Tài liệu Freitage.busamr.complexanalysis

Eberhard Freitag Rolf Busam Complex Analysis ABC Professor Dr. Eberhard Freitag Dr. Rolf Busam Translator Dr. Dan Fulea Faculty of Mathematics Institute of Mathematics University of Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: [email protected] [email protected] Faculty of Mathematics Institute of Mathematics University of Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: [email protected] Mathematics Subject Classification (2000): 30-01, 11-01, 11F11, 11F66, 11M45, 11N05, 30B50, 33E05 Library of Congress Control Number: 2005930226 ISBN-10 3-540-25724-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25724-0 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005
Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11396024 40/TechBooks 543210 In Memoriam Hans Maaß (1911–1992) Preface to the English Edition This book is a translation of the forthcoming fourth edition of our German book “Funktionentheorie I” (Springer 2005). The translation and the LATEX files have been produced by Dan Fulea. He also made a lot of suggestions for improvement which influenced the English version of the book. It is a pleasure for us to express to him our thanks. We also want to thank our colleagues Diarmuid Crowley, Winfried Kohnen and Jörg Sixt for useful suggestions concerning the translation. Over the years, a great number of students, friends, and colleagues have contributed many suggestions and have helped to detect errors and to clear the text. The many new applications and exercises were completed in the last decade to also allow a partial parallel approach using computer algebra systems and graphic tools, which may have a fruitful, powerful impact especially in complex analysis. Last but not least, we are indebted to Clemens Heine (Springer, Heidelberg), who revived our translation project initially started by Springer, New York, and brought it to its final stage. Heidelberg, Easter 2005 Eberhard Freitag Rolf Busam Contents I Differential Calculus in the Complex Plane C . . . . . . . . . . . . I.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 Convergent Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . I.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.4 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.5 The Cauchy–Riemann Differential Equations . . . . . . . . . . 9 9 24 36 42 48 II Integral Calculus in the Complex Plane C . . . . . . . . . . . . . . . . II.1 Complex Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.2 The Cauchy Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . II.3 The Cauchy Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . 71 72 79 94 III Sequences and Series of Analytic Functions, the Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 III.1 Uniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 III.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 III.3 Mapping Properties for Analytic Functions . . . . . . . . . . . . . . 126 III.4 Singularities of Analytic Functions . . . . . . . . . . . . . . . . . . . . . 136 III.5 Laurent Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A Appendix to III.4 and III.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 III.6 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 III.7 Applications of the Residue Theorem . . . . . . . . . . . . . . . . . . . 174 IV Construction of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . 195 IV.1 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 IV.2 The Weierstrass Product Formula . . . . . . . . . . . . . . . . . . . 214 IV.3 The Mittag–Leffler Partial Fraction Decomposition . . . 223 IV.4 The Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . 228 A Appendix : The Homotopical Version of the Cauchy Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 B Appendix : The Homological Version of the Cauchy Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 X Contents C Appendix : Characterizations of Elementary Domains . . . . 249 V Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 V.1 The Liouville Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 A Appendix to the Definition of the Periods Lattice . . . . . . . . 265 V.2 The Weierstrass ℘-function . . . . . . . . . . . . . . . . . . . . . . . . . 267 V.3 The Field of Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 274 A Appendix to Sect. V.3 : The Torus as an Algebraic Curve . 279 V.4 The Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 V.5 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 V.6 Abel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 V.7 The Elliptic Modular Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 V.8 The Modular Function j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 VI Elliptic Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 VI.1 The Modular Group and Its Fundamental Region . . . . . . . . 328 VI.2 The k/12-formula and the Injectivity of the j-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 VI.3 The Algebra of Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . 345 VI.4 Modular Forms and Theta Series . . . . . . . . . . . . . . . . . . . . . . . 348 VI.5 Modular Forms for Congruence Groups . . . . . . . . . . . . . . . . . 362 A Appendix to VI.5 : The Theta Group . . . . . . . . . . . . . . . . . . . 374 VI.6 A Ring of Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 VII Analytic Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 VII.1 Sums of Four and Eight Squares . . . . . . . . . . . . . . . . . . . . . . . 392 VII.2 Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 VII.3 Dirichlet Series with Functional Equations . . . . . . . . . . . . 418 VII.4 The Riemann ζ-function and Prime Numbers . . . . . . . . . . . 431 VII.5 The Analytic Continuation of the ζ-function . . . . . . . . . . . . . 439 VII.6 A Tauberian Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 VIII Solutions to the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 VIII.1 Solutions to the Exercises of Chapter I . . . . . . . . . . . . . . . . . . 459 VIII.2 Solutions to the Exercises of Chapter II . . . . . . . . . . . . . . . . . 471 VIII.3 Solutions to the Exercises of Chapter III . . . . . . . . . . . . . . . . 476 VIII.4 Solutions to the Exercises of Chapter IV . . . . . . . . . . . . . . . . 488 VIII.5 Solutions to the Exercises of Chapter V . . . . . . . . . . . . . . . . . 496 VIII.6 Solutions to the Exercises of Chapter VI . . . . . . . . . . . . . . . . 505 VIII.7 Solutions to the Exercises of Chapter VII . . . . . . . . . . . . . . . 513 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 Symbolic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 Introduction The complex numbers have their historical origin in the 16th century when they were created during attempts to solve algebraic equations. G. Cardano √ (1545) has already introduced formal expressions as for instance 5 ± −15, in order to express solutions of quadratic and cubic equations. Around 1560 R. Bombelli computed systematically using such expressions and found 4 as a solution of the equation x3 = 15x + 4 in the disguised form

√ √ 3 3 4 = 2 + −121 + 2 − −121 . Also in the work of G.W. Leibniz (1675) one can find equations of this kind, e.g.

