a region, and consider two analytic functions f, g : Ω C. Suppose that S is a subset of Ω that

More information

Homework 27. Homework 28. Homework 29. Homework 30. Prof. Girardi, Math 703, Fall 2012 Homework: Define f : C C and u, v : R 2 R by

Homework 27. Homework 28. Homework 29. Homework 30. Prof. Girardi, Math 703, Fall 2012 Homework: Define f : C C and u, v : R 2 R by Homework 27 Define f : C C and u, v : R 2 R by f(z) := xy where x := Re z, y := Im z u(x, y) = Re f(x + iy) v(x, y) = Im f(x + iy). Show that 1. u and v satisfies the Cauchy Riemann equations at (x, y)

More information

FINAL EXAM MATH 220A, UCSD, AUTUMN 14. You have three hours.

FINAL EXAM MATH 220A, UCSD, AUTUMN 14. You have three hours. FINAL EXAM MATH 220A, UCSD, AUTUMN 4 You have three hours. Problem Points Score There are 6 problems, and the total number of points is 00. Show all your work. Please make your work as clear and easy to

More information

Chapter 6: The metric space M(G) and normal families

Chapter 6: The metric space M(G) and normal families Chapter 6: The metric space MG) and normal families Course 414, 003 04 March 9, 004 Remark 6.1 For G C open, we recall the notation MG) for the set algebra) of all meromorphic functions on G. We now consider

More information

Part IB Complex Analysis

Part IB Complex Analysis Part IB Complex Analysis Theorems Based on lectures by I. Smith Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

More information

Paul-Eugène Parent. March 12th, Department of Mathematics and Statistics University of Ottawa. MAT 3121: Complex Analysis I

Paul-Eugène Parent. March 12th, Department of Mathematics and Statistics University of Ottawa. MAT 3121: Complex Analysis I Paul-Eugène Parent Department of Mathematics and Statistics University of Ottawa March 12th, 2014 Outline 1 Holomorphic power Series Proposition Let f (z) = a n (z z o ) n be the holomorphic function defined

More information

Qualifying Exams I, 2014 Spring

Qualifying Exams I, 2014 Spring Qualifying Exams I, 2014 Spring 1. (Algebra) Let k = F q be a finite field with q elements. Count the number of monic irreducible polynomials of degree 12 over k. 2. (Algebraic Geometry) (a) Show that

More information

REVIEW OF COMPLEX ANALYSIS

REVIEW OF COMPLEX ANALYSIS REVIEW OF COMPLEX ANALYSIS KEITH CONRAD We discuss here some basic results in complex analysis concerning series and products (Section 1) as well as logarithms of analytic functions and the Gamma function

More information

Lecture 7 Local properties of analytic functions Part 1 MATH-GA Complex Variables

Lecture 7 Local properties of analytic functions Part 1 MATH-GA Complex Variables Lecture 7 Local properties of analytic functions Part 1 MATH-GA 2451.001 omplex Variables 1 Removable singularities 1.1 Riemann s removable singularity theorem We have said that auchy s integral formula

More information

f (n) (z 0 ) Theorem [Morera s Theorem] Suppose f is continuous on a domain U, and satisfies that for any closed curve γ in U, γ

f (n) (z 0 ) Theorem [Morera s Theorem] Suppose f is continuous on a domain U, and satisfies that for any closed curve γ in U, γ Remarks. 1. So far we have seen that holomorphic is equivalent to analytic. Thus, if f is complex differentiable in an open set, then it is infinitely many times complex differentiable in that set. This

More information

Notes on uniform convergence

Notes on uniform convergence Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean

More information

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

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

More information

MA30056: Complex Analysis. Revision: Checklist & Previous Exam Questions I

MA30056: Complex Analysis. Revision: Checklist & Previous Exam Questions I MA30056: Complex Analysis Revision: Checklist & Previous Exam Questions I Given z C and r > 0, define B r (z) and B r (z). Define what it means for a subset A C to be open/closed. If M A C, when is M said

More information

Spring 2010 Exam 2. You may not use your books, notes, or any calculator on this exam.

