An Introduction to Analytic Functions : With Theoretical Implications [1st ed.] 9783030420840, 9783030420857

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John Sheridan Mac Nerney William E. Kaufman Ryan C. Schwiebert Editors

An Introduction to Analytic Functions With Theoretical Implications Revised Edition

An Introduction to Analytic Functions

John Sheridan Mac Nerney Author

William E. Kaufman Ryan C. Schwiebert Editors

An Introduction to Analytic Functions With Theoretical Implications Revised Edition

John Sheridan Mac Nerney Mathematics (deceased) Houston, TX, USA Editors William E. Kaufman Mathematics (emeritus) Athens, OH, USA

Ryan C. Schwiebert Seegrid Corporation Coraopolis, PA, USA

Previous ‘printings’ by University of North Carolina, Chapel Hill 1959 and University of Houston 1968. Copyright presumably remained with the author, now deceased. Estate passed to the University of Houston and Trinity College.

ISBN 978-3-030-42084-0 ISBN 978-3-030-42085-7 (eBook) https://doi.org/10.1007/978-3-030-42085-7 Mathematics Subject Classification (2020): 30-01 © University of Houston and Trinity College 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To students of mathematics —J.S.M. To Virginia Ann and Mac —W.E.K. For my friend and teacher, Gene Kaufman —R.C.S.

Preface by the Editors

Mac Nerney’s Introduction to Analytic Functions reflects his craftsmanlike choice of presentation, designed to enable the student the opportunity to engage in mathematical problem solving at the highest level. However, modern conventions and notation have evolved and drifted considerably from the standards Mac Nerney used, and this may act as a barrier for students and instructors considering the original text as an option. Accordingly, this revision seeks to preserve Mac Nerney’s intent while adapting terminology and notation to conform with modern usage, which has now become remarkably consistent throughout the mathematical community. Mac Nerney’s approach is unique, and we hope you find the revision to be a refreshing and readable approach to analytic function theory. Athens, OH, USA Coraopolis, PA, USA December 2019

William E. Kaufman Ryan C. Schwiebert

vii

Preface by the Author

The ten chapters of this guidebook are the basis of a two-semester course which my successive beginning classes in function-theory have helped me to develop at the University of North Carolina and the University of Houston. There are two features of this development which I regard as essential in mathematical training: 1. The concise approach wherein attention is focused on correct mathematical procedures as tools with which one builds upon a few simple facts about numbers, and 2. The pedagogical device whereby the traditional student-teacher relationship is gradually replaced by that of investigators concurrently exploring a logical pattern of ideas. Exercises, lemmas, and theorems (together with proofs thereof) are intended for classroom presentation and discussion. Insofar as student activity permits, the instructor expects to play the role of moderator—rather than that of lecturer. The reader is cautioned to be critical, and is expected to detect and correct for himself various misprints and misstatements of fact some initially inadvertent and some deliberate on my part. Occasionally, I have raised the question with a class as to whether or not some of these misstatements should be eliminated for subsequent classes: invariably, I have been answered in the negative. In view of the foregoing comments on pedagogy, one should not expect to find in these notes any proof of any theorem. Likewise, I have chosen neither to include any bibliographical references, nor to label theorems with the names of mathematicians commonly associated therewith. From time to time, after theorems have been proved in class, the instructor may supply such information. It has been my frequent experience, however, that an impressive label on a theorem is likely to have the psychological effect of preventing a student from obtaining a simple contextual proof which is available to him.

ix

x

Preface by the Author

I wish to thank my students, and my former colleagues, especially F. Burton Jones and W. M. Whyburn, and my teacher, H. S. Wall, for their encouragement to me during the development of this course. The University of North Carolina Chapel Hill, NC, USA June 1959 University of Houston Houston, TX, USA August 1968

John Sheridan Mac Nerney

About This Book

What Type of Text Is This, and From Whence Did It Come? J. S. Mac Nerney was a student of H. S. Wall, the author of Creative Mathematics Wall.1 Their teaching styles included elements derived from the Moore Method.2 They both taught by posing problems ranging in difficulty from those one would expect in the usual lectures and texts to others which many might presume to be too difficult for students to solve for themselves. This textbook was produced to facilitate a course including problems at every level, including many that are quite challenging. We believe it can also be used by some independent learners, and in tutorial settings. In this book, the method is not geared toward teaching the direct application of the results (in say, a field like engineering), but rather seeks to foster the development of mathematical thinking. To the extent that a student has success with this mode of learning, he or she could go on to further challenges in mathematics, and quite possibly create original mathematics. A prominent feature of Mac Nerney’s style is a strong do-it-yourself work ethic. As Mac Nerney says in his preface, “one should not expect to find in these notes any proof of any theorem.” They are purposefully omitted to encourage the student to discover them for themselves. This sets it apart from almost every other book on the topic, and indeed from most expositional mathematics books. In a class like Mac Nerney’s, students would volunteer to present their proofs, with classmates free to ask questions and the instructor acting as a moderator. In this way, he cultivated students’ ability to develop and rely on their own mathematical skill.

1 Wall,

H.S.: Creative Mathematics. Classroom resource materials. Mathematical Association of America, Washington (2009). 2 The R.L. Moore Legacy Project: Robert Lee Moore and the Moore Method (2017). http:// legacyrlmoore.org/method.html. xi

xii

About This Book

Mac Nerney and Wall’s implementation of Moore’s style also employed other strategies for isolating student attention. For example, it was made it clear to students that they were not to consult any outside sources such as books or other people—all presentation and discussion was to be conducted only in the classroom. In such a course, the only reading material that was fair game outside of class would have been a single text like this. Some of the insularity manifested itself in Mac Nerney and Wall’s text material also. Wall deliberately used “nonstandard” notation and terminology which made it difficult to connect it to more usual notation in other books—e.g. denoting the natural exponential and logarithmic functions by ‘E’ and ‘L’ respectively. Mac Nerney followed Wall’s practices for notation extensively in the original editions.

How Is This Book Different from Previous Versions? Why Change It? The main changes in this edition are (1) extensive translation to modern symbols and notation; and (2) revision of the some of the language to lower the cognitive load on the reader. The organization of content was preserved: this virtue of Mac Nerney’s text will probably not show age for a very, very long time. We hope that the benefits of the two changes are fairly self-explanatory. It should improve readability and comprehension, and enhance portability of ideas learned here. That being said, we have striven to retain desirable difficulty at a level that is pedagogically useful without the benefit of a teacher. Indeed, the book was originally designed to accompany Mac Nerney’s instruction, but now that his lecture will not be available with the book, we have taken steps to adapt and modify exercises to fill out what was originally supplied in class. So, in order to make Mac Nerney’s approach available to new generations of readers and accomplish the goals outlined above, we present this revision of Mac Nerney’s book.

Suggestions for Using This Book An independent learner with the mathematical maturity of an advanced undergraduate should be able to manage most of the material of the main chapters. In the spirit of the do-it-yourself method, we would like to encourage the reader to attempt to do as much of the book as possible without consulting other sources. A student with the diligence and perseverance to do this will reap the rewards of the exercise later.

About This Book

xiii

To avoid interruptions in the main body of the text, we limited brief pertinent information to footnotes. See the end of this section for brief chapter notes which were too big for footnotes. One important general note: it’s common practice in mathematics texts to state a theorem, then supply a few lemmas that may be useful in proving the theorem, then proceeding with the proof of the theorem. This is usually perfectly clear, but it is less clear if your book omits proofs! So, when confronted with a theorem followed by lemmas, tackle the lemmas first, and be aware that the notation of the preceding theorem may bleed over into the statements of the lemmas.

Chapter Notes Chapter 0 outlines much prerequisite modern notation and ideas used in the rest of the book. Chapters 1–7 provide the fundamentals of a course in analytic function theory. Chapter 1 introduces the basic ideas of the complex plane, as well as the modern concept of a relation, especially that of a function. Chapter 2 includes certain topological facts about the real line, which then provide an approach to generalizing these facts to the complex plane. It also provides material for understanding necessary topological and metric properties of complex functions. This is the first chapter containing some arguably quite difficult challenges called theorems. Chapter 3 introduces the (Riemann-)Stieltjes integral, which is a generalization of the Riemann integral that does not require presentation of the theory of the Lebesgue integral. It meshes especially well when the functions of interest are continuous functions of bounded variation. Conway’s text,3 which is an excellent conventional text on this material, also uses this integral. This chapter also provides us with the all-important ideas of paths and the Cauchy integral. Chapter 4, after some preliminaries, defines the notion of an analytic function in a simple way that is only possible for complex functions. That is to say, the definition does not suffice for real functions or those involving higher dimensions. This begins to reveal the extraordinarily central position which the complex plane occupies in analysis. Chapter 5 leads us to several astonishing facts about analytic functions, starting with, of all things, a theorem about triangles. Then Mac Nerney continues by revealing the computational power of Cauchy integrals which were first encountered in Theorem 4.11.

3 Conway,

J.B.: Functions of One Complex Variable I. Springer, Berlin (1978).

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About This Book

Chapter 6 acquaints us with simple regions, the most general form of the unique extension theorem, and definitions of uniformly Cauchy and uniform limit for sequences of complex functions. We also learn of contours and winding numbers, how to compute analytic functions using power-series, and the transcendental elementary complex functions: the principal logarithmic, exponential, sine, cosine and tangent functions, as well as the general power function zw . This chapter challenges the student to prove a great many of the facts about these topics which are usually presented in applied courses with little or no proof. Chapter 7 deals with absolute convergence of power series, and includes a littleknown but completely general criterion for the radius of convergence. It also covers the annulus theorem, residues, order, zeros and poles, and essential singularities. For the most part, this concludes the portion of the the book which may be termed “the fundamentals of analytic function theory,” but we should probably extend that border to include the Open Mapping Theorem at the beginning of Chap. 8. Chapters 8–10 constitute an extensive presentation of advanced topics; there are thirteen deep theorems in Chap. 8 alone. An outline of the ideas covered can be found in the table of contents. After Chap. 10, there are five appendices. The ideas are intended as extensions of the main part of the text. Mac Nerney suggested that the appendices could be a framework for a second-year course. Appendices A and B give the student a taste of homotopy groups and the special case of the fundamental group of a region. This provides a jumping-off point for further study in algebraic topology. Appendix C presents the reader with a series of exercises, leading up to the Uniformization Theorem. Appendix D is an excursion into set theory; more precisely, well-ordered sets and an outline of how to prove the well-ordering theorem. Mac Nerney preferred a “natural, although long” classical approach, rather than the “fast but tricky” proof found in some modern texts, which is unlikely to be accessible to anyone not already extensively familiar with advanced modern set theory. Appendix E provides a jumping-off point for learning about Riemann surfaces, with which Bernhard Riemann revolutionized complex analysis. Each of the ten main chapters is prefaced with a reproduction of one of Mac Nerney’s original, simple, hand-drawn diagrammatic pictures depicting an idea from that chapter, one which will hopefully inspire the reader to develop his own geometric intuition by drawing such pictures. The observant reader may notice that, unlike many texts, there are no theorems eponymously named for famous mathematicians, or many named theorems at all, for that matter. Mac Nerney believed that such labeling did more harm than good by intimidating the student before they had attempted to prove the theorem. We hope successful students of this text are pleasantly surprised as they gradually discover, in other sources, the “named” versions of famous theorems they have proved here independently.

About This Book

xv

Acknowledgements This edition of the text would not have been possible without the help several people. John W. Neuberger (a colleague of Mac Nerney’s) gave invaluable advice and encouragement and recommended us to Elizabeth Loew at Springer, who stuck with us throughout the process. Thanks also goes to Mary Dougherty the officers and staff at the University of Houston and Trinity College for helping to determine permissions necessary for use of the material. This manuscript was developed using www.overleaf.com, and benefited greatly from their technical support.

Contents

0

Conventions, Set Theory, Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

The Complex Plane, Relations, Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2

Boundedness, Convergence, Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3

Paths, Integrals, Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4

Connectedness, Convexity, Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5

Triangles, Polygons, Simple Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6

Extensions, Contours, Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7

Power-Series, Residues, Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8

Analytic Inverses, Standard Regions, Convergence Continuation . . . . . 47

9

Extended Complex Plane, Linear-Fractional Transformations, Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10

Analytic Relations, Analytic Continuation, Functional Boundaries, Branch-Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A

Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B

Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

C

Excepted Values and Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

D

Well-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

E

Analytic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Selected Works by Mac Nerney. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xvii

About the Author

John Sheridan Mac Nerney was born in New York City in 1923, and died in Houston in 1979. He attended Trinity College (1939–1941) and then worked worked as a Vibration Analyst at United Aircraft Corporation in Connecticut (1941–1943). He also served in the United States Army Air Force (1943– 1946) during which, in 1945, he married Kathleen Mary O’Connor. After his service, he attended University of Texas in Austin, attaining his PhD in 1951 under Hubert Stanley Wall. From there, he taught at Northwestern University (1951–1952), the University of North Carolina (1952–1967), and the University of Houston (1967–1979). Mac Nerney specialized in functional analysis, partial differential and integral equations, and Hilbert space operator theory. The reader will find a list of selected works by Mac Nerney near the end of the book. He was a member of numerous scientific and honor societies, including Phi Beta Kappa, the North Carolina Academy of Sciences,and Sigma Xi. He was President of the North Carolina chapter of Sigma Xi, 1966–1967. An extensive recorded interview with Mac Nerney that was taken in 1970 is available online at the Briscoe Center for American History.4 In them, one can hear his characteristic way of speaking. He describes his first experiences with the Socratic method with Hyman Ettlinger and H. S. Wall, and later formative educational experiences that led to the development of the notes in this text. As of the printing of this book, the Math Genealogy Project catalogues a total of 18 PhDs directly supervised by Mac Nerney, and 30 mathematical descendants beyond those.

