Basic Complex Analysis a Comprehensive Course in Analysis, Part 2A
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Informal Lecture Notes for Complex Analysis
Informal lecture notes for complex analysis Robert Neel I'll assume you're familiar with the review of complex numbers and their algebra as contained in Appendix G of Stewart's book, so we'll pick up where that leaves off. 1 Elementary complex functions In one-variable real calculus, we have a collection of basic functions, like poly- nomials, rational functions, the exponential and log functions, and the trig functions, which we understand well and which serve as the building blocks for more general functions. The same is true in one complex variable; in fact, the real functions we just listed can be extended to complex functions. 1.1 Polynomials and rational functions We start with polynomials and rational functions. We know how to multiply and add complex numbers, and thus we understand polynomial functions. To be specific, a degree n polynomial, for some non-negative integer n, is a function of the form n n−1 f(z) = cnz + cn−1z + ··· + c1z + c0; 3 where the ci are complex numbers with cn 6= 0. For example, f(z) = 2z + (1 − i)z + 2i is a degree three (complex) polynomial. Polynomials are clearly defined on all of C. A rational function is the quotient of two polynomials, and it is defined everywhere where the denominator is non-zero. z2+1 Example: The function f(z) = z2−1 is a rational function. The denomina- tor will be zero precisely when z2 = 1. We know that every non-zero complex number has n distinct nth roots, and thus there will be two points at which the denominator is zero. -
MATH 305 Complex Analysis, Spring 2016 Using Residues to Evaluate Improper Integrals Worksheet for Sections 78 and 79
MATH 305 Complex Analysis, Spring 2016 Using Residues to Evaluate Improper Integrals Worksheet for Sections 78 and 79 One of the interesting applications of Cauchy's Residue Theorem is to find exact values of real improper integrals. The idea is to integrate a complex rational function around a closed contour C that can be arbitrarily large. As the size of the contour becomes infinite, the piece in the complex plane (typically an arc of a circle) contributes 0 to the integral, while the part remaining covers the entire real axis (e.g., an improper integral from −∞ to 1). An Example Let us use residues to derive the formula p Z 1 x2 2 π 4 dx = : (1) 0 x + 1 4 Note the somewhat surprising appearance of π for the value of this integral. z2 First, let f(z) = and let C = L + C be the contour that consists of the line segment L z4 + 1 R R R on the real axis from −R to R, followed by the semi-circle CR of radius R traversed CCW (see figure below). Note that C is a positively oriented, simple, closed contour. We will assume that R > 1. Next, notice that f(z) has two singular points (simple poles) inside C. Call them z0 and z1, as shown in the figure. By Cauchy's Residue Theorem. we have I f(z) dz = 2πi Res f(z) + Res f(z) C z=z0 z=z1 On the other hand, we can parametrize the line segment LR by z = x; −R ≤ x ≤ R, so that I Z R x2 Z z2 f(z) dz = 4 dx + 4 dz; C −R x + 1 CR z + 1 since C = LR + CR. -
MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009)
MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009) Required Topics • Multivariable functions • 3-D space o Distance o Equations of Spheres o Graphing Planes Spheres Miscellaneous function graphs Sections and level curves (contour diagrams) Level surfaces • Planes o Equations of planes, in different forms o Equation of plane from three points o Linear approximation and tangent planes • Partial Derivatives o Compute using the definition of partial derivative o Compute using differentiation rules o Interpret o Approximate, given contour diagram, table of values, or sketch of sections o Signs from real-world description o Compute a gradient vector • Vectors o Arithmetic on vectors, pictorially and by components o Dot product compute using components v v v v v ⋅ w = v w cosθ v v v v u ⋅ v = 0⇔ u ⊥ v (for nonzero vectors) o Cross product Direction and length compute using components and determinant • Directional derivatives o estimate from contour diagram o compute using the lim definition t→0 v v o v compute using fu = ∇f ⋅ u ( u a unit vector) • Gradient v o v geometry (3 facts, and understand how they follow from fu = ∇f ⋅ u ) Gradient points in direction of fastest increase Length of gradient is the directional derivative in that direction Gradient is perpendicular to level set o given contour diagram, draw a gradient vector • Chain rules • Higher-order partials o compute o mixed partials are equal (under certain conditions) • Optimization o Locate and -