Spring 2010 Exam 2. You may not use your books, notes, or any calculator on this exam. MTH 282 final Spring 2010 Exam 2 Time Limit: 110 Minutes Name (Print): Instructor: Prof. Houhong Fan This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages are

More information

Dirichlet s Theorem. Calvin Lin Zhiwei. August 18, 2007

Dirichlet s Theorem. Calvin Lin Zhiwei. August 18, 2007 Dirichlet s Theorem Calvin Lin Zhiwei August 8, 2007 Abstract This paper provides a proof of Dirichlet s theorem, which states that when (m, a) =, there are infinitely many primes uch that p a (mod m).

More information

1 Discussion on multi-valued functions

1 Discussion on multi-valued functions Week 3 notes, Math 7651 1 Discussion on multi-valued functions Log function : Note that if z is written in its polar representation: z = r e iθ, where r = z and θ = arg z, then log z log r + i θ + 2inπ

More information

Math 715 Homework 1 Solutions

Math 715 Homework 1 Solutions . [arrier, Krook and Pearson Section 2- Exercise ] Show that no purely real function can be analytic, unless it is a constant. onsider a function f(z) = u(x, y) + iv(x, y) where z = x + iy and where u

More information

Complex Analysis. Travis Dirle. December 4, 2016

Complex Analysis. Travis Dirle. December 4, 2016 Complex Analysis 2 Complex Analysis Travis Dirle December 4, 2016 2 Contents 1 Complex Numbers and Functions 1 2 Power Series 3 3 Analytic Functions 7 4 Logarithms and Branches 13 5 Complex Integration

More information

Complex Variables Notes for Math 703. Updated Fall Anton R. Schep

Complex Variables Notes for Math 703. Updated Fall Anton R. Schep Complex Variables Notes for Math 703. Updated Fall 20 Anton R. Schep CHAPTER Holomorphic (or Analytic) Functions. Definitions and elementary properties In complex analysis we study functions f : S C,

More information

Simply Connected Domains

Simply Connected Domains Simply onnected Domains Generaliing the losed urve Theorem We have shown that if f() is analytic inside and on a closed curve, then f()d = 0. We have also seen examples where f() is analytic on the curve,

More information

Solutions to practice problems for the final

Solutions to practice problems for the final Solutions to practice problems for the final Holomorphicity, Cauchy-Riemann equations, and Cauchy-Goursat theorem 1. (a) Show that there is a holomorphic function on Ω = {z z > 2} whose derivative is z

More information

COMPLEX ANALYSIS Spring 2014

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

More information

Complex Analysis, Stein and Shakarchi Meromorphic Functions and the Logarithm

Complex Analysis, Stein and Shakarchi Meromorphic Functions and the Logarithm Complex Analysis, Stein and Shakarchi Chapter 3 Meromorphic Functions and the Logarithm Yung-Hsiang Huang 217.11.5 Exercises 1. From the identity sin πz = eiπz e iπz 2i, it s easy to show its zeros are

More information

2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α.

2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α. Chapter 2. Basic Topology. 2.3 Compact Sets. 2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α. 2.32 Definition A subset

More information

COMPLEX ANALYSIS MARCO M. PELOSO

COMPLEX ANALYSIS MARCO M. PELOSO COMPLEX ANALYSIS MARCO M. PELOSO Contents. Holomorphic functions.. The complex numbers and power series.2. Holomorphic functions 3.3. The Riemann sphere 7.4. Exercises 8 2. Complex integration and Cauchy

More information

Complex Analysis Problems

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

More information

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

(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

More information

ELEMENTARY PROPERTIES OF ANALYTIC FUNCTIONS OF SEVERAL VARIABLES

ELEMENTARY PROPERTIES OF ANALYTIC FUNCTIONS OF SEVERAL VARIABLES ELEMENTARY PROPERTIES OF ANALYTIC FUNCTIONS OF SEVERAL VARIABLES MARTIN TAMM. Basic Definitions Let C n denote the set of all n-tuples of compex numbers, i.e. the set of all elements z = (z, z,..., z n

More information

COMPLEX ANALYSIS HW 8

COMPLEX ANALYSIS HW 8 OMPLEX ANALYSIS HW 8 LAY SHONKWILE 23 For differentiable (but not necessarily holomorphic functions f, p, q : Ω, prove the following relations: (23.: dz d z 2idx dy d z dz. Proof. On one hand, dz d z (

More information

Week 5 Lectures 13-15

Week 5 Lectures 13-15 Week 5 Lectures 13-15 Lecture 13 Definition 29 Let Y be a subset X. A subset A Y is open in Y if there exists an open set U in X such that A = U Y. It is not difficult to show that the collection of all

More information

Fourth Week: Lectures 10-12

Fourth Week: Lectures 10-12 Fourth Week: Lectures 10-12 Lecture 10 The fact that a power series p of positive radius of convergence defines a function inside its disc of convergence via substitution is something that we cannot ignore

More information

Math Final Exam.