4 Jones, M.: Interview with John Mac Nerney Parts 1–3 (1970). http://av.cah.utexas.edu/index.php?

title=Math:E_math_02062&gsearch=MacNerney. xix

Chapter 0

Conventions, Set Theory, Number Systems

Conventions of terms and notations are introduced at the first convenient juncture wherever possible. The reader is encouraged to consult the index to locate, if necessary, notation for sequences, functions, composition, and the different integrals. This introduction discusses other items not having a convenient home elsewhere. We adopt the common conventions and symbols used in modern set theory. The words collection, family, and set are understood to be synonymous with the primitive notion of “set,” and ∈ denotes the primitive membership relation between objects and sets. The reader should find these ideas concerning sets familiar: 1. If x ∈ A, then this can be read x is in A, or x belongs to A, or x is a member of A. The negation of these statements, expressing that x is not in A, is denoted by x ∈ / A. 2. A is said to be a subset of the set B if x ∈ A implies x ∈ B, and then we write A ⊆ B. The sets are said to be equal if A ⊆ B and B ⊆ A, and then we write A = B. If two sets are not equal, we would write A = B. If A ⊂ B and A = B then A is said to be a proper subset of B. 3. The empty set is the set with no elements, and is denoted by ∅. 4. A singleton set is a set with exactly one member, say x, and in this case {x} denotes that set. Furthermore, there are several standard operations which we assume: 1. The union of two sets X and Y is the set X ∪ Y to which a belongs if and only if a ∈ X or a ∈ Y , and the intersection of X and Y is the set X ∩ Y to which a belongs if and only if a ∈ X and a ∈ Y . If X ∩ Y = ∅ then X and Y are said to be disjoint. 2. If C is a collection of (possibly infinitely many) sets,  we can still define union and intersection of the members of C. The notation C denotes the set of elements that belong to at least one member of C and is called the union of C. Likewise,

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_1

1

2

0 Conventions, Set Theory, Number Systems

 the notation C denotes the set of elements that are in all members of C and is called the intersection of C. 3. The complement of Y in X, often referred to simply as X without Y , is the set X \ Y to which a belongs if and only if a ∈ X and a ∈ / Y. We use standard set builder notation {x | P (x)} to denote the set to which x belongs if and only if the property P holds true for x. This is often read as “the set of all x such that P of x.” If it is known in advance that x is in some set S, this may be expressed to the left of the bar to simplify notation, e.g. {x ∈ S | P (x)}. Thus the   definitions of X ∪ Y , X ∩ Y , X \ Y , C, and C above may be written in ways like this: • • • • •

X ∪ Y = {a | a ∈ X or a ∈ Y } X ∩ Y = {a | a ∈ X and a ∈ Y } = {a ∈ X | a ∈ Y } X / Y } = {a ∈ X | x ∈ / Y}  \ Y = {a | a ∈ X and a ∈ C = {z | there exists S ∈ C such that z ∈ S}.  C = {z | for all S ∈ C, z ∈ S}

We assume the reader is familiar with ordered pairs: that for any (a, b) and (c, d), the equality (a, b) = (c, d) holds if and only if a = c and b = d. The order of the pair matters: for example, if a =  b then (a, b) = (b, a). For any sets X and Y the set of ordered pairs with first term in X and second term in Y is called the cross product of X with Y , or simply X cross Y . In set-builder notation, it can be written this way: X × Y = {(a, b) | a ∈ X and b ∈ Y } We consider the real number system as consisting of the set R whose members are called real numbers together with the usual linear ordering, the usual addition and multiplication operations, and satisfying the following axioms. Definition 0.1 (Axioms of the System of Real Numbers) For all x, y, z in R: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

x + (y + z) = (x + y) + z There is an element 0 with the property that 0 + x = x for all x ∈ R. There is an element −x ∈ R such that x + (−x) = 0. x+y =y+x x(yz) = (xy)z There is an element 1 with the property that 1x = x for all x ∈ R. If x = 0, then there is an element x1 with the property x x1 = 1. xy = yx x(y + z) = xy + xz 0 = 1 If x = y then either x < y or y < x. If x < y and y < z, then x < z. If x < y then x + z < y + z.

0 Conventions, Set Theory, Number Systems

3

14. If x < y and 0 < z then xz < yz. 15. If S is a nonempty subset of R and there is a number which is greater than or equal to every member of S then there is a least number which is greater than or equal to every member of S. Axioms 1–10 describe what is called a field in abstract algebra. Axioms 11–14 encompass the ordering properties and their compatibility with the field operations, and Axioms 1–14 are all that is necessary for an ordered field. The final Axiom 15 is a completeness axiom. In fact the real numbers are the only complete ordered field. That is, these fifteen axioms completely characterize the system of real numbers. Readers probably already have an adequate description of the natural numbers in mind. In this text, the natural numbers are denoted by N and include the number 0. Mac Nerney’s original description of N as a subset of R is interesting in its own right, and we summarize it here1 : 1. 2. 3. 4.

0 ∈ N. If x < 0, then x ∈ / N. If y ∈ N, then y + 1 ∈ N. If y ∈ N and y < z < y + 1, then z ∈ / N.

Although somewhat eclectic, it is in some ways the most direct definition, in the sense that with the aid of the least upper bound axiom (15) the reader can verify that N is unique, there is no upper bound of N, and the principle of mathematical induction holds true: if S ⊆ N, 0 ∈ S, and for every n ∈ S, n + 1 ∈ S, then S = N. The set Z of all integers and the set Q of all rational numbers can be defined respectively by Z = N ∪ {n ∈ R | −n ∈ N},   a  Q= ∈ R  a ∈ Z, b ∈ N \ {0} b = {q ∈ R | ∃p ∈ N \ {0} such that pq ∈ Z}. As a warm-up challenge, the reader may wish to determine which of the fifteen real number axioms apply or do not apply to N, Z and Q. We use “Let r > 0” or “Let r < 0” as a convenient shorthand for “Let r be a positive/negative real number”. As we will see, the expressions are meaningless for complex numbers, and if r is to be a rational or an integer, it will be mentioned. As a rule we will usually use letters from the beginning of the alphabet for real numbers, making exceptions for r, R, m and M, since they are useful for real numbers in later chapters. We’ll also use the Greek letters  and δ as real numbers in limit-type situations. These conventions break down at places in the book, so we urge the reader to always consult the context when in doubt.

1 Actually

start at 0.

Mac Nerney based his natural numbers at 1, and we have adjusted this description to

4

0 Conventions, Set Theory, Number Systems

The foregoing framework should be supplemented with the customary algebraic notation and terminology. For example, the positive numbers are {r ∈ R | r > 0}, the negative numbers are {r ∈ R | r < 0}. For convenience we may denote the positive natural numbers as N+ . One of the facts about the system of real numbers is that if y is a negative number then there is no number x such that x 2 = y. This observation leads us to the ideas of Chap. 1, wherein we learn how to embed R in a larger analytical structure.

Chapter 1

The Complex Plane, Relations, Functions

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_2

5

6

1 The Complex Plane, Relations, Functions

Definition 1.1 A complex number is an ordered pair of real numbers. If each of (a, b) and (c, d) is a complex number then addition and multiplication are defined on the set of complex numbers in the following way: (a, b) + (c, d) = (a + c, b + d) (a, b)(c, d) = (ac − bd, ad + bc) The set of all complex numbers is denoted by C. The symbol i is used to denote the complex number (0, 1). Exercise 1.1 (Algebraic Properties of the Complex Numbers) 1. In terms of ordered pairs above and real number axioms, show that C satisfies the same Axioms 1–10 of the real number system. What is the “0” element? What is the “1” element? 2. Show that there cannot be an ordering < on C satisfying axioms 11–14 of the real number axioms. (Hint: would (0, 0) be greater or less than (0, 1) in the order?) We identify each (a, 0) with the real number a ∈ R, and allow the two symbols to be interchanged as if R ⊆ C. Along with the notation i = (0, 1), these identifications allow an alternative notation for complex numbers laid out in the following exercises and used throughout the text. Exercise 1.2 In these exercises, a and b are real numbers. Show that the following translations are possible in terms of the operations and notation suggested above: 1. (0, b) can be rewritten as bi. 2. (a, b) can be rewritten as a + bi. 3. i 2 = −1 Definition 1.2 If (a, b) is a complex number then 1. 2. 3. 4.

Re(a + ib) = a and is called the real part of of a + ib, Im(a + ib) = b and is called the imaginary part of a + ib, (a + ib) = a + i(−b) and is called the conjugate of a + ib, |a + ib| = (a 2 + b2 )1/2 and is called the modulus of a + ib.

When choosing a symbol for an arbitrary complex number, we will typically use one from the end of the alphabet. (Recall earlier we said that for real numbers we would prefer letters from the beginning of the alphabet.) Exercise 1.3 Let w and z be complex numbers. 1. Show z + z¯ = 2 Re z and z − z¯ = 2i Im z. 2. Show wz = w¯ z¯ . 3. Show that the modulus of z has this property: |z|2 = z¯z = (Re x)2 + (Im x)2 . (In particular it is a nonnegative real number.)

1 The Complex Plane, Relations, Functions

7

Exercise 1.4 Let v, w, z denote complex numbers. 1. Given w and z, there is only one v such that w + v = z. We denote it by z − w, or simply by −w in case z = 0. 2. If w = 0, then there is only one v such that wv = z (we denote it by wz or by z/w ). (Hint: first determine v if w is a real number, then consider z¯ /|z|2 .) Remark 1.3 As an aid to the geometric intuition of complex numbers, we outline the “Cartesian plane” picture of the complex numbers: 1. a point is a complex number (a, b) thought of as lying in R × R. Due to this visualization, the collection of all such points is called the complex plane.1 2. the set of all real numbers, identified with the complex numbers of the form (a, 0), constitutes the real line. It is usually thought of as the horizontal axis of the complex plane, and that leads to the following conventions: 3. the right half-plane is {(a, b) ∈ C | a > 0}, and the left half-plane is similarly characterized as {(a, b) ∈ C | a < 0}; 4. the upper half-plane and lower half-plane are given respectively by {(a, b) ∈ C | b > 0} and {(a, b) ∈ C | b < 0}; 5. finally, we say that the distance from the point w to the point z is |z − w|. Exercise 1.5 If each of v, w and z is a complex number then 1. 2. 3. 4.

|w − z| = |z − w| and |wz| = |w||z| |w − z|2 = |w|2 − 2 Re(wz) ¯ + |z|2 . |v − z| ≤ |v − w| + |w − z|. ||v| − |z|| ≤ |v − z|

Exercise 1.6 Here are a few basic principles of reasoning with real numbers that will be indispensable in later proofs involving equality and inequalities. For any real numbers a and b 1. if a < b +  for every real number  > 0, then a ≤ b; and 2. if 0 < a < b +  for every real number  > 0, then a = b. Remark 1.4 The above two principles are often used in conjunction with the moduli of complex numbers, especially with identities in Exercise 1.5. Definition 1.5 A relation of the sets X and Y is a subset f ⊆ X × Y : 1. the domain of f is {x ∈ X | (x, y) ∈ f } and the range of f is {y ∈ Y | (x, y) ∈ f }. 2. if S is a subset of the domain of f then the f -image of S, denoted by f (S), is {y ∈ Y | (s, y) ∈ f such that s ∈ S}. 3. if f (S) ⊆ T ⊆ Y then f is said to map S into T . Also f is said to map S onto f (S).

1 Mac

Nerney used the term “number-plane” to reinforce the picture of complex numbers as points in a plane, and the reader is invited to bear this in mind.

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1 The Complex Plane, Relations, Functions

4. if g is another relation of X and Y , we say that g is an extension f if f ⊆ g. If T is the domain of g then g is an extension of f to T . 5. if S is a subset of the domain of f , then the subset {(x, y) ∈ f | x ∈ S} is another relation called the restriction of f to S. 6. the inverse relation of f , denoted by f −1 , is the relation {(y, x) ∈ Y × X | (x, y) ∈ f } . 7. If each of g ⊆ Y × Z and h ⊆ X × Y are relations, then the composite of g with h, denoted by g ◦ h, is the relation {(x, z) ∈ X × Z | ∃y ∈ Y, (x, y) ∈ h and (y, z) ∈ g} . Exercise 1.7 1. The relation A = {(a, b) ∈ R × R | a ≤ b} is an example of an order relation in the sense that (a) (b) (c) (d)

for every x ∈ R, (x, x) is in A; if each of (x, y) and (y, z) is in A, then (x, z) is in A; if each of (x, y) and (y, x) is in A then x = y; and for any x, y ∈ R, either (x, y) is in A or (y, x) is in A.

2. The relation B = {(u, v) ∈ C × C | Re u ≤ Re v} is an example of a partialorder relation in the sense that (a) for every x ∈ C, (x, x) is in A; (b) if each of (x, y) and (y, z) is in B then (x, z) is in B; and (c) if both (x, y) and (y, x) are in B then x = y. 3. The relation C = {(u, v) ∈ C × C | Re u = Re v} is an example of an equivalence relation in the sense that (a) if (x, y) is in C then (x, x) is in C. (b) if each of (x, y) and (y, z) is in C then (x, z) is in C; and (c) if (x, y) is in C then (y, x) is in C. 4. Given an equivalence relation C with domain A, denote [x] = {y ∈ A | (x, y) ∈ C} . The relation D = {(x, [x]) | x ∈ A} is an example of a functional relation in the sense that there are not two distinct members of D having the same first term. Definition 1.6 A function (or functional relation) is a relation f of which there are not two members having the same first term. That is, if (a, b) and (a, c) belong to f then b = c. If f is a function then

1 The Complex Plane, Relations, Functions

9

1. f is one-to-one provided there are not two members of f having the same second term. 2. f is said to be a function from its domain onto its range, and from its domain to (or into) each set of which its range is a subset. 3. The value of f at x, denoted by fx or by f (x), is the second term of that member of f of which x is the first term. For brevity of notation we adopt the following convention: it is permissible to write an expression in terms of a variable (say x or z) and call it a function as long as it is clear what the input is. So rather than writing “the function f (z) = z21−1 ” we may simply write “the function

1 .” z2 −1

Exercise 1.8 1. The equation {(x, ix) | x ∈ C} = {(−ix, x) | x ∈ C} holds true. This set is a one-to-one function which maps the right half-plane onto the upper half-plane, the upper half-plane onto the left half-plane, and the left half-plane onto the lower half-plane.2 2. Let U= {x ∈C  | |x| < 1}, P be the set of all points different from −1, and 1−x  f = x, 1+x  x ∈ P . Then f is a function from P onto P , and f (U ) is the right half-plane.     ¯ = 1 . Then 3. Let k be a point in U (as above) and t = x, x−k  x ∈ C and kx kx−1

t is a one-to-one function, t −1 = t, and t (U ) = U . if B is the set of all points with modulus 1 then t (B) = B. 4. The function {(x, x 2 ) | x ∈ C} is not one-to-one. 5. The restriction g of the function in Exercise 4 to the right half-plane is one-to-one and the range of g consists of all points z such that Im z = 0 or Re z > 0. (Note: z1/2 denotes g −1 (z) if either Im z = 0 or Re z > 0.) Definition 1.7 The statement that the function f is constant means that the range of f has only one member: if k is the only member of the range of f , then we may denote f simply by the symbol k if there is no ambiguity. We say f is constant on the set S provided the restriction of f to S is constant. Definition 1.8 If each of f and g is a function with a domain lying in C, and there is a member of the domain of f which belongs to the domain of g, then 1. the functions f + g, f − g, and f g consist respectively of all ordered pairs of the form (x, a + b), (x, a − b), and (x, ab) such that (x, a) ∈ f and (x, b) ∈ g. 2. the function −g consists of all ordered pairs (x, −b) such that (x, b) ∈ g. 3. the function fg consists of all ordered pairs of the form (x, g) such that (x, a) ∈ f and (x, b) ∈ g and b = 0. are welcome to convince themselves that the effect on the complex plane is that of a 90◦ counterclockwise rotation, but should not feel obligated to dwell on the definition or proof of such things for now. 2 Readers

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1 The Complex Plane, Relations, Functions

Definition 1.9 A complex function3 is a function of which the domain and range are nonempty subsets of C. The symbol I is used to denote4 the identity function given by {(z, z) | z ∈ C}.5

3 Mac

Nerney used “point-function” here, supporting his geometric perspective on complex analysis. 4 Mac Nerney employed the I notation frequently to suppress “variables” while writing functions, and permitting a concise notation for path integrals. Occasionally this resulted in eccentric versions of otherwise familiar expressions. An actual example: the function I 2 1−I [−I ] which we might 1+I 2 otherwise write as ( 1+z 1−z ) . In some places, but not all, we have replaced the function I with a “variable” z to make expressions more familiar to present-day readers. 5 Mac Nerney remarked here: Using I , the symbols iI , 1−I , I −k , and I 2 denote respectively the 1+I ktI −1 functions in the last exercise set. Although we shall not use the following terminology, we let it be a matter of record that: a variable is a function, a real variable is a function of which the range is a subset of R, a complex variable is a function of which the range is a subset of C, and the statement that y is a function of x means that x is a function and y is a function and there is a function f such that y = f ◦ x, i.e. y is the composite of f with x.