Lecture Notes in Mathematics
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universit~it und Max-Planck-lnstitut for Mathematik, Bonn - voL 5 Adviser: E Hirzebruch 1111 Arbeitstagung Bonn 1984 Proceedings of the meeting held by the Max-Planck-lnstitut fur Mathematik, Bonn June 15-22, 1984 Edited by E Hirzebruch, J. Schwermer and S. Suter I IIII Springer-Verlag Berlin Heidelberg New York Tokyo Herausgeber Friedrich Hirzebruch Joachim Schwermer Silke Suter Max-Planck-lnstitut fLir Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3, Federal Republic of Germany AMS-Subject Classification (1980): 10D15, 10D21, 10F99, 12D30, 14H10, 14H40, 14K22, 17B65, 20G35, 22E47, 22E65, 32G15, 53C20, 57 N13, 58F19 ISBN 3-54045195-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15195-8 Springer-Verlag New York Heidelberg Berlin Tokyo CIP-Kurztitelaufnahme der Deutschen Bibliothek. Mathematische Arbeitstagung <25. 1984. Bonn>: Arbeitstagung Bonn: 1984; proceedings of the meeting, held in Bonn, June 15-22, 1984 / [25. Math. Arbeitstagung]. Ed. by E Hirzebruch ... - Berlin; Heidelberg; NewYork; Tokyo: Springer, 1985. (Lecture notes in mathematics; Vol. 1t11: Subseries: Mathematisches I nstitut der U niversit~it und Max-Planck-lnstitut for Mathematik Bonn; VoL 5) ISBN 3-540-t5195-8 (Berlin...) ISBN 0-387q5195-8 (NewYork ...) NE: Hirzebruch, Friedrich [Hrsg.]; Lecture notes in mathematics / Subseries: Mathematischee Institut der UniversitAt und Max-Planck-lnstitut fur Mathematik Bonn; HST This work ts subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. -
Arxiv:1207.1472V2 [Math.CV]
SOME SIMPLIFICATIONS IN THE PRESENTATIONS OF COMPLEX POWER SERIES AND UNORDERED SUMS OSWALDO RIO BRANCO DE OLIVEIRA Abstract. This text provides very easy and short proofs of some basic prop- erties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor’s series, Prin- ciple of Identity, Principle of Isolated Zeros, and Binomial Series). This is done by simplifying the usual presentation of unordered sums of a (countable) family of complex numbers. All the proofs avoid formal power series, double series, iterated series, partial series, asymptotic arguments, complex integra- tion theory, and uniform continuity. The use of function continuity as well as epsilons and deltas is kept to a mininum. Mathematics Subject Classification: 30B10, 40B05, 40C15, 40-01, 97I30, 97I80 Key words and phrases: Power Series, Multiple Sequences, Series, Summability, Complex Analysis, Functions of a Complex Variable. Contents 1. Introduction 1 2. Preliminaries 2 3. Absolutely Convergent Series and Commutativity 3 4. Unordered Countable Sums and Commutativity 5 5. Unordered Countable Sums and Associativity. 9 6. Sum of a Double Sequence and The Cauchy Product 10 7. Power Series - Algebraic Properties 11 8. Power Series - Analytic Properties 14 References 17 arXiv:1207.1472v2 [math.CV] 27 Jul 2012 1. Introduction The objective of this work is to provide a simplification of the theory of un- ordered sums of a family of complex numbers (in particular, for a countable family of complex numbers) as well as very easy proofs of basic operations and properties concerning complex power series, such as addition, scalar multiplication, multipli- cation, division, rearrangement, composition, differentiation (see Apostol [2] and Vyborny [21]), Taylor’s formula, principle of isolated zeros, uniqueness, principle of identity, and binomial series. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0. -
Analytic Vortex Solutions on Compact Hyperbolic Surfaces