√ √ √ 1 + −3 + 1 − −3 = 6 . √ In the year 1777 L. Euler introduced the notation i = −1 for the imaginary unit. The terminology “complex number” is due to C.F. Gauss (1831). The rigorous introduction of complex numbers as pairs of real numbers goes back to W.R. Hamilton (1837). Sometimes it is already advantageous to introduce and make use of complex numbers in real analysis. One should for example think of the integration of rational functions, which is based on the partial fraction decomposition, und therefore on the Fundamental Theorem of Algebra: Over the field of complex numbers any polynomial decomposes as a product of linear factors. Another example for the fruitful use of complex numbers is related to Fourier series. Following Euler (1748) one can combine the real angular functions sine and cosine, and obtain the “exponential function” eix := cos x + i sin x . Then the addition theorems for sine and cosine reduce to the simple formula 2 Introduction ei(x+y) = eix eiy . In particular,  eix n = einx holds for all integers n . The Fourier series of a sufficiently smooth function f , defined on the real line, with period 1, can be written in terms of such expressions as f (x) = ∞  an e2πinx . n=−∞ Here it is irrelevant whether f is real or complex valued. In these examples the complex numbers serve as useful, but ultimatively dispensable tools. New aspects come into play when we consider complex valued functions depending on a complex variable, that is when we start to study functions f : D → C with two-dimensional domains D systematically. The dimension two is ensured when we restrict to open domains of definition D ⊂ C. Analogously to the situation in real analysis one introduces the notion of complex differentiability by requiring the existence of the limit lim f  (a) := z→a z=a f (z) − f (a) z−a for all a ∈ D. It turns out that this notion behaves much more drastically then real differentiability. We will show for instance that a (first order) complex differentiable function is automatically arbitrarily often complex differentiable. We will see more, namely that complex differentiable functions can always be developed locally as power series. For this reason, complex differentiable functions (defined on open domains) are also called analytic functions. “Complex analysis” is the theory of such analytic functions. Many classical functions from real analysis can be analytically extended to complex analysis. It turns out that these extensions are unique, as for instance in the case ex+iy := ex eiy . From the relation e2πi = 1 it follows that the complex exponential function is periodic with the purely imaginary period 2πi. This observation is fundamental for the complex analysis. As a consequence one can observe further phenomena: 1. The complex logarithm cannot be introduced as the unique inverse function of the exponential function in a natural way. It is a priori determined only up to a multiple of 2πi. Introduction 3 2. The function 1/z (z = 0) does not have any primitive in the punctured complex plane. A related fact is the following: the path integral of 1/z with respect to a circle line centered in the origin and oriented anticlockwise yields the non-zero value  |z|=r 1 dz = 2πi z (r > 0) . Central results of complex analysis, like e.g. the Residue Theorem, are nothing but a highly generalized version of these statements. Real functions often show their true nature first after considering their analytic extensions. For instance, in the real theory it is not directly transparent why the power series representation 1 = 1 − x2 + x4 − x6 ± · · · 1 + x2 is valid only for |x| < 1. In the complex theory this phenomenon becomes more understandable, simply because the considered function has singularities in ±i. Then its power series representation is valid in the biggest open disk excluding the singularities, namely the unit disk. In the real theory it is also hard to understand why the Taylor series around 0 of the C ∞ function  2 e−1/x , x = 0 , f (x) = 0, x=0, converges for all x ∈ R, but does not represent the function in any point other than zero. In the complex theory this phenomenon becomes understandable, 2 because the function e−1/z has an essential singularity in zero. Less trivial examples are more impressive. Here, one should mention the Riemann ζ-function ∞  n−s , ζ(s) = n=1 which will be extensively studied in the last chapter of the book as a function of the complex variable traditionally denoted by s using the methods of complex analysis, which will be presented throughout the preceeding chapters. From the analytical properties of the ζ-function we will deduce the Prime Number Theorem. Riemann’s celebrated work on