(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f
|
|
- Holly Park
- 5 years ago
- Views:
Transcription
1 . Holomorphic Harmonic Functions Basic notation. Considering C as R, with coordinates x y, z = x + iy denotes the stard complex coordinate, in the usual way. Definition.1. Let f : U C be a complex valued function defined on some open subset U C. We say that f(z) is differentiable at z 0, with derivative f (z 0 ), if f (z 0 ) = lim z z0 f(z) f(z 0 ) z z 0. We say that f is holomorphic if it is differentiable at every point of U It is easy to prove, as in the real case, that the sum, difference, product, quotient, composition of holomorphic functions is holomorphic that the usual formulas apply. The obvious first step in analyzing the condition that the derivative exists, is to see what happens if we approach z 0 along a line, in particular along a horizontal vertical line. Along a horizontal line, we fix y = y 0 vary x. Using the obvious notation z 0 = x 0 + iy 0, f(z) = f(x, y) setting z = x + iy 0 we get f f(z) f(z 0 ) (z 0 ) = lim z z0 z z 0 f(x, y 0 ) f(x 0, y 0 ) = lim x x0 x x 0 = Similarly along a vertical line, we fix x = x 0 vary y. Setting z = x 0 + iy, we get f f(z) f(z 0 ) (z 0 ) = lim z z0 z z 0 f(x 0, y) f(x 0, y 0 ) = lim y y0 i(y y 0 ) = 1 = i i (x0,y 0 ) In other words, if f is holomorphic then = i (x0,y 0 ) 1
2 There is another way to interpret all of this. Suppose we start with f(x, y), a function f : R R. Now we can express x y in terms of z z, x = 1 (z + z) y = 1 i (z z) = i (z z). Thus we can consider f(x, y) as a function f(z, z) of z z. Now formally differentiating formally applying the chain rule we would have z = z + z = 1 i, z = z + z = 1 + i This suggests that we introduce operators z = 1 ( i ) z = 1. ( + i ). In these terms, the condition above becomes z = 0. In other words, a function f : C C is holomorphic if only if it does not depend on z. Separating into real imaginary parts, we get the famous Cauchy- Riemann equations. Suppose that we write f = u + iv. Then Lemma.. If is a C 1 -function then = v u: U R R v =. ɛ = u(x + h, y + k) h k goes to zero faster than (h, k) goes to zero, for any (x, y) U. Proof. By assumption δ = u(x + h, y) u(x, y) h. (x,y). goes to zero faster than h. On the other h, η = u(x + h, y + k) u(x + h, y) k.
3 goes to zero faster than k. As the partials are continuous, the difference goes to zero as h goes to zero. So (x,y) ( ζ = (x,y) ) k goes to zero faster than k. Thus ɛ = u(x + h, y + k) u(x, y) h k = u(x + h, y + k) u(x + h, y) k + u(x + h, y) u(x, y) h = u(x + h, y + k) u(x + h, y) + ζ + δ = η + ζ + δ, which goes to zero faster than (h, k). Proposition.3. A function f(z) is holomorphic if only if its real imaginary parts u(x, y) v(x, y) are C 1 (that is, the first derivatives exist are continuous) satisfy the Cauchy-Riemann equations. Proof. We have already seen that the derivatives of u v exist satisfy the Cauchy-Riemann equations. We will see later that if f is holomorphic then the derivative of f is holomorphic (so that f is infinitely differentiatible) this will imply that u v are C 1. Suppose that u v are C 1. (.) implies that u(x + h, y + k) u(x, y) = h + k + ɛ 1 v(x + h, y + k) v(x, y) = v h + v k + ɛ 3
4 where ɛ 1 ɛ go to zero faster than h + ik. f(x + h, y + k) f(x, y) = (u(x + h, y + k) u(x, y)) + i(v(x + h, y + k) v(x, y)) = h + k + ɛ 1 + i v h + i v k + ɛ ( ) ( ) = + i v h + + i v k + ɛ 1 + ɛ ( ) = + i v (h + ik) + ɛ 1 + ɛ As ɛ 1 + ɛ approaches zero faster than h + ik, it follows that f is differentiable. It is interesting to further investigate the properties satisfied by u v. Recall that the Laplacian of a real function u(x, y): R R is defined to be u = u + u, that function u is said to be Harmonic if it is C the Laplacian vanishes, that is u = 0. Now suppose that f = u + iv, where f is holomorphic, so that u is the real part of a holomorphic function. Latter we will show that f is infinitely differentiable, so that in particular u v are both C. On the other h u = v = v = u, where we used the fact that if v is C then its mixed partials are equal. Thus u = u + u = 0, so u is harmonic. Similarly v is harmonic. Two functions u v that satisfy the Cauchy-Riemann equations, are called conjugate harmonic v is said to be the conjugate harmonic function of u. Note that this is a slight abuse of language, since v is only determined up to a constant. Given a harmonic function u it is instructive to try to construct the harmonic conjugate v. 4