Analytic vortex solutions on compact hyperbolic surfaces Rafael Maldonado∗ and Nicholas S. Mantony Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, U.K. August 27, 2018 Abstract We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 1 Introduction Consider the Abelian-Higgs field theory on a background surface M with metric ds2 = Ω(x; y)(dx2+dy2). At critical coupling the static energy functional satisfies a Bogomolny bound 2 1 Z B Ω 2 E = + jD Φj2 + 1 − jΦj2 dxdy ≥ πN; (1) 2 2Ω i 2 where the topological invariant N (the `vortex number') is the number of zeros of Φ counted with multiplicity [1]. In the notation of [2] we have taken e2 = τ = 1. Equality in (1) is attained when the fields satisfy the Bogomolny vortex equations, which are obtained by completing the square in (1). In complex coordinates z = x + iy these are 2 Dz¯Φ = 0;B = Ω (1 − jΦj ): (2) This set of equations has smooth vortex solutions. As we explain in section 2, analytical results are most readily obtained when M is hyperbolic, having constant negative cur- vature K = −1, a case which is of interest in its own right due to the relation between arXiv:1502.01990v1 [hep-th] 6 Feb 2015 hyperbolic vortices and SO(3)-invariant instantons, [3]. -
Defining Physics at Imaginary Time: Reflection Positivity for Certain
Defining physics at imaginary time: reflection positivity for certain Riemannian manifolds A thesis presented by Christian Coolidge Anderson [email protected] (978) 204-7656 to the Department of Mathematics in partial fulfillment of the requirements for an honors degree. Advised by Professor Arthur Jaffe. Harvard University Cambridge, Massachusetts March 2013 Contents 1 Introduction 1 2 Axiomatic quantum field theory 2 3 Definition of reflection positivity 4 4 Reflection positivity on a Riemannian manifold M 7 4.1 Function space E over M ..................... 7 4.2 Reflection on M .......................... 10 4.3 Reflection positive inner product on E+ ⊂ E . 11 5 The Osterwalder-Schrader construction 12 5.1 Quantization of operators . 13 5.2 Examples of quantizable operators . 14 5.3 Quantization domains . 16 5.4 The Hamiltonian . 17 6 Reflection positivity on the level of group representations 17 6.1 Weakened quantization condition . 18 6.2 Symmetric local semigroups . 19 6.3 A unitary representation for Glor . 20 7 Construction of reflection positive measures 22 7.1 Nuclear spaces . 23 7.2 Construction of nuclear space over M . 24 7.3 Gaussian measures . 27 7.4 Construction of Gaussian measure . 28 7.5 OS axioms for the Gaussian measure . 30 8 Reflection positivity for the Laplacian covariance 31 9 Reflection positivity for the Dirac covariance 34 9.1 Introduction to the Dirac operator . 35 9.2 Proof of reflection positivity . 38 10 Conclusion 40 11 Appendix A: Cited theorems 40 12 Acknowledgments 41 1 Introduction Two concepts dominate contemporary physics: relativity and quantum me- chanics. They unite to describe the physics of interacting particles, which live in relativistic spacetime while exhibiting quantum behavior. -
Real Proofs of Complex Theorems (And Vice Versa)
REAL PROOFS OF COMPLEX THEOREMS (AND VICE VERSA) LAWRENCE ZALCMAN Introduction. It has become fashionable recently to argue that real and complex variables should be taught together as a unified curriculum in analysis. Now this is hardly a novel idea, as a quick perusal of Whittaker and Watson's Course of Modern Analysis or either Littlewood's or Titchmarsh's Theory of Functions (not to mention any number of cours d'analyse of the nineteenth or twentieth century) will indicate. And, while some persuasive arguments can be advanced in favor of this approach, it is by no means obvious that the advantages outweigh the disadvantages or, for that matter, that a unified treatment offers any substantial benefit to the student. What is obvious is that the two subjects do interact, and interact substantially, often in a surprising fashion. These points of tangency present an instructor the opportunity to pose (and answer) natural and important questions on basic material by applying real analysis to complex function theory, and vice versa. This article is devoted to several such applications. My own experience in teaching suggests that the subject matter discussed below is particularly well-suited for presentation in a year-long first graduate course in complex analysis. While most of this material is (perhaps by definition) well known to the experts, it is not, unfortunately, a part of the common culture of professional mathematicians. In fact, several of the examples arose in response to questions from friends and colleagues. The mathematics involved is too pretty to be the private preserve of specialists. -
Mathematisches Forschungsinstitut Oberwolfach Reflection Positivity