Math Final Exam. Math 106 - Final Exam. This is a closed book exam. No calculators are allowed. The exam consists of 8 questions worth 100 points. Good luck! Name: Acknowledgment and acceptance of honor code: Signature:

More information

Quasi-conformal maps and Beltrami equation

Quasi-conformal maps and Beltrami equation Chapter 7 Quasi-conformal maps and Beltrami equation 7. Linear distortion Assume that f(x + iy) =u(x + iy)+iv(x + iy) be a (real) linear map from C C that is orientation preserving. Let z = x + iy and

More information

Review of complex analysis in one variable

Review of complex analysis in one variable CHAPTER 130 Review of complex analysis in one variable This gives a brief review of some of the basic results in complex analysis. In particular, it outlines the background in single variable complex analysis

More information

WEIERSTRASS THEOREMS AND RINGS OF HOLOMORPHIC FUNCTIONS

WEIERSTRASS THEOREMS AND RINGS OF HOLOMORPHIC FUNCTIONS WEIERSTRASS THEOREMS AND RINGS OF HOLOMORPHIC FUNCTIONS YIFEI ZHAO Contents. The Weierstrass factorization theorem 2. The Weierstrass preparation theorem 6 3. The Weierstrass division theorem 8 References

More information

Chapter 4: Open mapping theorem, removable singularities

Chapter 4: Open mapping theorem, removable singularities Chapter 4: Open mapping theorem, removable singularities Course 44, 2003 04 February 9, 2004 Theorem 4. (Laurent expansion) Let f : G C be analytic on an open G C be open that contains a nonempty annulus

More information

Complex Analysis Qual Sheet

Complex Analysis Qual Sheet Complex Analysis Qual Sheet Robert Won Tricks and traps. traps. Basically all complex analysis qualifying exams are collections of tricks and - Jim Agler Useful facts. e z = 2. sin z = n=0 3. cos z = z

More information

COMPLEX ANALYSIS HW 3

COMPLEX ANALYSIS HW 3 COMPLEX ANALYSIS HW 3 CLAY SHONKWILER 6 Justify the swap of limit and integral carefully to prove that for each function f : D C continuous on an open set D C, and for each differentiable curve : [0, D,

More information

Fixed Points & Fatou Components

Fixed Points & Fatou Components Definitions 1-3 are from [3]. Definition 1 - A sequence of functions {f n } n, f n : A B is said to diverge locally uniformly from B if for every compact K A A and K B B, there is an n 0 such that f n

More information

7 Asymptotics for Meromorphic Functions

7 Asymptotics for Meromorphic Functions Lecture G jacques@ucsd.edu 7 Asymptotics for Meromorphic Functions Hadamard s Theorem gives a broad description of the exponential growth of coefficients in power series, but the notion of exponential

More information

THE WEIERSTRASS PREPARATION THEOREM AND SOME APPLICATIONS

THE WEIERSTRASS PREPARATION THEOREM AND SOME APPLICATIONS THE WEIERSTRASS PREPARATION THEOREM AND SOME APPLICATIONS XUAN LI Abstract. In this paper we revisit the Weierstrass preparation theorem, which describes how to represent a holomorphic function of several

More information

Math 425 Fall All About Zero

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

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

MATH 6322, COMPLEX ANALYSIS

MATH 6322, COMPLEX ANALYSIS Complex numbers: MATH 6322, COMPLEX ANALYSIS Motivating problem: you can write down equations which don t have solutions, like x 2 + = 0. Introduce a (formal) solution i, where i 2 =. Define the set C

More information

III.2. Analytic Functions

III.2. Analytic Functions III.2. Analytic Functions 1 III.2. Analytic Functions Recall. When you hear analytic function, think power series representation! Definition. If G is an open set in C and f : G C, then f is differentiable