5 For example u = x y is harmonic. = x = y. Thus the harmonic conjugate satisfies v = y v = x. Take the first equation integrate with respect to x. We get that v(x, y) = xy + g(y) where g is an arbitrary function of y. Now differentiate with respect to y. Then x + g (y) = x. Thus g(y) is constant. Thus the harmonic conjugate of u is v = xy. Of course in this case the corresponding holomorphic function is f(z) = u(x, y) + iv(x, y) = (x y ) + ixy = (x + iy) = z. In fact there is another way to compute harmonic conjugates, which does not depend on using integration. In other words, we will find a formula for a holomorphic function f(z) in terms of its real part u(x, y). The trick is to consider the function g(z, z) = f(z). Note that g z = ḡ z = z = 0, so that g(z, z) = g( z) is a function only of z (g is sometimes known as an antiholomorphic function). Then u(x, y) = 1 (f(x + iy) + g(x iy)). Now treat this equality formally extend it to the case where x = z/, y = z/i. Thus u(z/, z/i) = 1 (f(z) + g(0)). Now f(z) is determined up to an imaginary constant. Thus we may as well assume that f(0) is real. In this case, we may take f(0) = g(0) = u(0, 0). Thus f(z) = u(z/, z/i) u(0, 0). For example, suppose that u(x, y) = x y. Then f(z) = u(z/, z/i) u(0, 0) = (z/) (z/i) = z / + z / = z. Note that for this to work, the expression u(z/, z/i) has to make sense (for example, u is given as a power series). 5
Math 185 Homework Exercises II
Math 185 Homework Exercises II Instructor: Andrés E. Caicedo Due: July 10, 2002 1. Verify that if f H(Ω) C 2 (Ω) is never zero, then ln f is harmonic in Ω. 2. Let f = u+iv H(Ω) C 2 (Ω). Let p 2 be an integer.
More informationComplex Variables. Chapter 2. Analytic Functions Section Harmonic Functions Proofs of Theorems. March 19, 2017
Complex Variables Chapter 2. Analytic Functions Section 2.26. Harmonic Functions Proofs of Theorems March 19, 2017 () Complex Variables March 19, 2017 1 / 5 Table of contents 1 Theorem 2.26.1. 2 Theorem
More informationChapter 9. Analytic Continuation. 9.1 Analytic Continuation. For every complex problem, there is a solution that is simple, neat, and wrong.
Chapter 9 Analytic Continuation For every complex problem, there is a solution that is simple, neat, and wrong. - H. L. Mencken 9.1 Analytic Continuation Suppose there is a function, f 1 (z) that is analytic
More information= 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 information2. Complex Analytic Functions
2. Complex Analytic Functions John Douglas Moore July 6, 2011 Recall that if A and B are sets, a function f : A B is a rule which assigns to each element a A a unique element f(a) B. In this course, we
More informationChapter 13: Complex Numbers
Sections 13.3 & 13.4 1. A (single-valued) function f of a complex variable z is such that for every z in the domain of definition D of f, there is a unique complex number w such that w = f (z). The real
More informationMath 126: Course Summary
Math 126: Course Summary Rich Schwartz August 19, 2009 General Information: Math 126 is a course on complex analysis. You might say that complex analysis is the study of what happens when you combine calculus
More informationCHAPTER 3. Analytic Functions. Dr. Pulak Sahoo
CHAPTER 3 Analytic Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-4: Harmonic Functions 1 Introduction
More informationMATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic.