Mathematisches Forschungsinstitut Oberwolfach Report No. 55/2017 DOI: 10.4171/OWR/2017/55 Reflection Positivity Organised by Arthur Jaffe, Harvard Karl-Hermann Neeb, Erlangen Gestur Olafsson, Baton Rouge Benjamin Schlein, Z¨urich 26 November – 2 December 2017 Abstract. The main theme of the workshop was reflection positivity and its occurences in various areas of mathematics and physics, such as Representa- tion Theory, Quantum Field Theory, Noncommutative Geometry, Dynamical Systems, Analysis and Statistical Mechanics. Accordingly, the program was intrinsically interdisciplinary and included talks covering different aspects of reflection positivity. Mathematics Subject Classification (2010): 17B10, 22E65, 22E70, 81T08. Introduction by the Organisers The workshop on Reflection Positivity was organized by Arthur Jaffe (Cambridge, MA), Karl-Hermann Neeb (Erlangen), Gestur Olafsson´ (Baton Rouge) and Ben- jamin Schlein (Z¨urich) during the week November 27 to December 1, 2017. The meeting was attended by 53 participants from all over the world. It was organized around 24 lectures each of 50 minutes duration representing major recent advances or introducing to a specific aspect or application of reflection positivity. The meeting was exciting and highly successful. The quality of the lectures was outstanding. The exceptional atmosphere of the Oberwolfach Institute provided the optimal environment for bringing people from different areas together and to create an atmosphere of scientific interaction and cross-fertilization. In particular, people from different subcommunities exchanged ideas and this lead to new col- laborations that will probably stimulate progress in unexpected directions. 3264 Oberwolfach Report 55/2017 Reflection positivity (RP) emerged in the early 1970s in the work of Osterwalder and Schrader as one of their axioms for constructive quantum field theory ensuring the equivalence of their euclidean setup with Wightman fields. -
Partial Differential Equations
CALENDAR OF AMS MEETINGS THIS CALENDAR lists all meetings which have been approved by the Council pnor to the date this issue of the Nouces was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri· can Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and second announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meet ing. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society in Providence. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for ab stracts submitted for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions. MEETING ABSTRACT NUMBER DATE PLACE DEADLINE ISSUE 778 June 20-21, 1980 Ellensburg, Washington APRIL 21 June 1980 779 August 18-22, 1980 Ann Arbor, Michigan JUNE 3 August 1980 (84th Summer Meeting) October 17-18, 1980 Storrs, Connecticut October 31-November 1, 1980 Kenosha, Wisconsin January 7-11, 1981 San Francisco, California (87th Annual Meeting) January 13-17, 1982 Cincinnati, Ohio (88th Annual Meeting) Notices DEADLINES ISSUE NEWS ADVERTISING June 1980 April 18 April 29 August 1980 June 3 June 18 Deadlines for announcements intended for the Special Meetings section are the same as for News. -
Augustin-Louis Cauchy - Wikipedia, the Free Encyclopedia 1/6/14 3:35 PM Augustin-Louis Cauchy from Wikipedia, the Free Encyclopedia
Augustin-Louis Cauchy - Wikipedia, the free encyclopedia 1/6/14 3:35 PM Augustin-Louis Cauchy From Wikipedia, the free encyclopedia Baron Augustin-Louis Cauchy (French: [oɡystɛ̃ Augustin-Louis Cauchy lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic Cauchy around 1840. Lithography by Zéphirin principle of the Belliard after a painting by Jean Roller. generality of algebra exploited by earlier Born 21 August 1789 authors. He defined Paris, France continuity in terms of Died 23 May 1857 (aged 67) infinitesimals and gave Sceaux, France several important Nationality French theorems in complex Fields Mathematics analysis and initiated the Institutions École Centrale du Panthéon study of permutation École Nationale des Ponts et groups in abstract Chaussées algebra. A profound École polytechnique mathematician, Cauchy Alma mater École Nationale des Ponts et exercised a great Chaussées http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy Page 1 of 24 Augustin-Louis Cauchy - Wikipedia, the free encyclopedia 1/6/14 3:35 PM influence over his Doctoral Francesco Faà di Bruno contemporaries and students Viktor Bunyakovsky successors. His writings Known for See list cover the entire range of mathematics and mathematical physics. "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)."[1] Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict Bourbon royalist, and a close associate of the Jesuit order.