More information

Preliminary Exam 2018 Solutions to Morning Exam

Preliminary Exam 2018 Solutions to Morning Exam Preliminary Exam 28 Solutions to Morning Exam Part I. Solve four of the following five problems. Problem. Consider the series n 2 (n log n) and n 2 (n(log n)2 ). Show that one converges and one diverges

More information

Properties of Entire Functions

Properties of Entire Functions Properties of Entire Functions Generalizing Results to Entire Functions Our main goal is still to show that every entire function can be represented as an everywhere convergent power series in z. So far

More information

Math 220A Homework 4 Solutions

Math 220A Homework 4 Solutions Math 220A Homework 4 Solutions Jim Agler 26. (# pg. 73 Conway). Prove the assertion made in Proposition 2. (pg. 68) that g is continuous. Solution. We wish to show that if g : [a, b] [c, d] C is a continuous

More information

Theorem Let J and f be as in the previous theorem. Then for any w 0 Int(J), f(z) (z w 0 ) n+1

Theorem Let J and f be as in the previous theorem. Then for any w 0 Int(J), f(z) (z w 0 ) n+1 (w) Second, since lim z w z w z w δ. Thus, i r δ, then z w =r (w) z w = (w), there exist δ, M > 0 such that (w) z w M i dz ML({ z w = r}) = M2πr, which tends to 0 as r 0. This shows that g = 2πi(w), which

More information

MA30056: Complex Analysis. Exercise Sheet 7: Applications and Sequences of Complex Functions

MA30056: Complex Analysis. Exercise Sheet 7: Applications and Sequences of Complex Functions MA30056: Complex Analysis Exercise Sheet 7: Applications and Sequences of Complex Functions Please hand solutions in at the lecture on Monday 6th March..) Prove Gauss Fundamental Theorem of Algebra. Hint:

More information

f(w) f(a) = 1 2πi w a Proof. There exists a number r such that the disc D(a,r) is contained in I(γ). For any ǫ < r, w a dw

f(w) f(a) = 1 2πi w a Proof. There exists a number r such that the disc D(a,r) is contained in I(γ). For any ǫ < r, w a dw Proof[section] 5. Cauchy integral formula Theorem 5.. Suppose f is holomorphic inside and on a positively oriented curve. Then if a is a point inside, f(a) = w a dw. Proof. There exists a number r such

More information

Homework in Topology, Spring 2009.

Homework in Topology, Spring 2009. Homework in Topology, Spring 2009. Björn Gustafsson April 29, 2009 1 Generalities To pass the course you should hand in correct and well-written solutions of approximately 10-15 of the problems. For higher

More information

Math 213br HW 1 solutions

Math 213br HW 1 solutions Math 213br HW 1 solutions February 26, 2014 Problem 1 Let P (x) be a polynomial of degree d > 1 with P (x) > 0 for all x 0. For what values of α R does the integral I(α) = 0 x α P (x) dx converge? Give

More information

Key to Homework 8, Thanks to Da Zheng for the text-file

Key to Homework 8, Thanks to Da Zheng for the text-file Key to Homework 8, Thanks to Da Zheng for the text-file November 8, 20. Prove that Proof. csc z = + z + 2z ( ) n z 2 n 2, z 0, ±π, ±2π, π2 n= We consider the following auxiliary function, where z 0, ±π,

More information

Solutions Final Exam May. 14, 2014

Solutions Final Exam May. 14, 2014 Solutions Final Exam May. 14, 2014 1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the claim, derive a contradiction or give a counter-example). (a) (10

More information

MATH 566 LECTURE NOTES 4: ISOLATED SINGULARITIES AND THE RESIDUE THEOREM

MATH 566 LECTURE NOTES 4: ISOLATED SINGULARITIES AND THE RESIDUE THEOREM MATH 566 LECTURE NOTES 4: ISOLATED SINGULARITIES AND THE RESIDUE THEOREM TSOGTGEREL GANTUMUR 1. Functions holomorphic on an annulus Let A = D R \D r be an annulus centered at 0 with 0 < r < R

More information

Part IB. Complex Analysis. Year

Part IB. Complex Analysis. Year Part IB Complex Analysis Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section I 2A Complex Analysis or Complex Methods 7 (a) Show that w = log(z) is a conformal

More information

ANALYSIS QUALIFYING EXAM FALL 2016: SOLUTIONS. = lim. F n

ANALYSIS QUALIFYING EXAM FALL 2016: SOLUTIONS. = lim. F n ANALYSIS QUALIFYING EXAM FALL 206: SOLUTIONS Problem. Let m be Lebesgue measure on R. For a subset E R and r (0, ), define E r = { x R: dist(x, E) < r}. Let E R be compact. Prove that m(e) = lim m(e /n).