Chapter 2

Boundedness, Convergence, Continuity

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_3

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2 Boundedness, Convergence, Continuity

Definition 2.1 A nonempty subset S of C is said to be bounded if there is a b ∈ R such that |x| ≤ b for all x ∈ S. An element z ∈ C is a boundary-point of the set S provided that for every  > 0 there is an x ∈ S such that |x − z| <  and there is a y∈ / S such that |y − z| < . The boundary of S is the set of boundary-points of S. Remark 2.2 We recall the following property of R: if S is a nonempty subset of R then 1. if there is a number which is greater than or equal to every number in S then there is a least number which is greater than or equal to every number in S. This number is called the least upper bound of S; 2. if there is a number which is less than or equal to every number in S then there is a greatest number which is less than or equal to every number in S. This is called the greatest lower bound of S. Exercise 2.1 1. If S is a nonempty, bounded subset of R, then there is a boundary-point of S. 2. If B = {x ∈ C | |x| = 1} then B is the boundary of the unit disc U = {x ∈ C | |x| < 1}. Definition 2.3 A sequence is a function with domain S ⊆ N such that 0 ∈ S, and for any n ∈ S, if m is a natural number less than n, then m ∈ S. In other words, S must be an initial segment of N. The notation {tp }np=0 indicates a sequence t of which the domain contains n but does not contain any number greater than n. A sequence is an infinite sequence if S = N, and this will be denoted as {tp }∞ p=0 . We may also say a “sequence in T ” or “sequence of (T ’s elements)” when the range of t is contained in T , as in a “sequence in C” or a “sequence of complex numbers,” or even a “sequence of sets.“ If t is an infinite sequence in C then 1. a cluster-point of t is an element x ∈ C such that for any  > 0 and m ∈ N+ there is an n ∈ N greater than m such that |x − tn | < . 2. t converges to y if for any  > 0 there is an m ∈ N+ such that for every n ∈ N greater than m, we have |y − tn | < . 3. t is said to be a Cauchy sequence if for every  > 0 there is a positive integer m such that for any n ∈ N greater than m, we have |tm − tn | < . Theorem 2.4 If t is an infinite sequence in R with bounded range, then there is a real number which is a cluster-point of t. Definition 2.5 If f is a function from the subset A ⊆ R into R, then 1. f is increasing provided A has two members and if x and y are members of A and x < y then f (x) < f (y). 2. f is nondecreasing provided that if x and y are members of A and x < y then f (X) ≤ f (Y ). 3. f is decreasing provided −f is increasing. 4. f is nonincreasing provided −f is nondecreasing.

2 Boundedness, Convergence, Continuity

13

If t is a sequence, then s is a subsequence of t provided there is an increasing sequence u, with range contained in N, such that s = t ◦ u. Exercise 2.2 If t is a sequence in C and x ∈ C, the following statements are equivalent: 1. x is a cluster-point of t; 2. some subsequence of t converges to x. Theorem 2.6 If t is an infinite sequence in C with bounded range, then t has a cluster-point. Theorem 2.7 If t is a Cauchy sequence in C, then t converges to some point in C. Definition 2.8 A limit-point of the subset S ⊆ C is an element x ∈ C such that if  > 0 then there is a y ∈ S such that 0 < |y − x| < . The set S is closed if ¯ is the set of all S contains all of its limit-points. The closure of S, denoted by S, limit-points of S together with the elements of S. Theorem 2.9 If S is an infinite, bounded subset of C then there is a limit-point of S. Hint: if S is an infinite set then there is an infinite sequence t such that t is oneto-one and the range of t is a subset of S. Exercise 2.3 1. If S is a closed subset of C and t is an infinite sequence in S that converges to x, show that x ∈ S. 2. Show by example that even if an infinite sequence t has a cluster-point x, it is not necessarily true that x is a limit-point of the image of t. 3. Show that a boundary point of S is necessarily either a limit point of S or a limit point of C \ S. Theorem 2.10 Suppose S is a sequence of nonempty sets such that for each n ∈ N, ∞Sn is a closed and bounded subset of C, and furthermore Sn+1 ⊆ Sn . Then i=0 Si = ∅. Definition 2.11 If f is a complex function then 1. f is continuous at (p, q) if (p, q) belongs to f and the following property holds: for each number c > 0, there is a number b > 0 such that for all (x, y) ∈ f , if |x − p| < b, then |y − q| < c. 2. f is continuous provided that f is continuous at all (p, q) ∈ f . 3. f is continuous on S if S is a subset of the domain of f and the restriction of f to S is continuous. Remark 2.12 The statement that the complex function f is continuous at the point p is sometimes used to mean that p is in the domain of f and f is continuous at (p, f (p)).

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2 Boundedness, Convergence, Continuity

Exercise 2.4 1. Show the function (x, |x|) is continuous on C. 2. For S = R, show that the function which is 0 on the negative numbers and 1 on the nonnegative numbers is not continuous at 0 ∈ S. Exercise 2.5 In the following exercises, S is a nonempty subset of C. 1. If both of f and g are complex functions continuous on the subset S, then f + g and f g are continuous on S. 2. If the complex function f is continuous on S and 0 ∈ / f (S) then f1 is continuous on S. 3. If the complex function g is continuous on S and the complex function f is continuous on g(S) then f ◦ g is continuous on S. 4. If x ∈ S is a cluster point of a sequence {tp }∞ p=0 in S and f is a continuous function on S, then f (x) is a cluster point of the sequence {f (tp )}∞ p=0 . Theorem 2.13 Let S be a nonempty, closed, bounded subset of C. Then if the complex function f is continuous on S then 1. f (S) is closed and bounded. 2. there is an element x ∈ S such that for all y ∈ S, we have |f (y)| ≤ |f (x)|. 3. if f is one-to-one then f −1 is continuous on f (S). Definition 2.14 If f is a complex function with domain D then 1. f is uniformly continuous if, for each  > 0, there is a δ > 0 such that for all x, y ∈ D, if |x − y| < δ then |f (x) − f (y)| < . 2. f is uniformly continuous on S if S is a subset of D and the restriction of f to S is uniformly continuous. Theorem 2.15 If the complex function f is continuous on a nonempty, closed, and bounded set S, then f is uniformly continuous on S. Definition 2.16 If S is a nonempty subset of the domain of the complex function f and f (S) is bounded then 1. |f |S denotes the least upper bound of {|f (x)| | x ∈ S}. The quantity |f |S is called the modulus of f on S. 2. for each b ≥ 0, Cf (S, b) denotes the least real number c such that for all x, y ∈ S, if |x−y| ≤ b then |f (x)−f (y)| ≤ c. This is called the modulus of continuity of f on S relative to b. Exercise 2.6 In the following exercises, S is a nonempty subset of C. 1. If the complex function f is uniformly continuous on S and  > 0 then there is a number b > 0 such that Cf (S, b) < . 2. If every complex function which is continuous on S is uniformly continuous on S, then S is a closed set. (Extra credit: under the same hypotheses, give an example to show that S need not be bounded.) 3. The interval [0, 1] = {x ∈ R | 0 ≤ x ≤ 1} is a closed and bounded set.

2 Boundedness, Convergence, Continuity

15

4. Suppose that x is a complex function from S into T , and that x(S) is bounded, and that f is a complex function such that f (T ) is bounded. Then for b > 0, we have Cf ◦x (S  , b) ≤ Cf (T , Cx (S, b)). 5. If m ∈ N+ and {tp }np=0 is a sequence with subsequence {sp }m p=0 , and if s0 = t0 and sm = tn , then there is an increasing sequence v : N → N such that v0 = 0 and vm = n and s = t ◦ v.

Optional Project Find an extension  of C, embodying extensions of the addition and multiplication operations and of the definition of modulus, such that 1. 2. 3. 4.

each complex number belongs to , if each of x, y ∈  then each of x + y and xy is in , if x is a nonzero element of  then |x| is a positive number, and the following postulates are satisfied for every x, y, z ∈ : (a) (b) (c) (d) (e) (f) (g) (h)

x + (y + z) = (x + y) + z. −x ∈ , and 0 + x = x, and x + (−x) = 0. x + y = y + x. 1x = x and x(yz) = (xy)z. If c ∈ C then cx = xc. x(y + z) = xy + xz and (x + y)z = xz + yz. |x + y| ≤ |x| + |y| and |xy| ≤ |x||y|. Each Cauchy sequence with range contained in  converges to an element of .

Such a structure  is sometimes called a complete normed algebra over the complex numbers.

Chapter 3

Paths, Integrals, Derivatives

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_4

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3 Paths, Integrals, Derivatives

Definition 3.1 A simple subdivision of an interval [a, b] is a nondecreasing sequence {tp }np=0 with values in [a, b] such that t0 = a and tn = b. If [a, b] is in the domain of a complex function x, then x is of bounded variation on [a, b] if there is a number v such that for every simple subdivision {tp }np=0 , we have n p=1 |x(tp ) − x(tp−1 )| ≤ v. The least upper bound of such v is called the total b variation of x on [a, b] and denoted by a |dx|. Exercise 3.1 If the complex function x is of bounded variation on [a, b] then 1. for every  > 0, there is a simple subdivision {sp }m p=0 of [a, b] with the property that for every simple subdivision {tp }np=0 that has s as a subsequence,

b

n 

|dx| − 
0, there is a  Stieltjes subdivision s of [a, b] such that for any refinement t of s, we have |z − t y dx| < .

3 Paths, Integrals, Derivatives

19

b Remark 3.4 One might wish also to have a dx y for x and y with ranges contained in an algebra . (See the closing project of Chap. 2.) Exercise 3.3 1. If each of

b a

b

y dx and

a b

Y dx exists then

(y + Y ) dx =

a

2. If each of

b a

y dx and

b a

b

b

a

b

y dx +

Y dx .

a

a

y dX exists then

b

y d(x + X) =

a

3. If

b

b

y dx +

a

y dX . a

y dx exists and w is a complex number then

b

y d(wx) =

a

b

wy dx = w

a

b

y dx . a

b 4. If a y dx exists, y([a, b]) is bounded, and x is of bounded variation on [a, b], then  b 

b     ≤ |y|[a,b] y dx |dx| .   a

5. If

b a

a

y dx exists and a < c < b then

c

b

y dx +

a

c

6. If each of x and y is a complex function and

b a

b

y dx +

b

y dx =

y dx . a

b a

y dx exists then

x dy = y(b)x(b) − y(a)x(a)

a

(Hint: Think about what kind of subdivision would be needed to get y(a)x(a) to appear, and what kind would be needed to get y(b)x(b) to appear.) Theorem 3.5 If [a, b] is an interval on which the complex function y is continuous, b and the complex function x is of bounded variation, then a y dx exists, and for each Stieltjes subdivision s of [a, b],

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3 Paths, Integrals, Derivatives

 

b  b     ≤ C y dx − y dx ([a, b], |s|) |dx| .   y  a  a s

Lemma 3.6 (1) If t is a refinement of the Stieltjes subdivision s of [a, b] then  

b      y dx − y dx  ≤ Cy ([a, b], |s|) |dx| .    a s

t

Lemma 3.7 (2) If r and s are Stieltjes subdivisions of [a, b] then  

b      y dx − y dx  ≤ (Cy ([a, b], |r|)) + Cy ([a, b], |s|)) |dx| .    a r

s

Definition 3.8 If [a, b] lies in the domain of the complex function x and {sp }2m p=0 is  a Stieltjes subdivision of [a, b] then s |dx| denotes the number m 

|x(s2p ) − x(s2p−2 )|

p=1

Exercise 3.4 1. If u is an increasing function from [0, 1] onto [0, 1] then u is continuous. 2. If v is a complex function, then v is a decreasing function from [0, 1] onto [0, 1] only in case 1 − v is an increasing function from [0, 1] onto [0, 1]. 3. If the complex function x is of bounded variation on [0, 1] and u is an increasing 1 1 function from [0, 1] onto [0, 1] then 0 |d(x ◦ u)| = 0 |dx|. (Hint: If t is a Stieltjes subdivision of [0, 1] and ris a refinement oft and of u−1 ◦t then u ◦ r is a refinement of t and of u ◦ t, and r |d(x ◦ u)| = u◦r |dx|.) 4. If the complex function x mapping z → x(z) is of bounded variation on [0, 1] then the function x ◦ (1 − I ) mapping z → x(1 − z) is of bounded variation on [0, 1] also. (Hint: if {tp }2n of [0, 1] then {1 − t2n−p }2n p=0 is a Stieltjes subdivision p=0 is a   Stieltjes subdivision s of [0, 1] such that t |d(x ◦ (1 − I ))| = s |dx|.) 1 Theorem 3.9 If x and y are complex functions such that 0 y dx exists, and u is an increasing function from [0, 1] onto [0, 1], and v is a decreasing function from [0, 1] onto [0, 1], then

1 0

1

(y ◦ u) d(x ◦ u) = 0

1

y dx = − 0

(y ◦ v) d(x ◦ v) .