MATH 45 SAMPLE 3 SOLUTIONS May 3, 06. (0 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. Because f is holomorphic, u and v satisfy the Cauchy-Riemann equations:
More informationIII.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 informationMath 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 informationMAT389 Fall 2016, Problem Set 4
MAT389 Fall 2016, Problem Set 4 Harmonic conjugates 4.1 Check that each of the functions u(x, y) below is harmonic at every (x, y) R 2, and find the unique harmonic conjugate, v(x, y), satisfying v(0,
More informationi. v = 0 if and only if v 0. iii. v + w v + w. (This is the Triangle Inequality.)
Definition 5.5.1. A (real) normed vector space is a real vector space V, equipped with a function called a norm, denoted by, provided that for all v and w in V and for all α R the real number v 0, and
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Spring 208 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationAssignment 2 - Complex Analysis
Assignment 2 - Complex Analysis MATH 440/508 M.P. Lamoureux Sketch of solutions. Γ z dz = Γ (x iy)(dx + idy) = (xdx + ydy) + i Γ Γ ( ydx + xdy) = (/2)(x 2 + y 2 ) endpoints + i [( / y) y ( / x)x]dxdy interiorγ
More information2 Complex Functions and the Cauchy-Riemann Equations
2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationMATH MIDTERM 1 SOLUTION. 1. (5 points) Determine whether the following statements are true of false, no justification is required.
MATH 185-4 MIDTERM 1 SOLUTION 1. (5 points Determine whether the following statements are true of false, no justification is required. (1 (1pointTheprincipalbranchoflogarithmfunctionf(z = Logz iscontinuous
More informationSynopsis 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 informationAnalytic Functions. Differentiable Functions of a Complex Variable
Analytic Functions Differentiable Functions of a Complex Variable In tis capter, we sall generalize te ideas for polynomials power series of a complex variable we developed in te previous capter to general
More informationLeplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE
Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math EECE 3640 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as
More informationf (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 informationHomework 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 informationIV. Conformal Maps. 1. Geometric interpretation of differentiability. 2. Automorphisms of the Riemann sphere: Möbius transformations
MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 IV. Conformal Maps 1. Geometric interpretation of differentiability We saw from the definition of complex differentiability that if f
More informationR- and C-Differentiability
Lecture 2 R- and C-Differentiability Let z = x + iy = (x,y ) be a point in C and f a function defined on a neighbourhood of z (e.g., on an open disk (z,r) for some r > ) with values in C. Write f (z) =
More informationTheorem [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 informationComplex Differentials and the Stokes, Goursat and Cauchy Theorems
Complex Differentials and the Stokes, Goursat and Cauchy Theorems Benjamin McKay June 21, 2001 1 Stokes theorem Theorem 1 (Stokes) f(x, y) dx + g(x, y) dy = U ( g y f ) dx dy x where U is a region of the
More information4.6 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 informationLecture 14 Conformal Mapping. 1 Conformality. 1.1 Preservation of angle. 1.2 Length and area. MATH-GA Complex Variables
Lecture 14 Conformal Mapping MATH-GA 2451.001 Complex Variables 1 Conformality 1.1 Preservation of angle The open mapping theorem tells us that an analytic function such that f (z 0 ) 0 maps a small neighborhood
More informationMATH243 First Semester 2013/14. Exercises 1
Complex Functions Dr Anna Pratoussevitch MATH43 First Semester 013/14 Exercises 1 Submit your solutions to questions marked with [HW] in the lecture on Monday 30/09/013 Questions or parts of questions
More informationMath Homework 1. The homework consists mostly of a selection of problems from the suggested books. 1 ± i ) 2 = 1, 2.
Math 70300 Homework 1 September 1, 006 The homework consists mostly of a selection of problems from the suggested books. 1. (a) Find the value of (1 + i) n + (1 i) n for every n N. We will use the polar
More informationComplex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
More informationLeplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math 16.
Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math 16.364 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as heat
More information1 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 informationCOMPLEX ANALYSIS AND RIEMANN SURFACES
COMPLEX ANALYSIS AND RIEMANN SURFACES KEATON QUINN 1 A review of complex analysis Preliminaries The complex numbers C are a 1-dimensional vector space over themselves and so a 2-dimensional vector space
More informationComplex Functions (1A) Young Won Lim 2/22/14
Complex Functions (1A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationMA3111S 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 informationConformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.
Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the
More informationMapping 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,
More informationComplex Analysis Homework 4: Solutions
Complex Analysis Fall 2007 Homework 4: Solutions 1.5.2. a The function fz 3z 2 +7z+5 is a polynomial so is analytic everywhere with derivative f z 6z + 7. b The function fz 2z + 3 4 is a composition of
More informationSolutions 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,
More informationChapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
More informationComplex Analysis review notes for weeks 1-6
Complex Analysis review notes for weeks -6 Peter Milley Semester 2, 2007 In what follows, unless stated otherwise a domain is a connected open set. Generally we do not include the boundary of the set,
More informationKey to Complex Analysis Homework 1 Spring 2012 (Thanks to Da Zheng for providing the tex-file)
Key to Complex Analysis Homework 1 Spring 212 (Thanks to Da Zheng for providing the tex-file) February 9, 212 16. Prove: If u is a complex-valued harmonic function, then the real and the imaginary parts
More informationExercises for Part 1
MATH200 Complex Analysis. Exercises for Part Exercises for Part The following exercises are provided for you to revise complex numbers. Exercise. Write the following expressions in the form x + iy, x,y
More informationMath 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 informationComplex Variables, Summer 2016 Homework Assignments
Complex Variables, Summer 2016 Homework Assignments Homeworks 1-4, due Thursday July 14th Do twenty-four of the following problems. Question 1 Let a = 2 + i and b = 1 i. Sketch the complex numbers a, b,
More informationTopic 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 informationAnalysis 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 informationFunctioning in the Complex Plane
Functioning in the Complex Plane Alan Donald Philip D artagnan Kane Senior Thesis Saint Mary s College of California supervised by Dr.Machmer-Wessels May 17, 2016 1 Abstract In this paper we will explore
More informationTHE GREEN FUNCTION. Contents
THE GREEN FUNCTION CRISTIAN E. GUTIÉRREZ NOVEMBER 5, 203 Contents. Third Green s formula 2. The Green function 2.. Estimates of the Green function near the pole 2 2.2. Symmetry of the Green function 3
More information26.2. Cauchy-Riemann Equations and Conformal Mapping. Introduction. Prerequisites. Learning Outcomes
Cauchy-Riemann Equations and Conformal Mapping 26.2 Introduction In this Section we consider two important features of complex functions. The Cauchy-Riemann equations provide a necessary and sufficient
More informationChapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs
2.6 Limits Involving Infinity; Asymptotes of Graphs Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs Definition. Formal Definition of Limits at Infinity.. We say that
More informationMATH 31BH Homework 5 Solutions
MATH 3BH Homework 5 Solutions February 4, 204 Problem.8.2 (a) Let x t f y = x 2 + y 2 + 2z 2 and g(t) = t 2. z t 3 Then by the chain rule a a a D(g f) b = Dg f b Df b c c c = [Dg(a 2 + b 2 + 2c 2 )] [
More informationComplex 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
More informationQualifying 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 informationHence, (f(x) f(x 0 )) 2 + (g(x) g(x 0 )) 2 < ɛ
Matthew Straughn Math 402 Homework 5 Homework 5 (p. 429) 13.3.5, 13.3.6 (p. 432) 13.4.1, 13.4.2, 13.4.7*, 13.4.9 (p. 448-449) 14.2.1, 14.2.2 Exercise 13.3.5. Let (X, d X ) be a metric space, and let f
More informationThe result above is known as the Riemann mapping theorem. We will prove it using basic theory of normal families. We start this lecture with that.
Lecture 15 The Riemann mapping theorem Variables MATH-GA 2451.1 Complex The point of this lecture is to prove that the unit disk can be mapped conformally onto any simply connected open set in the plane,
More informationWeak 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
More informationMath212a1413 The Lebesgue integral.
