Complex Variables

PREFACE Take care of the sense, and the sounds will take care of themselves. Alice in Wonderland After having lectured for several decades on complex variables to prospective engineers and physicists, I have definite and, I hope, not unrealistic ideas about their requirements and preferences. Students are students. Since they are required to take several courses, they may study some subjects just for the examination, with the intention of forgetting what they have learned after the examination. Yet they may (and the more intelligent and purposeful students do) ask pertinent questions about the subject: Is it interesting? Can I use it? These questions are fully justified. The instructor of a more advanced branch of mathematics-such as the theory of complex variables-who is a mathematician should try to put himself into the position of his students who are prospective physicists or engineers. Before going into heavy definitions and lengthy proofs, the student wants to satisfy himself that the subject is interesting and useful enough to expend his time and effort on definitions and proofs. Having realized these points as I taught successive generations of students, I adapted my lectures to their standpoint. I evolved the following guidelines. Start from something that is familiar, useful, or challenging-from some connection with the world around us, from the prospect of some application, or from an intuitive idea. Do not be afraid of using colloquial language when it is more suggestive than the conventional precise terminology. In fact, do not introduce technical terms before the student can understand the need for them. Do not enter too early or too far into the heavy details of a proof. First, give a general idea or just the intuitive germ of the proof. Generally, realize that the natural way to learn is to learn by stages. First, we want to see an outline of the subject in order to perceive a concrete source or a possible use. Then, gradually, as we can see use, connections, and interest, we accept more willingly the responsibility of filling in the details. The ideas just stated influenced the organization of this book. Whenever the mathematical context offers a natural opportunity, there are a few words inserted about concrete phenomena or connected general ideas. Before the introduction of a formal definition, the intervening ideas may be previously discussed by examples or in more colloquial language. The proofs emphasize the main points and may leave to the student, now and then, more intermediate points than usual. The most notable departure from the usual is to be found, however, in the "Examples and Comments" that follow most sections and each chapter. There are, of course, the examples of the standard kind that offer an opportunity to practice what has been explained in the text. Yet there is what is not usual-a definite effort to let the student learn the subject by stages and by his own work. Some problems or comments ask the student to reconsider the definitions and proofs given in the text, directing his attention to more subtle points. Other problems introduce new material: a proof different from the one given in the text, or generalizations of (or analogues to) the facts considered, encouraging the student toward further study. Moreover, even the simpler problems, insofar as is possible, are arranged to give the student an opportunity to face a variety of research situations that will awaken his curiosity and initiative. I hope that this book is useful not only to future engineers and physicists but also to future mathematicians. Mathematical concepts and facts gain in vividness and clarity if they are well connected with the world around us and with general ideas, and if we obtain them by our own work through successive stages instead of in one lump. The course presented here has been taught several times at Stanford by me and by my friend and colleague, Gordon E. Latta, who shares my pedagogical ideas. I am grateful to him for sharing the writing which, because of other interests and duties, I was not able to do alone. We may have achieved less than we hoped for at various points and in various respects, yet we still think that this book is a modest concrete contribution to the widespread debate about the lines along which the instruction in the universities should evolve. George Polya Stanford, August 1974 Preface ......Page 4 Hints to the reader ......Page 6 Contents ......Page 10 1.1 Real numbers......Page 16 1.3 Complex numbers as marks in a plane......Page 