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21

Definition 3.10 A curve in the complex plane is a function f : [a, b] → C where [a, b] ⊆ R is an interval upon which f is continuous and of bounded variation.1 Remark 3.11 Exercise 3 above and Theorem 3.9 suggest that x and x ◦ u do not need to be distinguished, and motivate the following definition. Definition 3.12 Let B be the set of functions from [0, 1] to C which are continuous and of bounded variation on [0, 1]. Define a relation on B in this way: for all x, y ∈ B, we say that x ∼ y if there exists an increasing function u of [0, 1] onto [0, 1] such that x ◦ u = y. This relation is an equivalence relation. The statement that K is a path means that K is an equivalence class of this equivalence relation on B.2 Definition 3.13 If x is a member of the path K then 1. K is a path from x(0) to x(1). We also say that K begins at x(0) and ends at x(1). 2. K is closed provided x(0) = x(1). 1 3. the length of K, denoted by λ(K), is the number 0 |dx|. 4. K  denotes the set x([0, 1]), called the carrier of K. 5. K is a path in S provided that K  ⊆ S. 6. −K denotes the path containing the function x ◦ (1 − I ). Exercise 3.5 1. If x and y are members of the path K and the complex function f is continuous 1 1 on K  then 0 f ◦ x dx = 0 f ◦ y dy. 2. If x belongs to the path K then y belongs to −K only in case there is a decreasing function v from [0, 1] onto [0, 1] such that y = x ◦ v. 3. If x belongs to the path K and y belongs to −K and the complex function f is 1 1 continuous on K  then 0 f ◦ y dy = − 0 f ◦ x dx. Definition 3.14 Let K1 and K2 be paths such that K2 begins at the point K1 ends. We define the sum of K1 and K2 in the following way. Select members x ∈ K1 and y ∈ K2 , and consider the function z defined like this: if 0 ≤ t ≤ 12 then z(t) = x(2t), but if 12 ≤ t ≤ 1 then z(t) = y(2t − 1). The function z is a curve of bounded variation on [0, 1], and the path to which z belongs is denoted by K1 ⊕ K2 . Definition 3.15 If K is a path and f is a complex function with domain containing 1 K  then K f denotes that point that is the common value of 0 f ◦ x dx shared by all the x ∈ K. The point K f is called the Cauchy integral over the path K of the function f . Remark 3.16 One might wish also to have K f for a function f with range lying in an algebra .

1 Such

curves were classically called “rectifiable curves,” but we will simply call them “curves.”

2 Another wrinkle: some authors use “path” for what we call curves, and “curve” for the equivalence

classes. We stick with Mac Nerney’s choice here.

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Exercise 3.6 Let K, K1 , and K2 denote paths. 1. | K f | ≤ |f |K  λ(K) provided f is continuous on K  . 2. If K = K1 ⊕ K2 then λ(K) = λ(K1 ) + λ(K2 ) and, if the complex function f is continuous on K  ,

f = f+ f. K

K1

K2

3. K ⊕ −K is a closed path. Definition 3.17 The complex function f is said to have slope at (p, q) if (p, q) ∈ f and there is a unique s ∈ C with the following property: For every  > 0 there is a δ > 0 such that for all (u, v) ∈ f    v − q  − s  <  . 0 < |u − p| < δ ⇒  u−p In this case the s is called the slope of f at (p, q). Exercise 3.7 1. If the complex function f has slope at (p, q) then p is a limit-point of the domain of f and f is continuous at (p, q); 2. If p is a limit-point of the domain of the complex function f and there exists a complex number w having the property referenced in Definition 3.17, then f has slope w at (p, f (p)). Definition 3.18 If A is the domain of the complex function f and B = {z ∈ A | f has slope at (z, f (z))}, then 1. the derivative of f , denoted by f  , is the complex function {(z, s) | z ∈ B and f has slope s at (z, f (z))}. 2. f is a function from A × B to C such that if (x, y) ∈ A × B then f (y, y) = 0 and f (x) − f (y) = f  (y)(x − y) + (x − y)f (x, y) 3. f is said to have a derivative at z if z ∈ B. Exercise 3.8 (Differential Calculus) 1. The derivative of the identity function is the constant function 1. 2. If the complex function f has a derivative then (−f ) = −f  . 3. If f (z) = 1z , then f  (z) = − z12 .

3 Paths, Integrals, Derivatives

23

4. If f and g are complex functions and S is the set of limit points of the domain of f + g at which f and g both have derivatives, then the restriction of f  + g  to S is a subset of (f + g) and the restriction of f g  + f  g to S is a subset of (f g) . 5. Suppose f and g are complex functions, and let T be the set of every point p satisfying three requirements: (a) g has a derivative at p; (b) f has a derivative at g(p); and (c) p is a limit-point of the domain of f ◦ g. Then the restriction of (f  ◦ g)g  to T lies in (f ◦ g) . Theorem 3.19 (Integral Calculus, Part 1) If f is a complex function and K is a path from A to B in the domain of f and f  is continuous on K  , then K f  = f (B) − f (A). Lemma 3.20 Let [a, b] be a subinterval of [0, 1], let and x ∈ K and c > 0. Then define the increasing sequence s such that s0 = a and for each n ∈ N such that sn < b, the value of sn+1 is the least upper bound of {u ∈ [sn , b] | |f (x(u), x(sn ))| ≤ c} The sequence s has the following properties: 1. If n ∈ N and sn+1 ≤ b then |f (x(sn+1 ), x(sn ))| ≤ c. 2. There is a positive integer m such that sm = b. Definition 3.21 If the domain of the complex function h contains the carrier of the path K, h/K denotes {h ◦ x | x ∈ K}. The set h/K is called the image of K under h. Exercise 3.9 If K is a path and h is a point function such that h is continuous on K  then h/K is a path, (h/K) = h(K  ), and λ(h/K) ≤ |h |K  λ(K). Theorem 3.22 (Integral Calculus, Part 2) If each of g and h is a complex function    and K is a path such that h is continuous on K and g is continuous on h(K ) then h/K g = K (g ◦ h)h . Definition 3.23 If each of A and B is a set and  is a relation with domain A × B and each of g and h is a relation, and there is a member (p, x) of g and a member (p, y) of h such that x is in A and y is in B, then the composite of  with (g, h), denoted by [g, h], is the relation to which (p, q) belongs only in case there is a member (p, x) of g and a member (p, y) of h such that ((x, y), q) belongs to . Remark 3.24 If the complex function f has a derivative at the point w then (w) f [I, w] is a complex function and consists of {(z, f (z)−f − f  (w) | z ∈ z−w C} ∪ {(w, 0)}. (This function is continuous at (w, 0).)

Chapter 4

Connectedness, Convexity, Analyticity

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_5

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4 Connectedness, Convexity, Analyticity

Definition 4.1 Two subsets of C are mutually separated provided neither of them contains a point or a limit-point of the other. A subset of C is connected provided it is not the union of two nonempty and mutually separated sets. A component of the subset S ⊆ C is a connected subset of S which is not a proper subset of any connected subset of S. In other words, when the collection of connected subsets of S is partially ordered by containment, the components are maximal elements. Exercise 4.1 In the following problems, S is a nonempty subset of C. 1. The interval [0, 1] is connected. 2. If S contains at least two points and S is connected, then each point of S is a limit-point of S. 3. If T is a nonempty proper subset of S and S is connected, then T has a boundary point in S. 4. If S has the property that if x and y are points of S then there is a connected subset of S containing both x and y, then S is connected. 5. If x ∈ S then x belongs to a component of S. 6. The union of all members of the complete set of components of S is equal to S, and any two components are mutually separated. Theorem 4.2 If the complex function f is continuous on a connected set S then f (S) is connected. Definition 4.3 If A and B are points, then the linear path from A to B, denoted by [A; B], is the path containing the restriction of the function (1 − z)A + zB to [0, 1]. A set S of complex numbers is said to be convex if whenever A and B are in S then [A; B] is a subset of S. We say is a polygonal path if there exists a sequence K n n {Ap }n+1 ⊆ C such that K = p=0 [Ap ; Ap+1 ], i.e. K = Sn where {Sp }p=0 is a p=0 sequence such that S0 = [A0 ; A1 ] and Sp+1 = Sp ⊕ [Ap+1 ; Ap+2 ] for all integers p with 0 ≤ p < n. Exercise 4.2 1. 2. 3. 4. 5. 6.

[0; 1] is the set of all increasing functions from [0, 1] onto [0, 1]. If A and B are points then [B; A] = −[A; B] and λ([A; B]) = |B − A|. For p ∈ C, the {x ∈ C | |x − p| < 4} is convex. If K is a path and A is a point then K = K ⊕ [A; A]. Every convex set of complex numbers is connected. If K1 is a path from A to B, K2 is a path from B to C, and K3 is a path from C to D, then K1 ⊕ (K2 ⊕ K3 ) = (K1 ⊕ K2 ) ⊕ K3 .

Definition 4.4 A subset S ⊆ C is open if C \ S is closed.1 A region is a nonempty, open, and connected subset of C. Exercise 4.3 Let S ⊆ C be an open set.

1 Mac

Nerney’s preferred formulation ran this way: S is said to be open if there is not a point in S which is a limit-point of a set of points not in S.

4 Connectedness, Convexity, Analyticity

27

1. If p ∈ S then there is an r > 0 such that if x is a point and |x − p| < r then x belongs to S. 2. Each component of S is a region. Theorem 4.5 If A and B are points in the region R, there is a polygonal path from A to B in R. Definition 4.6 An analytic function is a complex function of which the domain is a region and which has slope at each of its members. The statement that the complex function f is 1. analytic in R means that the restriction of f to R is an analytic function. 2. analytic at w means that w belongs to a region in which f is analytic. An entire function is an analytic function having domain C. Remark 4.7 One might wish further to consider analytic functions from regions to an algebra . Theorem 4.8 If f is an analytic function and f  has only the value 0 then f is constant. Theorem 4.9 If R is a region and f is a continuous complex function with domain R then the following two statements are equivalent: 1. There is an analytic function g such that g  = f . 2. If K is a closed path in R then K f = 0. Exercise 4.4 If the complex function f is continuous on the interval [a, b], then b a f dI = [a;b] f . Definition 4.10 If w is a point and r > 0 then the principal square contour with center w and radius r, denoted by Sr (w), is the path [w+r−ir; w+r +ir]⊕[w+r+ir; w−r+ir]⊕[w−r +ir; w−r−ir]⊕[w−r−ir; w+r−ir].

Theorem 4.11 If w is a point and r > 0 and K = Sr (w) then Lemma 4.12 (1) If h is the restriction of the function

z+i 1+iz



1 K I −w

= 2π i.

to [−1, 1], then

1. h is one-to-one with range Q such that w belongs to Q only in case |w| = 1 and Im w ≥ 0, and in this case h−1 (w) =

Re w w−i = 1 − iw 1 + Im w

2. The number π , which is defined to be

1

−1 |dh|,

is 2

1

1 −1 1+I 2

dI .

4.13 (2) If K is one of the four paths indicated in defining Sr (w) then Lemma π 1 = K I −w 2 i.

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4 Connectedness, Convexity, Analyticity

Corollary 4.14 There is an analytic function which is not the derivative of an analytic function. Definition 4.15 If w is a nonzero complex number, then the principal argument of w, denoted by Arg w, is defined as follows: 1. 2. 3. 4.

if Im w if Im w if Im w if Im w

= 0 and Re w > 0 then Arg w = 0. = 0 and Re w < 0 then Arg w = π . 1 Re w > 0 and t = |w|+Im w then Arg w = 2 t < 0 then Arg w = −Arg w. ¯

1 1+I 2

dI .

Exercise 4.5 If −π < u ≤ π then there is only one point w such that |w| = 1 and Arg w = u. Definition 4.16 If S ⊆ C and r > 0 then S + r denotes {w ∈ C | ∃z ∈ S such that |w − z| ≤ r}. Exercise 4.6 If S is a closed and bounded set lying in the region R, then there is an r > 0 such that S + r is a subset of R, and if |A − B| ≤ 2r then [A; B] lies in S + r. Theorem 4.17 Suppose the complex function f is continuous on the region R and K is a path in R. For any y ∈ K and

 > 0, there is a positive integer n n p−1 p such that if M = p=1 y( n ); y( n ) then M is a polygonal path in R and | K f − M f | < .

Chapter 5

Triangles, Polygons, Simple Regions

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Definition 5.1 The statement that K is a triangular path means that there is a sequence {Ap }2p=0 ⊆ C such that K = [A0 ; A1 ] ⊕ [A1 ; A2 ] ⊕ [A2 ; A0 ]. Another notation is K = [A0 ; A1 ; A2 ; A0 ]. Theorem 5.2 (Triangle Theorem) Suppose A, B, and C are points, and let D = {w ∈ C | ∃P ∈ [B; C] such that w ∈ [A, P ] } and let f be a complex function with derivative at each point of D. Then for the triangular path K = [A; B; C; A], the integral K f = 0. Lemma 5.3 (1) The set D in Theorem 5.2 is closed, bounded, and convex. Lemma 5.4 (2) Let r = M in D such that

A+B 2 ,

s=

B+C 2 ,

and t =

C+A 2 .

There is a triangular path

1. M is [A; r; t; A] or [r; B; s; r] or [s; C; t; s] or [t; r; s; t], 2. if x ∈ K  and y ∈ M  then |x − y| ≤ λ(M), and 3. λ(M) = λ(K)/2 and | K f | ≤ 4| M f |. Lemma 5.5 (3) There is a sequence T such that T1 = K and for each positive integer n: 1. 2. 3.

Tn+1 is a triangular path in D and λ(Tn+1 ) = λ(K)/2n ,  then |x − y| ≤ λ(Tn+1 ), and if x ∈ Tn and y ∈ Tn+1 n | K f | ≤ 4 | Tn+1 f |.

 Lemma 5.6 (4) Let y be a sequence in C such that yn ∈ Tn+1 for each positive integer n. Then each of the following holds for each positive integer m:

1. 2. 3. 4.

For every integer n ≥ m, |ym − yn | ≤ λ(Tm+1 ) . The sequence y converges to a point x ∈ D, and |ym − x| ≤ λ(Tm+1 ).  If w ∈ Tm+1 then |w − x| ≤ λ(Tm+1 ) | K f | ≤ 4m | Tm+1 (I − x)[I, x]|.