the ζ function [Ri2] is a brilliant example for the thesis he already presented eight years in advance in his dissertation [Ri1] 4 Introduction “Die Einführung der complexen Grössen in die Mathematik hat ihren Ursprung und nächsten Zweck in der Theorie einfacher durch Grössenoperationen ausgedrückter Abhängigkeitsgesetze zwischen veränderlichen Grössen. Wendet man nämlich diese Abhängigkeitsgesetze in einem erweiterten Umfange an, indem man den veränderlichen Grössen, auf welche sie sich beziehen, complexe Werthe giebt, so tritt eine sonst versteckt bleibende Harmonie und Regelmäßigkeit hervor.” In translation: “The introduction of complex variables in mathematics has its origin and its proximate purpose in the theory of simple dependency rules for variables expressed by variable operations. If one applies these dependency rules in an extended manner by associating complex values to the variables referred to by these rules, then there emerges an otherwise hidden harmony and regularity.” Complex numbers are not only useful auxiliary tools, but even indispensable in many applications, like e.g physics and other sciences: The commutation relations in quantum mechanics for impulse and coordinate operators h I, and respectively the Schrödinger equation H Ψ (x, t) = P Q − QP = 2πi h i 2π ∂t Ψ (x, t) contain the imaginary unit i. Here, H is the Hamilton operator. Already before the appearance of the first German edition there existed a series of good textbooks on complex analysis, so that a new attempt in this direction needed a special justification. The main idea of this book, and of a second forthcoming volume was to give an extensive description of classical complex analysis, whereby “classical” means that sheaf theoretical and cohomological methods are omitted. Obviously, it was not possible to include all material that can be considered as classical complex analysis. If somebody is especially interested in the value distribution theory, or in applications of conformal maps, then she or he will be quickly disappointed and might put this book aside. The line pursued in this text can be described by keywords as follows: The first four chapters contain an introduction to complex analysis, roughly corresponding to a course “complex analysis I” (four hours each week). Here, the fundamental results of complex analysis are treated. After the foundations of the theory of analytic functions have been laid, we proceed to the theory of elliptic functions, then to elliptic modular functions – and after some excursions to analytic number theory – in a second volume we move on to Riemann surfaces, the local theory of analytic functions of several variables, to abelian functions, and finally we discuss modular functions for several variables. Great importance is attached to completeness in the sense that all required notions and concepts are carefully developed. Except for basics in real analysis and linear algebra, as they are nowadays taught in standard introductory Introduction 5 courses, we do not want to assume anything else in this first book. In a second volume some simple topological concepts will be compiled without proof and subsequently used. We made efforts to introduce as few notions as possible in order to quickly advance to the core of the studied problem. A series of important results will have several proofs. If a special case of a general proposition will be used in an important context, we strived to give a simpler proof for this special case as well. This is in accordance with our philosophy, that a thorough understanding can only be achieved if one turns things around and over and highlights them from different points of view. We hope that this comprehensive presentation will convey a feeling for the way the treated topics are related with each other, and for their roots. Attempts like this are not new. Our text was primarily modelled on the lectures of H. Maass, to whom we both owe our education in complex analysis. In the same breath, we would also like to mention the elaborations of the lectures of C.L. Siegel. Both sources are attempts to trace a great historical epoch, which is inseparably connected with the names of A.