Giving rise to the residue theorem and the winding number. If the limit exists, we say that f is complex differentiable at the point z 0. Mcmullen harvard university this course covers some basic material on both the geometric and analytic aspects of complex analysis in one variable. You may have seen something similar in a first course on complex analysis, where the winding number was defined using a contour integral. Then, from this, the idea of contour integration is examined. Important mathematicians associated with complex numbers include euler, gauss, riemann, cauchy, weierstrass, and many more in the 20th century.
Background in real analysis and basic differential topology, and a first course in complex analysis. The argument of a nonzero complex number is only defined modulo \2\pi. This book is based on a firstyear graduate course i gave three times at the university of chicago. The winding number, or index, of a plane closed curve. Chapter 16, on homology versions of cauchys theorem and cauchys residue theorem, linking back to geometric intuition. One of the new features of this edition is that part of the book can be fruitfully used for a semester course for engineering students, who have a good calculus background. Its clear, concise writing style and numerous applications make the basics easily accessible to students, and serves as an excellent resource for selfstudy. From real to complex analysis springer undergraduate.
Most topics are covered thoroughly, though certain more complicated subjects such as winding number are left out for simplicity. In anticipation of the argument principle, we study the winding number of a closed rectifiable curve. The first half, more or less, can be used for a onesemester course addressed to undergraduates. Transfer of the theorems on integration argument and winding numbers. In this video we introduce a special integer called the winding number. Within each chapter there were a number of sections each containing some descriptive material, theorems with proofs, some examples and each ended with a. The final two chapters of the book take us back to algebraic topology. Instead, basic complex analysis can be usefully read by nonmaths majors, especially those in physics and engineering. A continuous argument of on is a continuous realvalued function on that for each is an argument of, i. This new work presents a muchneeded modern treatment of the subject, incorporating the latest developments and providing a rigorous yet accessible introduction to the concepts and proofs of this fundamental branch of mathematics.
This is a textbook for an introductory course in complex analysis. An introduction to complex function theory undergraduate. We formalise this approximation in the isabelle theorem prover, and provide a tactic to evaluate winding numbers through cauchy indices. It measures the number of times a moving point \p\ goes around a fixed point \q\, provided that \p\ travels on a path that never goes through \q\ and that the final position of \p\ is the same as its starting position. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Plane curves, polar coordinates and winding numbers jstor. This makes it ideal for a first course in complex analysis. Complex analysisintegration over chains wikibooks, open. A concise course in complex analysis and riemann surfaces. Complex analysis in one variable edition 2 by raghavan. As a particular case of the cauchy formula of complex analysis, the winding number is equal to this line. If one looks at the proof of cauchys integral formula, the key point is to determine the integral z 1 z a dz where is a closed path that does not contain a.
Complex analysis spring 2014 cauchy and runge under the same roof. The index or winding number of a curve about a point. Develops cauchys theorem, presenting the powerful and easytouse winding number version contains over 100 sophisticated graphics to provide helpful examples and. Complex numbers, complex functions, elementary functions, integration, cauchys theorem, harmonic functions, series, taylor and laurent series, poles, residues and argument principle. Free complex analysis books download ebooks online textbooks.
The unifying concept that links all these ideas is that of the winding number. This book is designed for students who, having acquired a good working knowledge of the calculus, desire to become acquainted with the theory of functions of a complex variable, and with the principal applications of that theory. Since the complex numbers are not orderedthereisnosimpleanswertothisquestion,asintherealcase. The winding number and the residue theorem springerlink. If the flow is c 2 and the winding number is irrational, then all trajectories of the flow are dense 38. The second half can be used for a second semester, at either level. The main result is a theorem linking the index of a toeplitz operator to a certain winding number. The argument principle in analysis and topology, wiley 1979. Winding numbers, the generalized version of cauchys theorem, moreras. Formal properties of the winding number 8 lecture 2 10 1. This volume is an enlarged edition of a classic textbook on complex analysis. This includes complex differentiability, the cauchyriemann equations, cauchys theorem, taylors and liouvilles theorem, laurent expansions. The student mathematical library publication year 2015. Apr 16, 2016 in anticipation of the argument principle, we study the winding number of a closed rectifiable curve.
Some of the new material has been described in research papers only or appears here for the first time. The winding number is one of the most basic invariants in topology. Somewhat more material has been included than can be. Numerous examples have been given throughout the book, and there is also a set of miscellaneous examples, arranged. Recent decades have seen profound changes in the way we understand complex analysis. Table of contents i complex analysis in one variable. Chapter 15, on infinitesimals in real and complex analysis. The threepart treatment provides geometrical insights by covering angles, basic complex analysis, and interactions with plane topology while focusing on the concepts of angle and winding numbers. Rudolf wegmann, in handbook of complex analysis, 2005. Such functions can be found, and if, are two continuous arguments, then they differ by a constant integral multiple of. Complex analysis serge lang now in its fourth edition, the first part of this book is devoted to the basic material of complex analysis, while the second covers many special topics, such as the riemann mapping theorem, the gamma function, and analytic continuation. Part of the monographs in computer science book series mcs.