Theorem 5.7 Suppose the complex function f is analytic in R. If K is a closed   polygonal path in R and there is a point V such that if P ∈ K then [V ; P ] lies in R, then K f = 0. Theorem 5.8 If z is a point in the region R, and the complex function f is continuous on R and analytic at each point in R different from z, then the following statements are true: 1. If A, B, and C are points such that if P ∈ [B; C] then [A; P ] ⊆ R and K = [A; B; C; A] then K f = 0. 2. If K is a closed polygonal path in R and there is a point V such that if P is in K  then [V ; P ] ⊆ R, K f = 0. Definition 5.9 The open disc with radius r and center w, denoted Dr (w), is {z ∈ C | |w − z| < r}.

5 Triangles, Polygons, Simple Regions

31

Theorem 5.10 Suppose z is a point in the region R and the complex function f is continuous on R and analytic at each point in R different from z. If K is a closed path in R and there is a point V such that if P ∈ K  then [V ; P ] ⊆ R, then K f = 0. Lemma 5.11 For each P ∈ K  there is an r > 0 such that for all Q ∈ Dr (P ), we have [V ; Q] ⊆ R. Lemma 5.12 There is an r > 0 such that for each P ∈ K  and Q ∈ Dr (P ), we have [V ; Q] ⊆ R. Theorem 5.13 Let K be a path, and let the complex function h be continuous  on K  . Define a sequence {fn }∞ n=0 of complex functions with domains C \ K in the following way:

fn (w) = n! K

h for all w ∈ C \ K  (I − w)n+1

Then for every n ∈ N, fn = fn+1 . Theorem 5.14 Suppose the complex function f is analytic in R, K is a closed path in R, and there is a point V such that if P ∈ K  then [V ; P ] ⊆ R. There is1a point f in R which is not in K  , and if w is such a point then K I −w = f (w) K I −w . Theorem 5.15 Suppose K is a closed path and g : C \ K  → C is the complex 1 function defined by g(w) = K I −w . 1. The function g is constant on each component of C \ K  . 2. g(z) = 0 for each z in the unbounded component of C \ K  . Corollary 5.16 If the complex function f is analytic in the region R then f  is analytic in R. Theorem 5.17 If the entire function f has bounded range then f is constant. Theorem 5.18 Suppose n is a positive integer, {Ap }np=0 is a sequence in C such  that An = 0, and F is the complex function A0 + np=1 Ap zp (a polynomial of degree n). 1. F is an entire function.  n−1 . 2. If z ∈ C and |z| > 1 + n−1 k=0 |Ak /An | then |F (z)| > |An ||z| 3. There exists z ∈ C such that F (z) = 0. Definition 5.19 If f is a complex function and k ∈ N, then we define the k’th derivative of f (denoted f (k) ) recursively in the following way: f (0) = f and f (k+1) = (f (k) ) . Theorem 5.20 Let f be a complex function which is analytic in R, and w ∈ R. Suppose r > 0 such that if z ∈ C and |z − w| ≤ 2r, then z ∈ R. Let D be the component of C \ Sr (w) containing w.

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5 Triangles, Polygons, Simple Regions

1. If either | Re(z − w)| > r or | Im(z − w)| > r then z belongs to the unbounded component of C \ Sr (w) . 2. D = {z ∈ R | | Re(z − w)| < r and | Im(z − w)| < r}. 3. If P ∈ Sr (w) then [w; P ] ⊆ R. 4. If k ∈ N and x ∈ D then f (k) (x) =

k! 2π i

Sr (w)

f . (I − x)k+1

5. If n is a positive integer and x ∈ D then f (x) =

n−1  k=0

f

(k)

1 (x − w)k + (w) k! 2π i

Sr (w)



x−w I −w

n

f . I −x

6. If 0 < s < r and  > 0, there is a positive integer m such that for any positive integer n > m and for any x ∈ C such that |x − w| ≤ s, we have   n   (x − w)k   (k) f (w) f (x) −  0 then

5 Triangles, Polygons, Simple Regions

33

there is a δ > 0 such that if (x, y) is in R × R and |x − z| < δ and |y − w| < δ then |f (x, y) − f (z, w)| < . 2. Suppose  is such an algebra as indicated at the close of Chap. 2, and x1 exists for each x in  different from 0, i.e., if x is in  and x = 0 then there is a member y of  such that xy = yx = 1. If w is a member of  which is not a complex 1 number then w−I is an “analytic function from the plane to ” with bounded range.

Chapter 6

Extensions, Contours, Elementary Functions

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Theorem 6.1 If f is an analytic function with domain R, the following are equivalent: 1. There is a set S ⊆ C such that R contains a limit-point of S, and for all z ∈ S, f (z) = 0. 2. There is w ∈ C such that for all k ∈ N, f (k) (w) = 0. 3. There is w ∈ R and a positive number r such that for all z ∈ Dr (w), f (z) = 0. 4. For all z ∈ R, f (z) = 0. Exercise 6.1 (Unique Extension Theorem) If each of f and g is an analytic function which is an extension to the region R of the function h and R contains a limit-point of the domain of h, then f = g.1 Definition 6.2 The principal logarithmic function, denoted by “Log”, is the function    1  Re x > 0 or Im x =  0 . x,  [1;x] I Exercise 6.2

1. If K is a path from 1 to x in the domain of Log then Log(x) = K I1 . 2. Log is analytic, and (Log) is a proper subset of the function 1z , but Log is not a proper subset of an analytic function. 3. If z is in the domain of Log and a > 0 then Log(a) + Log(z) = Log(az). 4. Suppose z is in the domain of Log, and h is the function to which (u, v) belongs only in case each of u and uz is in the domain of Log and v = Log(u)+Log(z)− Log(uz). If D is the component of the domain of h to which 1 belongs, then z¯ ∈ D and h(u) = 0 for each u ∈ D. 5. If z is in the domain of Log then Log(z) = Log(¯z), Re Log(z) = Log(|z|), and Log(z) = Log(|z|) + Log(z/|z|). 6. If |w| = 1 and Im w > 0 then Log(w) = iArg w. 7. The  point w belongs to the range of Log only in case −π < Im w < π . 2   1+z is a one-to-one analytic function from the unit disc U onto the 8. Log 1−z range of Log which has slope 4 at (0, 0).

1 Mac Nerney remarks here: In continuation of a footnote at the close of Chap. 1, concerning unused terminology, it is believed that the following interpretation of common usage may be appropriate:

1. w is a single valued analytic function of the complex variable z provided there is an analytic function f such that w = f ◦ z. 2. If the range S of the function z lies in the domain R of the analytic function f and R contains  a limit-point of S and w = f ◦ z, then dw dz denotes f ◦ z.

6 Extensions, Contours, Elementary Functions

37

Definition 6.3 Below, S denotes a nonempty subset of C and f = {fn }∞ n=0 is a sequence of complex functions. 1. For x, y ∈ C, we say the sequence f of functions is Cauchy at x if the sequence {fn (x)}∞ n=0 is a Cauchy sequence. We say f has the limit y at x if for every  > 0 there is a positive integer m such that for any integer n > m, we have |y − fn (x)| < . 2. We say f is pointwise Cauchy on S if f is Cauchy at all points of S. Also, f has the limit g on S if g is a complex function and if x ∈ S then f has the limit g(x) at x. 3. We say f is uniformly Cauchy on S if for every  > 0 there is a positive integer N such that for all integers m, n greater than N and every x ∈ S, we have |fm (x) − fn (x)| < . 4. f  is defined to be the sequence of derivatives {(fn ) }∞ n=0 . Exercise 6.3 1 1. Let f = {fn }∞ n=0 be the sequence of complex functions given by fn (z) = n−z for each nonnegative integer n. Then f has the limit 0 on C\{0}, but is not uniformly Cauchy on any set of which 0 is a limit-point. 2. If f = {fn }∞ n=0 is a sequence of complex functions and f is pointwise Cauchy on S then there is a complex function g such that f has the limit g on S. If, moreover, f is uniformly Cauchy on S, then f has the limit g uniformly on S in the sense that for any  > 0 there is a positive integer m such that for any integer n > m we have |g − fn |S < . 3. Suppose K is a path, f = {fn }∞ n=0 is a sequence of complex functions which  are each continuous on K  , and f has the limit g uniformly on K . Then the  g is continuous on K , and K g is the limit of the sequence complex function ∞ . f = { f } K K n n=0

Theorem 6.4 Suppose R is a region, and f = {fn }∞ n=0 is a sequence of complex functions which are analytic in R, and f has the limit g on R, and f is uniformly Cauchy on each closed and bounded subset of R. The following statements are true: 1. The complex function g is analytic in R. 2. f  has the limit g  on R. 3. f  is uniformly Cauchy on each closed and bounded subset of R. Definition 6.5 The statement that f is a power-series means that there is a point w and an infinite sequence A ⊆ C such that f =

⎧ n ⎨ ⎩

p=0

Ap (z − w)p

⎫∞ ⎬ ⎭ n=0

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6 Extensions, Contours, Elementary Functions

That is f is a sequence and f0 is the constant A 0 with domain C and if n is a positive integer then fn is the complex function A0 + np=1 Ap (z − w)p ; in this case, f is said to be a power-series about w with coefficient sequence A.2 Remark 6.6 If w is in the domain of the analytic h then, by Theorem 5.20, ∞ function n h(p) (w) p there is a b > 0 such that the power-series p=0 p! (z − w) n=0 has the limit h uniformly on {x ∈ C | |x − w| ≤ b}. Exercise 6.4 1.  If B is an infinite sequence in C with bounded range, then the power-series n Bp p ∞ about 0 is uniformly Cauchy on each closed and bounded p=0 p! z n=0 subset of C. 2. If f is an analytic function and there is a number M such that for all k ∈ N it is true that |f (k) (0)| ≤ M, then there is only one entire function of which f is a subset. Definition 6.7 The exponential function, denoted by “exp” is that entire function f such that f  = f and f (0) = 1. Exercise 6.5 1. exp(z) is the limit on C of the power-series ⎧ ⎫∞ n ⎨ 1 p⎬ z ⎩ p! ⎭ p=0

n=0

about 0. 2. exp(¯z) = exp(z) and exp(z) = 1 + [0;z] exp. 3. If each of v, w ∈ C, then exp(v) exp(w) = exp(v + w). 4. The restriction of exp to R is an increasing function with range the set of all positive real numbers. 5. If a ∈ R then Log(exp(a)) = a and, if a > 0, exp(Log(a)) = a. 6. If w is in the domain of Log, then exp(Log(w)) = w. 7. exp(−iπ ) = −1 and exp(iπ ) = −1. 8. If m ∈ Z, then exp(z + 2mπ i) = exp(z). 9. If exp(z) = 1, then 2πz i is an integer. (Hint: if not, then there is an integer m such that −π < zi + π − 2mπ < π .) 10. | exp(z)| = exp(Re z).

2 In

what follows, notice that Mac Nerney distinguishes between a power-series and a function that is the limit In many texts a “power-series” about w would be an expression  of the power-series. p f (z) = ∞ p=0 Ap (z − w) treated as a function of z, and referring to the limit of the sequence.

6 Extensions, Contours, Elementary Functions

39

Definition 6.8 If w is a point and r > 0 then the principal circular contour with center w and radius r, denoted by Cr (w), is the path which contains the restriction to [0; 1] of the function w + r exp((2z − 1)π i). Exercise 6.6 Let x, w ∈ C and r > 0. 1. 2. 3. 4.

C1 (0) = exp /[−iπ ; iπ ] and λ(C1 (0)) = 2π . Cr (w) = (w + rI )/C1 (0). If |x − w| = r, then x ∈ Cr (w) . If |x − w| = r, then x ∈ / Cr (w) . Furthermore, accordingly as |x − w| > r or |x − w| < r.

1 2π i



1 Cr (w) x−I

is 0 or 1,

Theorem 6.9 If K is a path from A to B and h is acomplex function such that h  is continuous on K  and 0 ∈ / h(K  ), then exp K hh = h(B) h(A) . Definition 6.10 If K is a closed path and w ∈ C \ K  then the winding number of 1 . If W (K, w) ∈ {0, 1} K about w, denoted by W (K, w), is the integer 2π1 i K I −w  for all w ∈ C \ K , the path K is called a contour. Remark 6.11 If the complex function f is analytic in R, K is a closed path in R,  and w ∈ / K  then W (f/K, w) = 2π1 i K f f−w . Theorem 6.12 If the complex function f is analytic w ∈ R, and r > 0 such  in R, ∞ f (p) (w) n p that the disc Dr (w) ⊆ R, then the power-series p=0 p! (z − w) n=0 about w has the limit f on Dr (w) and is uniformly Cauchy on each closed point set which lies in Dr (w). Definition 6.13 If w ∈ C then zw denotes an analytic function which contains exp ◦(w Log) but is not a proper subset of an  analytic function. The binomial w sequence of w, whose values we denote by w p , is defined recursively by 0 = 1,  w  w−p w and p+1 = p+1 p for each integer p > 0.  w p ∞ n Exercise 6.7 If w ∈ C, the binomial power-series has the limit p=0 p z n=0 w (1 + z) on each nonempty U ⊆ C, and is uniformly Cauchy on U if U is closed. Theorem 6.14 Suppose g is an analytic function with domain R1 and R2 = {¯z | z ∈ R1 } and R3 = R1 ∩ R2 and R = R1 ∪ R2 . The following statements are true: 1. There is an analytic function h such that if z ∈ R2 then h(z) = g(¯z). 2. Each component of R3 is a region which contains a real number. 3. If there is a component D of R3 and a subset S of D with a limit-point in D such that if z ∈ S then g(z) = g(¯z), then there is an analytic function f which is an extension of g to R. Definition 6.15 The sine function, denoted by “sin”, is the entire function exp(iz) − exp(−iz) , 2i

40

6 Extensions, Contours, Elementary Functions

the cosine function, denoted by “cos”, is the entire function exp(iz) + exp(−iz) , 2 and the tangent function, denoted by “tan”, is the analytic function sin . cos Exercise 6.8 1. 2. 3. 4. 5. 6. 7. 8.

If z¯ = z then cos(z) = Re exp(iz) and sin(z) = Im exp(iz). If |w|  = 1 and  u = Arg w then cos(u) = Re w and sin(u) = Im w. cos π2 − z = sin(z). sin(z) = 0 implies πz is an integer. tan(z + π ) = tan(z). sin(2z) = 2 sin(z) cos(z) and cos[2z] = (cos(z))2 − (sin(z))2 . (sin) (z) = cos(z) and (cos) (z) = − sin(z) and (tan) (z) = 1 + (tan(z))2 . If A is the function     1  Re(x) = 0 or − 1 < Im(x) < 1 , x, 2  [0;x] 1 + I then A−1 is a subset of tan.