-L. Cauchy, N.H. Abel, C.G.J. Jacobi, B. Riemann and K. Weierstrass, and to introduce results developed by themselves. Our objectives and contents are very similar to both mentioned examples, however methodically our approach differs in many aspects. This will emerge especially in the second book, where we will again dwell on the differences. The present volume presents a comparatively simple introduction to the complex analysis in one variable. The content corresponds to a two semester course with accompanying seminars. The first three chapters contain the standard material up to the Residue Theorem, which must be covered in any introduction. In the fourth chapter – we rank it among the introductory lectures – we treat problems that are less obligatory. We present the gamma function in detail in order to illustrate the learned methods by a beautiful example. We further focus on the Theorems of Weierstrass and Mittag–Leffler about the construction of analytic functions with prescribed zeros and poles. Finally, as a highlight, we prove the Riemann Mapping Theorem which claims that any proper subdomain of the complex plane C “without holes” is conformally equivalent to the unit disk. Only now, in an appendix to chapter IV we will treat the question of simply connectedness and we will give different equivalent characterizations for simply connected domains, which, roughly speaking, are domains without holes. In this context different versions, namely the homotopical and homological versions, of the Cauchy Integral Formula will be deduced. However fruitful these results are for insights into the theory, and however important they are for later developments in the book, they have minor significance in order to develop the standard repertoire of complex analysis. Among simply connected domains we will only need star-shaped domains (and some 6 Introduction domains that can be constructed from star-shaped domains). Consequently one needs the Cauchy Integral Theorem merely for star-shaped domains, which can be reduced to triangular paths by an idea of A. Dinghas without any topological complications. Therefore we will deliberately content ourselves with star-shaped domains a longer time and we will avoid the notion of simply connectedness. There is a price to be paid for this approach, namely that we have to introduce the concept of an elementary domain. By definition it is a domain where the Cauchy Integral Theorem holds without exception. We will be content to know that star-shaped domains are elementary domains, and postpone their final topological identification to the appendix of the fourth chapter, where this is done in an extensive but basically simple manner. For the sake of a lucid methodology we have postponed this to a possibly later point. In principle it is possible to proceed without it in this first volume. The subject of the fifth chapter is the theory of elliptic functions, i.e. meromorphic functions with two linearly independent periods. Historically these functions appeared as inverse functions of certain elliptic integrals, as for example the integral  x 1 √ y= dt . 1 − t4 ∗ It is easier to follow the converse approach, and to obtain the elliptic integrals as a byproduct of the impressively beautiful and simple theory of elliptic functions. One of the great achievements of complex analysis is the simple and transparent construction of the theory of elliptic integrals. As usual nowadays, we will choose the Weierstrass approach to the ℘-function. In connection with Abel’s Theorem we will also give a short account of the older approach via the Jacobi theta function. We finish the fifth chapter by proving that any complex number is the absolute invariant of a period lattice. This fact is needed to show, that one indeed obtains any elliptic integral of the first kind as the inverse function of an elliptic function. At this point the elliptic modular function j(τ ) appears. As simple as this theory may be, it remains highly obscure how an elliptic integral gives rise to a period lattice, and thus to an elliptic function. In a second volume, the more complicated theory of Riemann surfaces will allow a deeper insight. In the sixth chapter we will further systematically introduce – as a continuation of the end of fifth chapter – the theory of modular functions and modular forms. In the center of our interest will be structural results, the detection of all modular forms for the full modular group, and for certain subgroups. Other important examples of modular forms are Eisenstein and theta series, which have arithmetical significance. One of the most beautiful applications of complex analysis can be found in analytical number theory. For instance, the Fourier coefficients of modular Introduction 7 forms have arithmetic meaning: The Fourier coefficients of the theta series are representation numbers associated to quadratic forms, those of the Eisenstein are sums of divisor powers. Identities between modular forms worked out in complex analysis then give rise to number theoretical applications. Following Jacobi we determine the number of representations of a natural number as a sum of four and respectively eight squares of integers. The