Winding numbers play a very important role throughout complex analysis c. This book is helping me a lot in the mission of reconciliation with math after being taught for many years about how to use many aspects of the complex numbers framework in physics and electrical engineering by intelligent people that knew real analysis well but couldnt explain well the confusing aspects as they surfaced on and on as the topics were presented. Complex analysisintegration over chains wikibooks, open. The amount of material in it means it should suit a one semester course very well. A complex number is a number comprising area land imaginary part. If the flow is c 2 and the winding number is irrational, then all trajectories of the flow are dense 38.
The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of. If the limit exists, we say that f is complexdifferentiable at the point z 0. Ive never had any complex analysis, but id like to teach myself. I learned real analysis with pugh, so id like a complex analysis book on a similar level or maybe higher. This makes it ideal for a first course in complex analysis.
The argument principle in analysis and topology, wiley 1979. In the context of complex analysis, the winding number of a closed curve. In anticipation of the argument principle, we study the winding number of a closed rectifiable curve. This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. Free complex analysis books download ebooks online textbooks. This is a textbook for an introductory course in complex analysis. 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. 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. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. 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 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. This invariant is rational if and only if the flow has periodic orbits. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates.
The winding number describes the number of twists performed by the curve about a fixed. The second part includes various more specialized topics as the argument principle, the schwarz lemma and hyperbolic. Complex analysis springer undergraduate mathematics. The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. 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. Complex numbers, complex functions, elementary functions, integration, cauchys theorem, harmonic functions, series, taylor and laurent series, poles, residues and argument principle.
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