Chapter 7

Power-Series, Residues, Singularities

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_8

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Definition 7.1 The sequence {tn }∞ absolutely if there is n=0 ⊆ C is said to converge  a β > 0 such that for each positive integer n we have np=1 |tp − tp−1 | ≤ β. If f is a sequence each value of which is a complex function then 1. For x ∈ C, f converges absolutely at x if the sequence {fn (x)}∞ n=0 converges absolutely. 2. For S ⊆ C, f converges absolutely on S if f converges absolutely at each point of S. Theorem 7.2 Suppose  ∞ A is an infinite sequence in C and S is the power-series n p about w. If S converges at a point y different from w, and p=0 Ap (z − w) n=0 if D = {x ∈ C | |x − w| < |y − w|}, then S converges absolutely on D and converges uniformly on each closed subset of D. Theorem 7.3 Suppose  ∞ A is an infinite sequence in C, and S is the power-series n p about w. One of the following statements is true: p=0 Ap (z − w) n=0

1. S is totally divergent, i.e., S converges only at w; 2. S is totally convergent, i.e., S converges at every point. 3. S has a radius of convergence, i.e., there is a real number r > 0 such that S converges at x when |x − w| < r and S does not converge at x when |x − w| > r. In this case, r is called the radius of convergence of S. Theorem 7.4 Suppose  ∞ A is an infinite sequence in C and S is the power-series n p about w, and {sn }∞ p=0 Ap (z − w) n=0 is a sequence in R such that if n is n=0

a positive integer then sn = |An |1/n . The three cases described in Theorem 7.3 are characterized respectively as follows: 1. S is totally divergent if the range of s is not bounded. 2. S is totally convergent if s has the limit 0. 3. S has the radius of convergence r if the range of s is bounded, s does not have the limit 0, and 1r is the greatest number which is a cluster-point of s. Definition 7.5 An annulus is a set of the form {z ∈ C | m < |z − w| < M}, where w ∈ C and m and M are real numbers such that 0 < m < M. This set can be denoted by ann(w; m, M). Exercise 7.1 Given the annulus A = ann(w; m, M), and the set S = {z ∈ C | Log(m) < Re z < Log(M)}, and the function h(z) = w + exp(z), show that h maps S onto A and, if m < r < R < M, 1. 2. 3. 4.

h/[Log(R) − iπ ; Log(R) + iπ ] = CR (w) h/[Log(R) + iπ ; Log(r) + iπ ] = [w − R; w − r]. h/[Log(r) + iπ ; Log(r) − iπ ] = Cr (w). h/[Log(r) − iπ ; Log(R) − iπ ] = [w − R; w − r].

7 Power-Series, Residues, Singularities

43

Theorem 7.6 (Annulus Theorem) Suppose the complex function f is analytic in the annulus A = ann(w; m, M). 1. If m < r < R < M, then CR (w) f = Cr (w) f . 2. If D1 is the disc {z ∈ C | |z −w| < m1 } and D2 is the disc {z ∈ C | |z −w| < M}, then there is only one ordered pair (g, h) satisfying all of the following: • • • •

g is an analytic function with domain D1 ; h is an analytic function with domain D2 ; g(w) = 0;   1 for all z ∈ A, f (z) = g w + z−w + h(z).

3. There is a function s with domain Z such that if m < r < M, then sn =

1 2π i

Cr (w)

f . (I − w)n+1

If, moreover, (g, h) is the ordered pair determined in part (2), then s0 = h(w) and for each positive integer n s−n =

g (n) (w) h(n) (w) and sn = . n! n!

Remark 7.7 Given a “bi-infinite” sequence of complex numbers {tn }n∈Z , it seems reasonable to define the limit of such as being a complex number z such that for every  > 0 thereis a positive integer n such that if each of u and v is a positive integer, then |z − n+v When the limit of a bi-infinite sequence exists p=−n−u tp | < .  in this sense, it is often written as ∞ n=−∞ tn . If the sequence is a sequence of functions rather than complex numbers, it is reasonable to consider the function  whose value at z is ∞ n=−∞ tn (z). In accordance with Theorem 6.12, the representation of f in the annulus A, in terms of g and h as in Theorem 7.6 (2, 3), suggests the following interpretation: the bi-infinite power-series {sn }n∈Z can have a limit at each set S lying  point of a closed p in the annulus A, and in that case the function f is ∞ n=−∞ sp (z − w) . In the notation of Theorem 7.6, if there is an entire function of which g is a subset then, of course, there is an analytic extension of f to a set containing {z ∈ C | 0 < |z − w| < M}, and in this case s−1 is called the residue of f at w, and is seen to k to A is the derivative of a be the only k ∈ C such that the restriction of f (z) − z−w function. Exercise 7.2 In the sense of thepreceding Remark, if theanalytic functions f ∞ p p and F have the representations ∞ p=−∞ bp (z − w) and p=−∞ Bp (z − w) , respectively; in A and m < r < M,

44

7 Power-Series, Residues, Singularities

=

1 2π

1 2π i

[−π ;π ]

Cr (w)

fF I −w

f (w + r exp(iI ))F (w + r exp(iI )) =

∞ 

bp Bp r 2p .

p=−∞

Theorem 7.8 If R is a region and {wp }np=0 is a one-to-one sequence in C with range S lying in R, and the complex function f is analytic in R \ S, then there is only one sequence {gp }n+1 p=0 such that 1. if p is an integer in [0, n] then gp is an entire function and gp (wp ) = 0, 2. gn+1 is an analytic function with domain R, and  n 1 + gn+1 (z). 3. if z ∈ R \ S then f (z) = p=0 gp wp + z−w p Theorem 7.9 (Residue Theorem) Suppose R is a simple region and {wp }np=0 is a one-to-one sequence with range S lying in R, and the complex function f is analytic in R \ S. If, for each integer p in [0, n], kp is the residue of f at wp , and if K is a closed path in R \ S, then 1 2π i

f = K

n 

kp W (K, wp ) .

p=0

Definition 7.10 The complex function f has order j at the point w if all of the following items are satisfied: 1. w is a limit-point of the domain of f ; 2. there is a b ∈ R for which there exist δ > 0 and   > 0 such that for all z in the  f (z)  domain of f , 0 < |z − w| < δ implies  (z−w) b  ≤ ; 3. j is the least upper bound of the set of all b’s described in the preceding item. Exercise 7.3 The complex function z1/2 has order

1 2

at the point 0.

Theorem 7.11 If the complex function f is analytic in R and w ∈ R, 1. f does not have negative order at w. 2. if f does not have order at w, then f (z) = 0 for each z ∈ R. 3. if f has order j at w, then j ∈ Z. Furthermore, there is a function g which is analytic in R such that g(w) = 0 and f (z) − f (w) = (z − w)j g(z) for every z ∈ R different from w. Theorem 7.12 Suppose g is an entire function, w ∈ C, and that b, δ and  are  g(z)  positive real numbers such that for all z ∈ C with |z−w| > δ we have  (z−w) b  ≤ . If n is an integer greater than b, then g (n) (w) = 0, so that either g is constant or g is a polynomial of degree less or equal to b.

7 Power-Series, Residues, Singularities

45

Theorem 7.13 Suppose f is a complex function and R is a region, and suppose there exists w ∈ R such that f has order j at w and f is analytic in R \ {w}. Then j is an integer and there is a function g, analytic in R, such that g(w) = 0 and f (z) = (z − w)j g(z) if z is in R and z = w. Theorem 7.14 Suppose R is a region and {wp }np=0 is a one-to-one sequence with range S lying in R, and the complex function f is analytic in R \ S. If, for each integer p ∈ [0, n], jp is the order of f at wp then 1. there is a function g which is analytic in R such that 0 ∈ / g(S) and, for all z ∈ R \ S, f (z) = g(z) np=0 (z − wp )jp ; 2. if K is a closed path in R \ S, and 0 ∈ / f (K  ), and R is simple, then 1 2π i

K

n  f = jp W (K, wp ) . f p=0

Definition 7.15 Suppose f is a complex function and R is a region, and that there exists w ∈ R such that f is analytic in R \ {w}. Then 1. w is a zero of f provided f has positive order at w. 2. w is a pole of f provided f has negative order at w. 3. w is an essential singularity of f if there do not exist  a b ∈ R, δ > 0, and  > 0  f (z)  such that if z ∈ R and 0 < |z − w| < δ then  (z−w)b  ≤ . Theorem 7.16 Suppose f is a complex function and R is a region, and that there exists w ∈ R such that f is analytic in R \{w}, and that w is an essential singularity of f . For every x ∈ C and positive real numbers  and δ, there exists a z ∈ R such that 0 < |z − w| < δ and |f (z) − x| < . Corollary 7.17 If the entire function g is not constant and g is not a polynomial then, for each x ∈ C and each pair of positive real numbers δ and , there exists a z ∈ C such that |z| > δ and |g(z) − x| < . Exercise 7.4 Now 0 is an essential singularity of exp(−1/z2 ), but (0, 0) together with the restriction of exp(−1/z2 ) to the set R \ {0} is “infinitely differentiable.”

Chapter 8

Analytic Inverses, Standard Regions, Convergence Continuation

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_9

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8 Analytic Inverses, Standard Regions, Convergence Continuation

Theorem 8.1 (Open Mapping Theorem) If the complex function f is analytic and R is a region lying in the domain of f , then either f is constant or f (R) is a region. Hint: if f is not constant and w ∈ R then there exists an integer n > 0 and a complex function g which is analytic in R such that g(w) = 0 and f (z) − f (w) = (z − w)n g(z) for each z ∈ R. Let r > 0 be such that if |z − w| ≤ r then z ∈ R and g(z) = 0, and show that f (Dr (w)) contains a component of C \ f (Cr (w) ). Exercise 8.1 1. If f is a nonconstant analytic function with domain R and S is a bounded open set such that S¯ lies in R then |f (z)| < |f |S for all z ∈ S. 2. Each one-to-one entire function is a polynomial. Theorem 8.2 If u and v are complex functions which are analytic in the region R, and is a closed path in R such that |u(z) − v(z)| < |v(z)| for each z ∈ K  , then uK  v K u = K v. Hint: if h = in U .

u−v v

then h is analytic in a region containing K  and h/K is a path

Theorem 8.3 If the complex function f is analytic in R and w ∈ R, the following two statements are equivalent: 1. f  (w) = 0. 2. For every  > 0 there exist distinct p, q ∈ C such that |p −w| ≤  and |q −w| ≤  and f (p) = f (q). Hint: if  has the property that if p and q are distinct points and |p − w| ≤  and |q − w| ≤  then p and q are in R and f (p) = f (q), then there is a p ∈ R such that 0 < |p − w| <  and if |z − w| =  then      f (z) − f (p) f (z) − f (w)   f (z) − f (w)      −   0 and S is a closed and bounded set lying in R, then there is a one-to-one complex function f which is analytic in R such that for all z ∈ S, we have |f (z) − g(z)| < . Either g is constant or g is one-to-one. Hint: if p and q are points in R such that g(p) = g(q) and g is not constant, there is a b > 0 such that if 0 < |z − p| ≤ b then z ∈ R and g(z) = g(p). Let f be a one-to-one complex function which is analytic in R such that if z is such that |z − p| = b then |[f (z) − f (q)] − [g(z) − g(q)]| < |g(z) − g(q)|. Theorem 8.5 If f is a one-to-one analytic function then f −1 is analytic. Lemma 8.6 Let w be in the domain R of f , and let r > 0 be such that Dr (w) ⊆ R.

8 Analytic Inverses, Standard Regions, Convergence Continuation

49

1. f (Dr (w)) is a component of C \ f (Cr (w) ), and  2. if x ∈ f (Dr (w)) then f −1 (x) = 2π1 i Cr (w) fzf−x . Theorem 8.7 If the complex function h is analytic in the simple region R and the restriction of h to R is one-to-one, then h(R) is simple. Moreover, for each closed path K in R, 1. if P ∈ / h(R) then W (h/K, P ) = 0, and 2. if w ∈ R \ K  then W (h/K, h(w)) = W (K, w). Theorem 8.8 Let R be a region, and A ∈ R, and B ∈ C \ R. Suppose that either R is simple or B belongs to an unbounded component of C \ R. 1. There is a one-to-one analytic function g with domain R such that if K is a path 1 from A to x in R then g(x) = (A − B) K I −B . 2. There is a one-to-one analytic function h having R as its domain and having slope 1 at (A, 0) such that h(R) is bounded. Theorem 8.9 Suppose A is a point in the region R, h is a one-to-one analytic function with domain R and slope 1 at (A, 0), and b > 0 such that h(R) = Db (0). If k is an analytic function with domain R and slope 1 at (A, 0) and |k|R ≤ b, then k = h. Lemma 8.10 If g is an analytic function with slope 1 at (0, 0) and |g(z)| ≤ b for each z such that |z| < b, then g is a restriction of the identity function. Theorem 8.11 Let h be a one-to-one analytic function with domain R which has slope 1 at (A, 0) and such that h(R) is bounded. Also suppose there exists B ∈ C \ h(R) such that |B| < |h|R . If b > 0 and b2 = |B|/|h|R , and either R is simple or B belongs to the unbounded component of C \ h(R), then 1. there is a one-to-one analytic function g with domain R such that if K is a path from A to z in R then g(z) = b exp

    1 1 b4 − 4 h , 2 K h−B b h−B

−1 2. g has slope b b2B at (A, b) and |g|R = 1, and 3. the function 4

f (z) =

1 2B g(z) − b b 1 + b2 bg(z) − 1

is one-to-one and analytic with domain R and slope 1 at (A, 0) such that |f |R =

2b |h|R < |h|R . 1 + b2

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8 Analytic Inverses, Standard Regions, Convergence Continuation

Theorem 8.12 For a region R, let GR be the collection of closed and bounded subsets of R. Suppose A is a function from GR to a set of positive numbers, and f is a sequence of complex functions which are analytic in R such that, for every n ∈ N and S ∈ GR , we have |fn |S ≤ A(S). 1. There is a function B from GR to a set of positive numbers such that, for each n ∈ N and S ∈ GR , we have |fn |S ≤ B(S). 2. If S ∈ GR and r > 0 and S + r ⊆ R then, if n ∈ N and each of w, z ∈ S, we have |fn (w) − fn (z)| ≤ |w − z|B(S + r). 3. There is a sequence t with range T a subset of R such that, for all z ∈ R and b > 0, there is an m ∈ N satisfying |z − tm | < b. 4. There is a subsequence of f which converges on T . 5. If S ∈ GR and b > 0, there is a positive integer n such that if z ∈ S then there is an integer p ∈ [0, n] such that |z − tp | ≤ b. 6. If f converges on T then f converges uniformly on each member of GR . Exercise 8.2 If R is a region and f is a sequence of complex functions whose domains contain R, the following two statements are equivalent: 1. There is a continuous complex function g with domain R such that f has the limit g uniformly on each closed and bounded set lying in R. 2. f is continuously convergent on R in the sense that if z is a sequence with range lying in R and with a limit in R then the sequence w given by wn = fn (zn ) converges. Theorem 8.13 Suppose R is a region, A ∈ R and B ∈ C\R, and either R is simple or B belongs to an unbounded component of C \ R. Let H be the collection of oneto-one analytic functions which have domain R and bounded range, and which have slope 1 at (A, 0). Let b be the greatest lower bound of {|h|R | h ∈ H}. Then b > 0 and there is a one-to-one analytic function h from R onto a bounded region such that 1. h has slope 1 at (A, 0) and |h|R = b, 2. the unbounded component of C \ h(R) is C \ Db (0), and 3. if R is simple then h(R) = Db (0). Remark 8.14 The number b which is described in the statement of Theorem 8.13 is sometimes called the analytic radius of R with respect to A. Exercise 8.3 1. If w ∈ C and r > 0 then r is the analytic radius of Dr (w) with respect to w, and the restriction of the function z − w to Dr (w) is that one-to-one analytic function h with domain Dr (w) and slope 1 at (w, 0) such that h(Dr (w)) is a disc with center 0. 2. If k ∈ U then 1 − |k|2 is the analytic radius of U with respect to k, and the restriction of (1 − |k|2 ) z−k to U is that one-to-one analytic function h with 1−kz domain U and slope 1 at (k, 0) such that h(U ) is a disc with center 0.