Isolated singularities and residue theorem brilliant math. A complex number is a sum of a real number and an imaginary number. In most treatments of complex analysis in which the winding number is. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. For additional information and updates on this book, visit. In complex analysis, the winding number measures the number of times a path counterclockwise winds around a point, while the cauchy index can approximate how the path winds. Winding number and signed curvature crux sancti patris. In addition to the classical material of the first edition it provides a concise and accessible treatment of loewner theory, both in the disc and in the halfplane. They assume the theorem on winding numbers of the notes on. We use complex analysis for mathematics and engineering by john h. Other applications of winding numbers can be found in the very readable book of. In particular, the limit is taken as the complex number z approaches z 0, and must have the same value for any sequence of complex values for z that approach z 0 on the complex plane. Beyond the material of the clarified and corrected original edition, there are three new chapters. A geometric algorithm for winding number computation with.
Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Complex analysis presents a comprehensive and studentfriendly introduction to the important concepts of the subject. Somewhat more material has been included than can be covered at leisure in one term, to give opportunities for the instructor to exercise his taste, and lead the course in whatever direction strikes his fancy at the time. Winding numbers play a very important role throughout complex analysis c. Suppose we want to measure how often a moving object rotates about us ie. Part i takes a critical look at the concept of an angle, illustrating that because a nonzero complex number varies continuously, one may select a. Bifurcation diagrams top and winding numbers bottom for k 0. The second part includes various more specialized topics as the argument principle, the schwarz lemma and hyperbolic. The amount of material in it means it should suit a one semester course very well. Analytic functions, functions of a complex variable, cauchy riemann equations, complex integration, theorems on complex integration, cauchys integral formula, series of complex numbers, residue integration, taylor series, computation of residues at poles, zeros of analytic functions, evaluation of improper integrals.
Starting with the laurant series, which generalises the taylor series. The winding number describes the number of twists performed by the curve about a fixed point. There is more on degree theory in the book by deimling and much of the presentation here follows this reference. Winding numbers and cauchys formula, so i begin by repeating this theorem. Homotopy version of cauchys theorem and cauchy formulae please hand solutions in at the lecture on monday 9th march. Evaluating winding numbers and counting complex roots.
Palkas discussion of the winding number was good, but it would have been even better if he would have elobarted more on how the uniform. The final chapter develops the theory of complex analysis, in which emphasis is placed on the argument, the winding number, and a general homology version of cauchys theorem which is proved using the approach due to dixon. Which of the two possibilities are we to denote by the symbol p w. The winding number is then, in chapter 7, applied to functional analysis, specifically to fredholm and toeplitz operators on a hilbert space. Complex analysisspring 2014 1 winding numbers fau math. Assume we are given a closed contour supported in, and suppose that we are an observer located at the origin. They assume the theorem on winding numbers of the notes on winding numbers and cauchys formula, so i begin by repeating this theorem and consequences here. For flows with a global crosssection on a two dimensional torus, a fundamental invariant is the winding number, or equivalently the rotation number of a return map 90. The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The winding number in topology, geometry, and analysis about this title. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the riemann mapping theorem, the gamma function, and analytic continuation. Prior to looking at this book, i had at least heard of the winding number, but i must confess that it was something of a hazy memory.
Oct 17, 2008 4 rudins ral and complex analysis is not good beacuse it does neglect series developments an integral part of complex. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. Palka could have developed chapter 4, section 4, a little more, but this is a minor gripe. This is the fourth edition of serge langs complex analysis. It is defined via an integral and counts the number of times a curves wraps around a given point. A comprehensive course in analysis, part 2a by barry simon. These notes can be used as an alternative to section 5. Among other things i explained how to compute winding number by counting intersections with a ray with signs. Some other nice books at an advanced undergraduate to beginning graduate level are complex function theory by sarason, complex analysis by lang, functions of one complex variable i by conway, complex analysis by steinshakarchi, and basic complex analysis.
The resulting number is called the winding number of the given closed contour. See books on topology, metric spaces, real and complex analysis, for the details. The complex number system 3 by i and call it the imaginary unit. There will be no dropin session on tuesday 3rd march. As it was addressed to graduate students who intended to specialize in mathematics, i tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of view which relate the subject to other branches of mathematics. Jul 17, 2003 the book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or firstyear graduate level. This time around, the author again discusses some geometry, but also throws in topology, complex analysis, and abstract algebra. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number. The second part includes various more specialized topics as the argument. The reader of this book will learn how the winding number can help us show that every polynomial equation has a root the fundamental theorem of algebra, guarantee a fair division of three objects in space by a single planar cut the ham sandwich theorem, explain why every simple closed curve has an inside and an outside the jordan curve.
This invariant is rational if and only if the flow has periodic orbits. This book definitely prepared me for tackling the dense, theoretical, and exceptional complex analysis by ahlfors. An introduction to complex function theory undergraduate texts in mathematics hardcover december 1. Winding numbers we want to generalise cauchys integral formula to a more general path. In mathematics, the winding number of a closed curve in the plane around a given point is an. John roe, pennsylvania state university, state college, pa.
This book evolved from a series of lectures at the university of sussex and is. Let be an arc in the complex plane and let be a point not on. The original reference for the approach given here, based on analysis, is and dates from 1959. This is a rigorous introduction to the theory of complex functions of one complex variable. The authors have made an effort to present some of the deeper and more interesting results, for example, picards theorems, riemann mapping theorem, runges theorem in the first few chapters.