necessary complex analysis identities will be deduced independently from the structure theorems for modular forms. A special section was dedicated to Hecke’s theory on the connection between Fourier series satisfying a transformation rule with respect to the transformation and Dirichlet series satisfying a functional equation. This theory is a brige between modular functions and Dirichlet series. However, the theory of Hecke operators will not be discussed, merely in the exercises we will go into it. Afterwards we will concentrate in detail on the most famous among the Dirichlet series, the Riemann ζ-function. As a classical application we will give a complete proof of the Prime Number Theorem with a weak estimate for the error term. In all chapters there are numerous exercises, easy ones at the beginning, but with increasing chapter number there will also be harder exercises complementing the main text. Occasionally the exercises will require notions from topology or algebra not introduced in the text. The present material originates in the standard lectures for mathematicians and physicists at the Ruprecht–Karls University of Heidelberg. Heidelberg, Easter April 2005 Eberhard Freitag Rolf Busam I Differential Calculus in the Complex Plane C In this chapter we shall first give an introduction to complex numbers and their topology. In doing so we shall assume that this is not the first time the reader has encountered the system C of complex numbers. The same assumption is made for topological notions in C (convergence, continuity etc.). For this reason we shall not dwell on these matters. In Sect. I.4 we introduce the notion of complex derivative. One can begin reading directly with this section if one is already sufficiently familiar with the algebra, geometry and topology of complex numbers. In Sect. I.5 the relationship between real differentiability and complex differentiability will be covered (the Cauchy–Riemann differential equations). The story of the complex numbers from their early beginnings in the 16th century until their eventual full acceptance in the course of the 19th century — probably in the end thanks to the scientific authority of C.F. Gauss — as well as the lengthy period of uncertainty and unclarity about them, is an impressive example of the history of mathematics. The historically interested reader should read [Re2]. For more historical remarks about the complex numbers see also [CE]. I.1 Complex Numbers It is well known that not every polynomial with real coefficients has a real root (or zero), e.g. the polynomial P (x) = x2 + 1 . There is, for instance, no real number x with x2 + 1 = 0. If, nonetheless, one wishes to arrange that this and similar equations have solutions, this can only be achieved if one goes on to make an extension of R, in which such solutions exist. One extends the field R of real numbers to the field C of the complex numbers. In fact, in this field, every polynomial equation, not just the equation 10 I Differential Calculus in the Complex Plane C x2 + 1 = 0, has solutions. This is the statement of the “Fundamental Theorem of Algebra”. Theorem I.1.1 There exists a field C with the following properties: (1) The field R of real numbers is a subfield of C, i.e. R is a subset of C, and addition and multiplication in R are the restrictions to R of the addition and multiplication in C. (2) The equation X2 + 1 = 0 has exactly two solutions in C. (3) Let i be one of the two solutions; then −i is the other. The map R × R −→ C , (x, y) → x + iy , is a bijection. We call C a field of the complex numbers . (Any other field isomorphic to C is also a field of complex numbers.) Proof. The proof of existence is suggested by (3). One defines on the set C := R × R the following composition laws, (x, y) + (u, v) := (x + u, y + v), (x, y) · (u, v) := (xu − yv, xv + yu) and then first shows that the field axioms hold. These are: (1) The associative laws (z + z  ) + z  = z + (z  + z  ) , (zz  )z  = z(z  z  ) . (2) The commutative laws z + z = z + z , zz  = z  z . (3) The distributive laws z(z  + z  ) = zz  + zz  , (z  + z  )z = z  z + z  z . I.1 Complex Numbers 11 (4) The existence of neutral elements (a) There exists a (unique) element 0 ∈ C with the property z + 0 = z for all z ∈ C . (b) There exists a (unique) element 1 ∈ C with the property z · 1 = z for all z ∈ C and 1 = 0 . (5) The existence of inverse elements (a) For each z ∈ C there exists a (unique) element −z ∈ C with the property z + (−z) = 0 . (b) For each z ∈ C, z = 0, there exists a (unique) element z −1 ∈ C with the property z · z −1 = 1 . Verification of the field axioms The axioms (1) – (3) can be verified by direct calculation. (4) (a) 0 := (0, 0). (b) 1 := (1, 0). (5) (a) −(x, y) := (−x, −y). (b) Assume z = (x, y) = (0, 0). Then x2 + y 2 = 0. A direct calculation shows that  x y , − z −1 := x2 + y 2 x2 + y 2 is the inverse of z. Obviously (a, 0)(x, y) = (ax, ay) , and therefore, in particular, (a, 0)(b, 0) = (ab, 0) . In addition, we have (a, 0) + (b, 0) = (a + b, 0) . Therefore CR := { (a, 0) ; a ∈ R } is a subfield of C, in which the arithmetic is just the same as in R itself. 