More information

Complex Analysis Slide 9: Power Series

Complex Analysis Slide 9: Power Series Complex Analysis Slide 9: Power Series MA201 Mathematics III Department of Mathematics IIT Guwahati August 2015 Complex Analysis Slide 9: Power Series 1 / 37 Learning Outcome of this Lecture We learn Sequence

More information

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 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

More information

= 2 x y 2. (1)

= 2 x y 2. (1) COMPLEX ANALYSIS PART 5: HARMONIC FUNCTIONS A Let me start by asking you a question. Suppose that f is an analytic function so that the CR-equation f/ z = 0 is satisfied. Let us write u and v for the real

More information

X.9 Revisited, Cauchy s Formula for Contours

X.9 Revisited, Cauchy s Formula for Contours X.9 Revisited, Cauchy s Formula for Contours Let G C, G open. Let f be holomorphic on G. Let Γ be a simple contour with range(γ) G and int(γ) G. Then, for all z 0 int(γ), f (z 0 ) = 1 f (z) dz 2πi Γ z

More information

Introductory Complex Analysis

Introductory Complex Analysis Introductory Complex Analysis Course No. 100 312 Spring 2007 Michael Stoll Contents Acknowledgments 2 1. Basics 2 2. Complex Differentiability and Holomorphic Functions 3 3. Power Series and the Abel Limit

More information

MATH COMPLEX ANALYSIS. Contents

MATH COMPLEX ANALYSIS. Contents MATH 3964 - OMPLEX ANALYSIS ANDREW TULLOH AND GILES GARDAM ontents 1. ontour Integration and auchy s Theorem 2 1.1. Analytic functions 2 1.2. ontour integration 3 1.3. auchy s theorem and extensions 3

More information

Synopsis of Complex Analysis. Ryan D. Reece

Synopsis of Complex Analysis. Ryan D. Reece Synopsis of Complex Analysis Ryan D. Reece December 7, 2006 Chapter Complex Numbers. The Parts of a Complex Number A complex number, z, is an ordered pair of real numbers similar to the points in the real

More information

Math 185 Fall 2015, Sample Final Exam Solutions

Math 185 Fall 2015, Sample Final Exam Solutions Math 185 Fall 2015, Sample Final Exam Solutions Nikhil Srivastava December 12, 2015 1. True or false: (a) If f is analytic in the annulus A = {z : 1 < z < 2} then there exist functions g and h such that

More information

Complex Analysis. Chapter V. Singularities V.3. The Argument Principle Proofs of Theorems. August 8, () Complex Analysis August 8, / 7

Complex Analysis. Chapter V. Singularities V.3. The Argument Principle Proofs of Theorems. August 8, () Complex Analysis August 8, / 7 Complex Analysis Chapter V. Singularities V.3. The Argument Principle Proofs of Theorems August 8, 2017 () Complex Analysis August 8, 2017 1 / 7 Table of contents 1 Theorem V.3.4. Argument Principle 2

More information

or E ( U(X) ) e zx = e ux e ivx = e ux( cos(vx) + i sin(vx) ), B X := { u R : M X (u) < } (4)

or E ( U(X) ) e zx = e ux e ivx = e ux( cos(vx) + i sin(vx) ), B X := { u R : M X (u) < } (4) :23 /4/2000 TOPIC Characteristic functions This lecture begins our study of the characteristic function φ X (t) := Ee itx = E cos(tx)+ie sin(tx) (t R) of a real random variable X Characteristic functions

More information

MATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD

MATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD MATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD TSOGTGEREL GANTUMUR 1. Introduction Suppose that we want to solve the equation f(z) = β where f is a nonconstant entire function and

More information

Math 421 Midterm 2 review questions

Math 421 Midterm 2 review questions Math 42 Midterm 2 review questions Paul Hacking November 7, 205 () Let U be an open set and f : U a continuous function. Let be a smooth curve contained in U, with endpoints α and β, oriented from α to

More information