Math212a1413 The Lebesgue integral. October 28, 2014 Simple functions. In what follows, (X, F, m) is a space with a σ-field of sets, and m a measure on F. The purpose of today s lecture is to develop the
More informationNATIONAL UNIVERSITY OF SINGAPORE. Department of Mathematics. MA4247 Complex Analysis II. Lecture Notes Part IV
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
More informationThe shortest route between two truths in the real domain passes through the complex domain. J. Hadamard
Chapter 6 Harmonic Functions The shortest route between two truths in the real domain passes through the complex domain. J. Hadamard 6.1 Definition and Basic Properties We will now spend a chapter on certain
More informationTransformations from R m to R n.
Transformations from R m to R n 1 Differentiablity First of all because of an unfortunate combination of traditions (the fact that we read from left to right and the way we define matrix multiplication
More informationSecond Midterm Exam Name: Practice Problems March 10, 2015
Math 160 1. Treibergs Second Midterm Exam Name: Practice Problems March 10, 015 1. Determine the singular points of the function and state why the function is analytic everywhere else: z 1 fz) = z + 1)z
More information(x x 0 ) 2 + (y y 0 ) 2 = ε 2, (2.11)
2.2 Limits and continuity In order to introduce the concepts of limit and continuity for functions of more than one variable we need first to generalise the concept of neighbourhood of a point from R to
More informationClass test: week 10, 75 minutes. (30%) Final exam: April/May exam period, 3 hours (70%).
17-4-2013 12:55 c M. K. Warby MA3914 Complex variable methods and applications 0 1 MA3914 Complex variable methods and applications Lecture Notes by M.K. Warby in 2012/3 Department of Mathematical Sciences
More informationFilters 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
More informationMath Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim
Mat 311 - Spring 013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, 013 Question 1. [p 56, #10 (a)] 4z Use te teorem of Sec. 17 to sow tat z (z 1) = 4. We ave z 4z (z 1) = z 0 4 (1/z) (1/z
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More informationMath 115 Spring 11 Written Homework 10 Solutions
Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,
More informationPart IB. Further Analysis. Year
Year 2004 2003 2002 2001 10 2004 2/I/4E Let τ be the topology on N consisting of the empty set and all sets X N such that N \ X is finite. Let σ be the usual topology on R, and let ρ be the topology on
More informationQ You mentioned that in complex analysis we study analytic functions, or, in another name, holomorphic functions. Pray tell me, what are they?
COMPLEX ANALYSIS PART 2: ANALYTIC FUNCTIONS Q You mentioned that in complex analysis we study analytic functions, or, in another name, holomorphic functions. Pray tell me, what are they? A There are many
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationThus f is continuous at x 0. Matthew Straughn Math 402 Homework 6
Matthew Straughn Math 402 Homework 6 Homework 6 (p. 452) 14.3.3, 14.3.4, 14.3.5, 14.3.8 (p. 455) 14.4.3* (p. 458) 14.5.3 (p. 460) 14.6.1 (p. 472) 14.7.2* Lemma 1. If (f (n) ) converges uniformly to some
More informationComplex 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 informationTopic 2 Notes Jeremy Orloff
Topic 2 Notes Jeremy Orloff 2 Analytic functions 2.1 Introduction The main goal of this topic is to define and give some of the important properties of complex analytic functions. A function f(z) is analytic
More informationMATH GRE PREP: WEEK 2. (2) Which of the following is closest to the value of this integral:
MATH GRE PREP: WEEK 2 UCHICAGO REU 218 (1) What is the coefficient of y 3 x 6 in (1 + x + y) 5 (1 + x) 7? (A) 7 (B) 84 (C) 35 (D) 42 (E) 27 (2) Which of the following is closest to the value of this integral:
More informationINTRODUCTION 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 informationComplex 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 informationIII. Consequences of Cauchy s Theorem
MTH6 Complex Analysis 2009-0 Lecture Notes c Shaun Bullett 2009 III. Consequences of Cauchy s Theorem. Cauchy s formulae. Cauchy s Integral Formula Let f be holomorphic on and everywhere inside a simple
More informationAero III/IV Conformal Mapping
Aero III/IV Conformal Mapping View complex function as a mapping Unlike a real function, a complex function w = f(z) cannot be represented by a curve. Instead it is useful to view it as a mapping. Write
More informationProblem 1A. Use residues to compute. dx x
Problem 1A. A non-empty metric space X is said to be connected if it is not the union of two non-empty disjoint open subsets, and is said to be path-connected if for every two points a, b there is a continuous
More informationMATH 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 informationMath 312 Fall 2013 Final Exam Solutions (2 + i)(i + 1) = (i 1)(i + 1) = 2i i2 + i. i 2 1
. (a) We have 2 + i i Math 32 Fall 203 Final Exam Solutions (2 + i)(i + ) (i )(i + ) 2i + 2 + i2 + i i 2 3i + 2 2 3 2 i.. (b) Note that + i 2e iπ/4 so that Arg( + i) π/4. This implies 2 log 2 + π 4 i..