8 Analytic Inverses, Standard Regions, Convergence Continuation

51

3. If A is a point in the region R and f is a one-to-one analytic function from R onto U which has positive slope at (A, 0), then f  1(A) is the analytic radius of R with respect to A. 4. 2 is the analytic radius of the right half-plane with respect to 1, and 4 is the analytic radius of the range of Log with respect to 0. Theorem 8.15 Let R1 and R2 be simple regions, neither of which are C. If A1 ∈ R1 and A2 ∈ R2 , then there is only one one-to-one analytic function from R1 onto R2 which has positive slope at (A1 , A2 ). Exercise 8.4 1. If A is a point in the region R and h is a one-to-one analytic function from R  h−h(A) onto U and f = |hh (A)| (A) 1−h(A)h then f is the one-to-one analytic function from R onto U which has positive slope at (A, 0). 2. If f is a one-to-one analytic function from U onto U , then there exist a v ∈ C with modulus 1 and a k ∈ U such that f is a restriction of v¯ k−z . 1−kz 3. Suppose f is an analytic function from U into U satisfying f (0) = 0. If g is the  function f (z) z together with (0, f (0)), then g is analytic in U , and |g(0)| ≤ 1, and g(U ) is a subset of U unless g is constant. Theorem 8.16 (Convergence Continuation Theorem) If, with the suppositions of Theorem 8.12, the sequence f converges at each point of a set which has a limitpoint in R, then f is continuously convergent on R. Exercise 8.5 (*) 1. Let k be an infinite sequence with range lying in U ⊆ C such that if n ∈ N and |kn | = 1 then kn+p = 0 for each positive integer p. If S and t are sequences such that S0 = t0 and if n ∈ N then (a) tn is a function from U such that, if x ∈ U , tn (x) is the function kn +

(1 − |kn |2 )x kn x +

1 z

together with (0, kn )

and (b) Sn+1 is a function from U such that, if z ∈ U , Sn+1 (z) = Sn (z) ◦ tn+1 (z), then the relation f =U×



{Sn (U ) | n ∈ N}

is an analytic function from U into U . Conversely, if f is an analytic function from U into U then there is only one . . .

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8 Analytic Inverses, Standard Regions, Convergence Continuation

2. Suppose b ∈ R and q is a nondecreasing function from [−π, π ] to a set of numbers such that q(π ) − q(−π ) > 0, and let z denote the complex identity function. Then the function f defined by

f (x) =

π

−π

exp(iz) + x dq + ib exp(iz) − x

is an analytic function from U into the right half-plane. Complete and prove the following: conversely, if f is an analytic function from U into the right half-plane then there is . . . .

Chapter 9

Extended Complex Plane, Linear-Fractional Transformations, Meromorphic Functions

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7_10

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9 C∞ , Linear-Fractional Transformations, Meromorphic Functions

54

Definition 9.1 The unit sphere,1 denoted by S 2 , is {(u, v, w) ∈ R × R × R | u2 + v 2 + w 2 = 1}. The extended complex plane, denoted by C∞ , is formed by adjoining a new point ∞, which is not in C, to the set C. That is, C∞ = C ∪ {∞}. The ordered triple (0, 0, 1) ∈ S 2 is called the point at infinity and is also denoted by the symbol ∞. Remark 9.2 For readers who are uncomfortable with adjoining a new element to C as above, it should be helpful to revisit and extend the geometric picture emphasized for C earlier in the book into R × R × R in the following way. Instead of identifying a complex number with a point in the plane R × R, we can use a plane in R × R × R by identifying x ∈ C with (Re x, Im x, 0). Then it is convenient to choose (0, 0, 1) = ∞ to be the new point adjoined, so that we have the following model of C∞ in R × R × R: {(Re x, Im x, 0) | x ∈ C} ∪ {(0, 0, 1)} Defining C∞ in terms of C and ∞ as done above allows convenient definitions of functions from C∞ without the detour through R × R × R. Exercise 9.1 Let T be the relation in C∞ × S 2 given by 



2 Re x 2 Im x |x|2 − 1 x, , , |x|2 + 1 |x|2 + 1 |x|2 + 1

   x ∈ C {(∞, ∞)} 

1. T is a one-to-one function from C∞ onto S 2 , and if (u, v, w) is in the unit sphere other than ∞ then T −1 (u, v, w) =

u + iv 1+w and |T −1 (u, v, w)|2 = . 1−w 1−w

2. if z ∈ C and T (z) = (u, v, w) then 4 = (u − 0)2 + (v − 0)2 + (w − 1)2 . +1

|z|2

3. if x and y are complex numbers and T (x) = (u, v, w) and T (y) = (r, s, t) then 4|x − y|2 = (u − r)2 + (v − s)2 + (w − t)2 . (|x|2 + 1)(|y|2 + 1)

1 In

keeping with the geometric program, Mac Nerney used “number-sphere” here.

9 C∞ , Linear-Fractional Transformations, Meromorphic Functions

55

Definition 9.3 The statement that t is a linear-fractional transformation means that t is a subset of C∞ × C∞ and there is an ordered quadruple (a, b, c, d) ∈ C × C × C × C such that the following statements hold true: 1. ad − bc = 0,   2. if c = 0 then t = { x, ax+b | x ∈ C} ∪ {(∞, ∞)}, and d   ax+b 3. if c = 0 then t = { x, cx+d | x ∈ C} ∪ {(− dc , ∞), (∞, ac )}. Remark 9.4 If (a, b, c, d) is an ordered quadruple of complex numbers and ad − bc = 0, then there exactly one linear-fractional transformation which is an extension of the function az+b cz+d . Hence, it is customarily agreed that for x ∈ C, 1. x + ∞ = ∞ since (∞, ∞) belongs to the linear-fractional transformation which extends f (z) = x + z. 2. if x = 0 then x∞ = ∞ since (∞, ∞) belongs to the linear-fractional transformation which extends f (z) = xz. x 3. if x = 0 then x0 = ∞ and ∞ = 0 since both (0, ∞) and (∞, 0) belong to the linear-fractional transformation which extends f (z) = xz . In each of the three cases above, the reader is invited to work out exactly which tuple (a, b, c, d) corresponds to the function. Exercise 9.2 1. If each of t1 and t2 is a linear-fractional transformation then t1 ◦ t2 is a linearfractional transformation. 2. Each linear-fractional transformation t is a one-to-one function from C∞ onto C∞ , and t −1 is a linear-fractional transformation as well. 3. If (∞, ∞) belongs to the linear-fractional transformation t then there exist A, B ∈ C such that t is an extension of f (z) = (1 − z)A + zB. 4. If (∞, ∞) does not belong to the linear-fractional transformation t then there is an ordered triple (A, B, C) ∈ C × C × C such that B = 0 and t contains {(x, y) ∈ C × C | (y − C)(x − A) = B} Theorem 9.5 If a, b, and c are three distinct members of C∞ and each of A, B, and C is a member of C∞ , then there is only one linear-fractional transformation to which all of (a, A), (b, B), and (c, C) belong. Hint: consider the linear-fractional transformation which extends the function z−a b−c b−a z−c .

Exercise 9.3 If t is the linear-fractional transformation to which all of (−1, −1), (0, i), and (1, 1) belong then (∞, −i) belongs to t, and the t-image of the lower halfplane is the unit-disc, and the t-image of the real line is {x ∈ C | |x| = 1, x = −i}.

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Definition 9.6 The extended real line, denoted by R∞ 2 is the subset R ∪ {∞} of C∞ . The statement that K is a circle (in the extended number-plane) means that there is a linear-fractional transformation t such that K is the t-image of R∞ . The symbol J denotes the function of C∞ to C∞ given by {(x, x) ¯ | x ∈ C} ∪ {(∞, ∞)}. Exercise 9.4 1. If t is a linear-fractional transformation then so is J ◦ t ◦ J 2. If t is a linear-fractional transformation then, in order that t (R∞ ) = R∞ , it is necessary and sufficient that J ◦ t = t ◦ J . 3. If A, B, and C are three distinct members of C∞ then there is only one circle to which all of A, B, and C belong. 4. If t1 and t2 are linear-fractional transformations and t1 (R∞ ) = t2 (R∞ ), then t1 ◦ J ◦ t1−1 = t2 ◦ J ◦ t2−1 . Definition 9.7 If K is a circle, the inversion of C∞ in (or with respect to) K is a function T such that if t is a linear-fractional transformation and K = t (R∞ ), then T is t ◦ J ◦ t −1 . Exercise 9.5 1. If T is the inversion of C∞ in the circle K, then P ∈ K if (P , P ) ∈ T . 2. J is the inversion of C∞ in the extended real line R∞ . 3. If each of T1 and T2 is the inversion of C∞ in a circle, then T1 ◦ T2 is a linearfractional transformation. 4. If T is the inversion of C∞ in the circle K and t is a linear-fractional transformation, then t ◦ T ◦ t −1 is the inversion of C∞ in the circle t (K). Theorem 9.8 If t is a linear-fractional transformation, there is a sequence {Tp }7p=0 , each value of which is an inversion of C∞ in a circle, such that t = T1 ◦ T 2 ◦ T 3 ◦ T 4 ◦ T 5 ◦ T 6 ◦ T 7 ◦ T 0 . Lemma 9.9 (1) If b > 0, and T0 is the inversion of C∞ in the circle through the points {1, i, −1}, and T1 is the inversion of C∞ in the circle through the points {b, bi, −b}, then T1 ◦ T0 is the linear-fractional transformation which extends b2 z. Lemma 9.10 (2) If w ∈ C other than 0, T0 is the inversion of C∞ in the circle through the points {0, iw, ∞}, and T1 is the inversion of C∞ in the circle through the points {w, w+iw, ∞}, then T1 ◦T0 is the linear-fractional transformation which extends z + 2w. Lemma 9.11 (3) If u ∈ C with |u| = 1 and T is the inversion of C∞ in the circle through the points {0, u, ∞}, then T ◦J is the linear-fractional transformation which extends u2 z.

2 This can also be called the projectively extended real line, so as not to be confused with R adjoined with two new distinct elements −∞ and ∞.

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Lemma 9.12 (4) There is an inversion T of C∞ in a circle such that J ◦ T extends 1z . Definition 9.13 The chordal metric is a function from C∞ × C∞ , with value at (x, y) denoted by the symbol chd(x, y) and called the chordal distance from x to y. For each x, y ∈ C, 1. chd(x, y) = 2|x − y|(|1 + xy| ¯ 2 + |x − y|2 )−1/2 , 2. chd(x, ∞) = chd(∞, x) = 2(1 + |x|2 )−1/2 and 3. chd(∞, ∞) = 0. Remark 9.14 The chordal metric has the following properties (see Exercise 9.1): 1. 0 ≤ chd(x, y) ≤ 2, chd(x, y) = 0 only in case x = y, chd(x, y) = 2 only in case (x, y) = 0, ∞ or (x, y) = (∞, 0) or y = −1 x . 2. chd(x, y) = chd(y, x) and chd(x, z) ≤ chd(x, y) + chd(y, z). 3. if x = 0 and x = ∞ then chd (x, ∞) = chd( x1 , 0). Theorem 9.15 If t is a linear-fractional transformation then t is continuous with respect to the chordal metric in the sense that if (p, q) ∈ t then, for each positive  > 0, there is a δ > 0 such that if (x, y) ∈ t and chd(x, p) < δ then chd(y, q) < . Theorem 9.16 If t is a linear-fractional transformation, then chd(t (x), t (y)) = chd(x, y) for all x, y ∈ C∞ if and only if there are k, v ∈ C with |v| = 1 such that t contains     k−w k−z  . (z, w) ∈ C × C  =v 1 + kw 1 + kz   Definition 9.17 The complex function f has order j at ∞ provided f 1z has order j at 0. If R > 0 and the complex function f is analytic at each point with modulus greater than R then ∞ is a zero, a pole, or an essential singularity of f  accordingly as 0 is a zero, a pole, or an essential singularity of f 1z . Definition 9.18 The statement that F is a meromorphic function means that F is a function from a subset of C∞ into C∞ such that 1. either the domain of F is C∞ or there is a linear-fractional transformation t such that the domain of F is the t-image of a region, and 2. either F is constant or there is an analytic function g contained in F such that if (p, q) ∈ F then, for each real  > 0, there is a δ > 0 such that if x ∈ C∞ and 0 < chd(x, p) < δ then chd(g(x), q) < . Theorem 9.19 If F is a nonconstant meromorphic function with domain C∞ , there exist polynomials P and Q such that F is the meromorphic function with domain P C∞ which contains the function Q . Exercise 9.6 (*) Is it true that each one-to-one meromorphic function with domain C∞ is a linear-fractional transformation?