12 I Differential Calculus in the Complex Plane C More precisely: The map ι : R −→ CR , a → (a, 0) , is an isomorphism of fields. Thus we have constructed a field C, which does not actually contain R, but a field CR which is isomorphic to R. One could then easily construct by set– isomorphic to C which actually does contheoretical manipulations a field C tain the given field R as a subfield. We shall skip this construction and simply identify the real number a with the complex number (a, 0). To simplify matters further we shall use the Notation i := (0, 1) and call i the imaginary unit (L. Euler, 1777). Obviously then (a) i2 = i · i = (0, 1) · (0, 1) = (0 · 0 − 1 · 1, 0 · 1 + 1 · 0) = (−1, 0), (b) (x, y) = (x, 0) + (0, y) = (x, 0) · (1, 0) + (y, 0) · (0, 1) or, written more simply, (b) (x, y) = x + y i = x + iy. (a) i2 = −1, Thus each complex number can be written uniquely in the form z = x + iy with real numbers x and y. Therefore we have proved Theorem I.1.1. 2 It can be shown that a field C is “essentially” uniquely defined by properties (1) – (3) in Theorem I.1.1 (cf. Exercise 13 in I.1). In the unique representation z = x + iy we say x is the real part of z and y is the imaginary part of z. Notation. x = Re (z) and y = Im (z). If Re (z) = 0, then z is said to be purely imaginary. Remark. Note the following essential difference from the field R of real numbers: R is an ordered field, i.e. there is in R a special subset (“positive cone”) P of the so-called “positive elements”, such that the following holds: (1) For each real number a exactly one of the following cases occurs: (a) a ∈ P (b) a = 0 or (c) − a ∈ P . (2) For arbitrary a, b ∈ P , a+b∈P and ab ∈ P . However, it is easy to show that C cannot be ordered, i.e. there is no subset P ⊂ C, for which axioms (1) and (2) hold for any a, b ∈ P . (Else, if such a P would exist, then ±i ∈ P with a suitable choice of ±, thus −1 = (±i)2 ∈ P , and 1 = 12 ∈ P , therefore 0 = −1 + 1 ∈ P . Contradiction.) Passing to the conjugate complex is often useful in working with complex numbers: I.1 Complex Numbers 13 Let z = x + iy, x, y ∈ R. We put z = x − iy and call z the complex conjugate of z. It is easy to check the following arithmetical rules for the conjugation map : C −→ C , z −→ z . Remark I.1.2 For z, w ∈ C there hold: (1) z=z , (2) (3) z±w =z±w , Re z = (z + z)/2 , (4) z ∈ R ⇐⇒ z = z , zw = z · w , Im z = (z − z) / 2i , z ∈ iR ⇐⇒ z = −z . The map : C → C, z → z, is therefore an involutory field automorphism with R as its invariant field. Obviously zz = x2 + y 2 is a nonnegative real number. Definition I.1.3 The absolute value or modulus of a complex number z is defined by  √ |z| := zz = x2 + y 2 . Clearly |z| is the Euclidean distance of z from the origin. We have |z| ≥ 0 and |z| = 0 ⇐⇒ z=0. Remark I.1.4 For z, w ∈ C we have: (1) (2) (3) (4) |z · w| = |z| · |w| , |Re z| ≤ |z| , |Im z| ≤ |z| , |z ± w| ≤ |z| + |w| | |z| − |w| | ≤ |z ± w| (triangle inequality) , (triangle inequality) . 2 By using the formula z z̄ = |z| one also gets a simple expression for the inverse of a complex number z = 0: z −1 = z̄ 2 |z| Example. (1 + i)−1 = . 1−i . 2 Geometric visualization in the Gaussian number plane (1) The addition of complex numbers is just the vector addition of pairs of real numbers: 14 I Differential Calculus in the Complex Plane C Im z+w w z _ z Re (2) z̄ = x − iy results from z = x + iy by reflection through the real axis. (3) A geometrical meaning for the multiplication of complex numbers can be found with the help of polar coordinates. It is known from real analysis that any point (x, y) = (0, 0) can be written in the form (x, y) = r(cos ϕ, sin ϕ) , r>0. In this expression r is uniquely fixed,  r = x2 + y 2 , however, the angle ϕ (measured in radians) is only fixed up to the addition of an integer multiple of 2π.1 If we use the notation R•+ := { x ∈ R; x>0} for the set of positive real numbers, and C• := C \ {0} for the complex plane with the origin removed, then there holds Theorem I.1.5 The map R•+ × R −→ C• , (r, ϕ) → r(cos ϕ + i sin ϕ) , Im z = r (cos ϕ + i sin ϕ) is surjective. i Additional result. From |z |= r r(cos ϕ + i sin ϕ) = r (cos ϕ + i sin ϕ ), ϕ r, r > 0, 1 it follows that r = r and ϕ − ϕ = 2πk , 1 One also says: modulo 2π. k∈Z. Re