More informationcarries the circle w 1 onto the circle z R and sends w = 0 to z = a. The function u(s(w)) is harmonic in the unit circle w 1 and we obtain
4. Poisson formula In fact we can write down a formula for the values of u in the interior using only the values on the boundary, in the case when E is a closed disk. First note that (3.5) determines the
More informationTo the Math and Computer Science department for fostering. my curiosity; To Professor Treviño for his guidance and patience;
Abstract From our early years of education we learn that polynomials can be factored to find their roots. In 797 Gauss proved the Fundamental Theorem of Algebra, which states that every polynomial every
More information1 Adeles over Q. 1.1 Absolute values
1 Adeles over Q 1.1 Absolute values Definition 1.1.1 (Absolute value) An absolute value on a field F is a nonnegative real valued function on F which satisfies the conditions: (i) x = 0 if and only if
More informationTools 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 informationStudents need encouragement. So if a student gets an answer right, tell them it was a lucky guess. That way, they develop a good, lucky feeling.
Chapter 8 Analytic Functions Stuents nee encouragement. So if a stuent gets an answer right, tell them it was a lucky guess. That way, they evelop a goo, lucky feeling. 1 8.1 Complex Derivatives -Jack
More informationB Elements of Complex Analysis
Fourier Transform Methods in Finance By Umberto Cherubini Giovanni Della Lunga Sabrina Mulinacci Pietro Rossi Copyright 21 John Wiley & Sons Ltd B Elements of Complex Analysis B.1 COMPLEX NUMBERS The purpose
More informationMATH 311: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE
MATH 3: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE Recall the Residue Theorem: Let be a simple closed loop, traversed counterclockwise. Let f be a function that is analytic on and meromorphic inside. Then
More informationHarmonic Mappings and Shear Construction. References. Introduction - Definitions
Harmonic Mappings and Shear Construction KAUS 212 Stockholm, Sweden Tri Quach Department of Mathematics and Systems Analysis Aalto University School of Science Joint work with S. Ponnusamy and A. Rasila
More informationTEST CODE: PMB SYLLABUS
TEST CODE: PMB SYLLABUS Convergence and divergence of sequence and series; Cauchy sequence and completeness; Bolzano-Weierstrass theorem; continuity, uniform continuity, differentiability; directional
More informationMATH Complex Analysis. Charles Walkden
MATH200 Complex Analysis Charles Walkden 9 th January, 208 MATH200 Complex Analysis Contents Contents 0 Preliminaries 2 Introduction 5 2 Limits and differentiation in the complex plane and the Cauchy-Riemann
More informationGeometric Complex Analysis. Davoud Cheraghi Imperial College London
Geometric Complex Analysis Davoud Cheraghi Imperial College London May 9, 2017 Introduction The subject of complex variables appears in many areas of mathematics as it has been truly the ancestor of many
More informationQuasi-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 informationCHAPTER 2. CONFORMAL MAPPINGS 58
CHAPTER 2. CONFORMAL MAPPINGS 58 We prove that a strong form of converse of the above statement also holds. Please note we could apply the Theorem 1.11.3 to prove the theorem. But we prefer to apply the
More information18.04 Practice problems exam 2, Spring 2018 Solutions
8.04 Practice problems exam, Spring 08 Solutions Problem. Harmonic functions (a) Show u(x, y) = x 3 3xy + 3x 3y is harmonic and find a harmonic conjugate. It s easy to compute: u x = 3x 3y + 6x, u xx =
More informationNotes 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 informationComplex 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