18 1.5 Addition and subtraction......Page 21 1.6 Multiplication and division......Page 23 1.7 Summary and notation......Page 27 1.8 Conjugate numbers......Page 30 1.9 Vectorial operations......Page 32 1.10 Limits......Page 35 1.11 Additional examples and comments on Chapter one......Page 38 2.1 Extension to the complex domain......Page 50 2.2 Exponential function......Page 51 2.3 Trigonometric functions......Page 53 2.4 Consequences of Euler's theorem......Page 56 2.5 Further applications of Euler's theorem......Page 58 2.6 Logarithms......Page 61 2.7 Powers......Page 64 2.8 Inverse trigonometric functions......Page 67 2.9 General remarks......Page 68 2.10 Complex function of a real variable: kinematic representation......Page 70 2.11 Real functions of a complex variable: graphical representation......Page 72 2.12 Complex functions of a complex variable: graphical representation on two planes......Page 74 2.13. Complex functions of a complex variable: physical representation in one plane......Page 76 2.14 Additional examples and comments on Chapter two......Page 78 3.1 Derivatives......Page 90 3.2 Rules for differentiation......Page 92 3.3 Analytic condition for differentiability: the Cauchy-Riemann equations......Page 95 3.4 Graphical interpretation of differentiability: conformal mapping......Page 100 3.5 Physical interpretation of differentiability: sourceless and irrotational vector-fields......Page 103 3.6 Divergence and curl......Page 106 3.7 Laplace's equation......Page 110 3.8 Analytic functions......Page 112 3.9 Summary and outlook......Page 113 3.10 Additional examples and comments on Chapter three......Page 114 4.1 The stereographic or Ptolemy projection......Page 128 4.2 Properties of the stereographic projection......Page 132 4.3 The bilinear transformation......Page 135 4.4 Properties of the bilinear transformation......Page 137 4.5 The transformation w = z^2......Page 143 4.6 The transformation w = e^z......Page 144 4.7 The Mercator map......Page 146 4.8 Additional examples and comments on Chapter four......Page 147 5.1 Work and flux......Page 158 5.2 The main theorem......Page 161 5.3 Complex line integrals......Page 162 5.4 Rules for integration......Page 167 5.5 The divergence theorem......Page 170 5.6 A more formal proof of Cauchy's theorem......Page 172 5.7 Other forms of Cauchy's theorem......Page 173 5.8 The indefinite integral in the complex domain......Page 177 5.9 Geometric language......Page 182 5.10 Additional examples and comments on Chapter five......Page 184 6.1 Cauchy's integral formula......Page 192 6.2 A first application to the evaluation of definite integrals......Page 195 6.3 Some consequences of the Cauchy formula: higher derivatives......Page 199 6.4 More consequences of the Cauchy formula: the principle of maximum modulus......Page 202 6.5 Taylor's theorem, MacLaurin's theorem......Page 203 6.6 Laurent's theorem......Page 210 6.7 Singularities of analytic functions......Page 217 6.8 The residue theorem......Page 221 6.9 Computation of residues......Page 223 6.10 Evaluation of definite integrals......Page 225 6.11 Additional examples and comments on Chapter six......Page 234 7.1 Analytic continuation......Page 246 7.2 The gamma function......Page 250 7.3 Schwarz reflection principle......Page 255 7.4 The general mapping problem: Riemann's mapping theorem......Page 258 7.5 The Schwarz-Christoffel mapping......Page 260 7.6 A discussion of the Schwarz-Christoffel formula......Page 266 7.7 Degenerate polygons......Page 270 7.8 Additional examples and comments on Chapter seven......Page 275 8.1 The equations of hydrodynamics......Page 280 8.2 The complex potential......Page 282 8.3 Flow in channels: sources, sinks, and dipoles......Page 285 8.4 Flow in channels: conformal mapping......Page 287 8.5 Flows past fixed bodies......Page 293 8.6 Flows with free boundaries......Page 297 9.1 Asymptotic series......Page 306 9.2 Notation and definitions......Page 309 9.3 Manipulating asymptotic series......Page 311 9.4 Laplace's asymptotic formula......Page 317 9.5 Perron's extension of Laplace's formula......Page 322 9.6 The saddle-point method......Page 329 9.10 Additional examples and comments on Chapter nine......Page 337 Index......Page 340 Complex Numbers -- Complex Functions -- Differentiation: Analytic Functions -- Conformal Mapping By Given Functions -- Integration: Cauchy's Theorem -- Cauchy's Integral Formula And Applications -- Conformal Mapping And Analytic Continuation -- Hydrodynamics -- Asymptotic Expansions. [by] George Polya [and] Gordon Latta.