Chapter 10

Analytic Relations, Analytic Continuation, Functional Boundaries, Branch-Points

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Definition 10.1 An analytic relation is a relation F , each member of which belongs to an analytic function contained in F , such that if g and h are analytic functions contained in F then there is an ordered pair (x, y) such that 1. each of x and y is a continuous function from [0, 1] to C and x is not constant on any subinterval of [0, 1]; 2. there is a real number b satisfying 0 < b < 1 such that for all s ∈ [0, b] we have g(x(s)) = y(s), and for all t ∈ [1 − b, 1] we have h(x(t)) = y(t); and 3. if 0 < u < 1 then there is a c > 0 and an analytic function k contained in F such that for all v ∈ [u − c, u + c], we have k(x(v)) = y(v). Exercise 10.1 1. Every analytic function is an analytic relation. 2. If F and G are analytic relations, and H = F ∪ G, and there is an analytic function which is contained in F ∩ G, then H is an analytic relation. 3. If F is an analytic relation then there is an analytic relation which contains F and is not contained in any other analytic relation. In other words, there exists an analytic relation maximal with respect to containing F . 4. Is it true that if f is an analytic function then there is an analytic function which contains f and is not contained in any other analytic function? Theorem 10.2 If g and h are analytic functions contained in the analytic relation F , g(A1 ) = A2 , and h(B1 ) = B2 , then there is an n ∈ N and a sequence {fp }n+1 p=0 of analytic functions, all of which are contained in F , such that 1. (A1 , A2 ) ∈ f0 ⊆ g and (B1 , B2 ) ∈ fn+1 ⊆ h, and 2. if p ∈ [0, n] is an integer, then fp ∪ fp+1 is an analytic function. Exercise 10.2 Let {fp }n+1 p=0 be a sequence of analytic functions such that for every integer p ∈ [0, n], we have that fp ∪ fp+1 is an analytic function. Then for each A in the domain of f0 and each B in the domain of fn+1 there exist sequences {zp }n+2 p=0 and {Mp }n+1 p=0 such that 1. z0 = A, zn+2 = B, and if p ∈ [0, n] is an integer, then zp+1 belongs to the domain of fp and to the domain of fp+1 , and 2. if p ∈ [0, n + 1] is an integer, then Mp is a path from zp to zp+1 in the domain of fp . Definition 10.3 If g is an analytic function and K is a path, the statement that F is an analytic continuation of g along K means that F is an analytic relation containing g, K is a path from a point A in the domain of g to a point B, and there exist an n ∈ N and a sequence {fp }n+1 p=0 of analytic functions contained in F , and n+1 sequences {zp }n+2 p=0 and {Mp }p=0 such that

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1. f0 ⊆ g and if p ∈ [0, n] is an integer, then fp ∪ fp+1 is an analytic function; 2. z0 = A and is in the domain of f0 , zn+2 = B and is in the domain of fn+1 , and for every integer p ∈ [0, n], zp+1 belongs to the domain of fp and to the domain of fp+1 ; 3. for every integer p ∈ [0, n + 1], Mp is a path from zp to zp+1 in the domain of fp ; and  4. K = n+1 p=0 Mp . Exercise 10.3 Let g be an analytic function and (A1 , A2 ) ∈ g. There exists an analytic relation containing g and (B1 , B2 ) if and only if (B1 , B2 ) belongs to an analytic continuation of g along a path from A1 to B1 . Theorem 10.4 (Monodromy Theorem) Suppose that g is an analytic function, that A is in the domain of g, and that R is a simple region containing the domain of g. If, for each point B ∈ R and each path K from A to B in R, there is an analytic continuation of g along K, then there exists an analytic function with domain R which contains g. Hint: show that, by Theorem 8.13, it suffices to consider the case that A = 0 and R is either all of C or else a disc with center 0. In this case, it suffices to consider continuations of g along linear paths from 0 to points of R. Definition 10.5 If F is an analytic relation, the statement that w is an F -boundarypoint of R means 1. R is the domain of an analytic function g which is contained in F and w belongs to the boundary of R; and 2. there do not exist two analytic functions h1 and h2 , each of which is a subset of F , such that h1 ⊂ g ∩ h2 , w is in the boundary of the domain of h1 , and w is in the domain of h2 . ∞  n p Theorem 10.6 If the power-series A (z − w) about w has the p p=0 n=0 radius of convergence r and has the limit g on the disc Dr (w), and F is the analytic relation which contains g and is not contained in any other analytic relation, then there is a boundary-point of Dr (w) which is an F -boundary-point of Dr (w). Definition 10.7 If F is an analytic relation, the derivative of F , denoted by F  , is the function {(z, w) ∈ C × C | There exists an analytic function g ⊆ F such that g  (z) = w} . Exercise 10.4 If F is an analytic relation then F  is an analytic relation. Theorem 10.8 Every analytic relation is the derivative of an analytic relation. Remark 10.9 Compare this result with Theorem 4.9, and with Corollary 4.14. Theorem 10.10 Suppose R is a region, each of g and h is a complex function analytic in R, and 0 ∈ / g  (R). If F = {(g(z), h(z)) | z ∈ R}, then F is an analytic relation.

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Definition 10.11 A branch-point of the analytic relation F is a point B such that there exist three analytic functions g1 , g2 , and g3 , all contained in F , with respective domains R1 , R2 , and R3 , such that 1. B is a boundary-point of each of R1 , R2 , and R3 , 2. there is a real number r > 0 such that if z ∈ C and 0 < |z − B| < r then z ∈ R1 ∪ R2 ∪ R3 , 3. g1 ∪ g2 is an analytic function and g2 ∪ g3 is an analytic function, and 4. g3 ∪ g1 is not a function. Theorem 10.12 Suppose f is a nonconstant analytic function and that f has slope 0 at (A, B), and let g = {(C, D) ∈ f | the slope of f at (C, D) is not 0}. Then the set g −1 is an analytic relation, and B is a branch-point of g −1 . Lemma 10.13 (1) There exist an integer n > 1 and an analytic function g such that g(A) = 0 and f (z) = B + (z − A)n g(z) for all z in the domain of f. Lemma 10.14 (2) There exist an s > 0 and a one-to-one analytic function h such that h(B) = A and if z ∈ Ds (B) then f (h(z)) = B + (z − B)n . Hint: A belongs to a convex region D such that 0 ∈ / g(D). Let w ∈ C satisfy w n = g(A) and consider a complex function k such that for all z ∈ D   g 1 . k(z) = B + (z − A)w exp n [A;z] g Lemma 10.15 (3) Suppose there is an r > 0 such that 2r 2 < s 2 and R = {z ∈ C | 0 < |z − B| < s}. Consider the following three subsets: R1 = {z ∈ R | Re(z − B) > 0 or Im(z − B) > 0} , R2 = {z ∈ R | Re(z − B) < 0 or Im(z − B) > 0} , R3 = {z ∈ R | Re(z − B) < 0 or Im(z − B) < 0} . There exist one-to-one analytic functions 1 , 2 , and 3 , with respective domains R1 , R2 , and R3 , such that   1 , 1. if x ∈ R1 then 1 (x) = B + r 1/n exp n1 [B+r;x] I −B 2. 1 ∪ 2 is an analytic function and 2 ∪ 3 is an analytic function, and 3. 3 ∪ 1 is not a function. Exercise 10.5 1. Let log denote the inverse of the exponential function exp. Then log is an analytic relation (the natural logarithmic relation), and 0 is a branch-point of log.

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2. Suppose A, B, and C are points such that A = C, f (z) = B + (z − A)2 (z − C), and F is the inverse of {(z, w) ∈ f | f  (z) is defined and f  (z) = 0}. The point B is a branch-point of F belonging to the domain of F . 3. Use Theorem 10.10 to show that the set F = {(x, y) ∈ C × C | y = 0 and x 2 + y 2 = 1} is an analytic relation. Definition 10.16 A relation F ⊆ C × C is called a (planar) algebraic curve if there is an integer n > 0 and a complex matrix A, i,e., an (n + 1) × (n + 1) array of complex numbers where the entry in the pth row and qth column is denoted by Ap,q , such that there is a row Ap,− = [Ap,1 , Ap,2 , . . . , Ap,n , Ap,n+1 ] of A which is not uniformly zero, and ⎧ ⎨

 ⎫  n n ⎬  F = (x, y) ∈ C × C  Ap,q y p x q = 0 . ⎩ ⎭  p=0 q=0 Exercise 10.6 1. If F is an algebraic curve then the inverse relation F −1 is an algebraic curve. 2. If the algebraic curve F contains the analytic function g then F contains every analytic relation containing g. 3. If the analytic relation F is contained in an algebraic curve then F  is contained in an algebraic curve. 4. Extend the notion of branch-point to give meaning to the statement that ∞ is a branch-point of the analytic relation F . 5. Formulate a definition of meromorphic relation.

Appendix A

Homotopy Groups

Theorem A.1 If R is a simple region which is not all of C then C \ R does not have a bounded component. Lemma A.2 (1) Let h be a one-to-one analytic function from the unit-disc U onto the region R. If t is a real number in (0, 1), then C \ h(Ct (0) ) has only two components, and the bounded component is the h-image of the bounded component of C \ Ct (0) . Lemma A.3 (2) If S1 and S2 are disjoint closed and bounded subsets of C, then there exist disjoint open subsets T1 and T2 of C such that S1 ⊆ T1 and S2 ⊆ T2 . Lemma A.4 (3) If A ∈ C \ R and m > 0, then there exist B ∈ C such that |B − A| = m and a connected subset of C \ R which contains both A and B. Theorem A.5 Suppose R is a region, A ∈ R and B ∈ C \ R such that if K is a closed path in R then W (K, B) = 0. Then 1. there is a one-to-one analytic function with domain R and bounded range which has positive slope at (A, 0). 2. if A = 0 and b ∈ R such that |B| < |b|, then there exist c ∈ R and a one-to-one analytic function f with domain R and slope 1 at (0, 0) such that |f |R < |c|. Definition A.6 Let R be a region, A, B ∈ R, and K0 , K1 be two paths from A to B in R. We say K0 is homotopically equivalent to K1 in R if there exist p > 0 and a function F from [0, 1] × [0, 1] into R such that 1. F is continuous—in the sense that for all (s, t) ∈ [0, 1] × [0, 1] and  > 0, there is a δ > 0 such that if (u, v) ∈ [0, 1] × [0, 1] and |u − s| < δ and |v − t| < δ, then |F (u, v) − F (s, t)| < ;

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2. if t ∈ [0, 1] and x = F [I, t] then x(0) = A , x(1) = B , 3. F [I, 0] ∈ K0 and F [I, 1] ∈ K1 .1

1 0

|dx| ≤ p; and

Theorem A.7 If R is a region which is not all of C then the following statements are equivalent: 1. 2. 3. 4. 5.

R is simple. There is a one-to-one analytic function from R onto U . R is simply connected meaning that C \ R does not have a bounded component. If K is a closed path in R then W (K, P ) = 0 for each P ∈ / R. If A, B ∈ R and K0 , K1 are paths from A to B in R, then K0 is homotopically equivalent to K1 in R.

Theorem A.8 Let R be a region, and let A ∈ R, and let L be the set of paths from A to A in R. Define the relation TA = {(K0 , K1 ) ∈ L × L | K0 homotopically equivalent to K1 in R} . Furthermore, for a given path K, denote [K] = {K1 ∈ L | (K, K1 ) ∈ TA } , and let G = {[K] | K ∈ L}. Then the following statements are true: 1. TA is an equivalence relation. (The set [K] is called the equivalence class of K, and G is the set of TA ’s equivalence classes.) 2. If each of (K1 , K3 ) and (K2 , K4 ) is in TA , (K1 ⊕ K2 , K3 ⊕ K4 ) is in TA . 3. The relation {(([K0 ], [K1 ]), [K0 ⊕ K1 ]) | K0 , K1 ∈ G} is a functional relation from G×G → G. We write this function suggestively as [K0 ]⊕[K1 ] = [K0 ⊕K]. 4. The ordered pair (G, ⊕) is a group in the sense that a. if each of C1 , C2 , and C3 is in G then C1 ⊕ (C2 ⊕ C3 ) = (C1 ⊕ C2 ) ⊕ C3 ; b. there exists C0 ∈ G such that for all C ∈ G, C ⊕ C0 = C0 ⊕ C = C; and c. given C ∈ G, there exists C  ∈ G such that C ⊕ C  = C  ⊕ C = C0 . Remark A.9 The group (G, ⊕) described in Theorem A.8 is sometimes called the homotopy group of the region R (with respect to the point A). Exercise A.1 If A1 and A2 are points in the region R, (G1 , ⊕1 ) is the homotopygroup of R with respect to A1 , and (G2 , ⊕2 ) is the homotopy-group of R with respect to A2 , then (G2 , ⊕2 ) is isomorphic to (G1 , ⊕1 ) in in the sense that there is

1 This

is the composite of F with pairs of relations as defined on page 23.

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a one-to-one function f from G1 onto G2 such that for all X, Y ∈ G1 , f (X ⊕1 Y ) = f (X) ⊕2 f (Y ). Exercise A.2 Suppose R is the range of the exponential function exp, (G0 , ⊕0 ) is the homotopy-group of R with respect to 1, and G is the set of linear-fractional transformations which extend complex functions of the form f (z) = z + 2nπ i where n ∈ Z. 1. If s ∈ G then exp ◦s = exp. 2. If s1 , s2 ∈ G then s1−1 and s1 ◦ s2 are also in G. 3. If t is a linear-fractional transformation such that t (∞) = ∞ and exp ◦t = exp, then t ∈ G. 4. If z1 , z2 ∈ C and exp(z1 ) = exp(z2 ) then there is a unique t ∈ G such that t (z1 ) = z2 . 5. If K is a path from 1 in R, there is a path K0 from 0 such that K = exp /K0 . 6. If f is the relation {(s, [ exp /[0; s(0)] ] ) ∈ G × G0 | s ∈ G}, then f is a oneto-one function from G onto G0 such that if (s, t) is in G × G then f (s ◦ t) = f (s) ⊕0 f (t) Definition A.10 The hyperbolic metric is a function from U ×U into R. The value at the member (x, y) of U × U denoted by the symbol hyp(x, y), and it is called the hyperbolic distance from x to y. For x, y ∈ U , we define   1 |1 − xy| ¯ + |y − x| hyp(x, y) = Log 2 |1 − xy| ¯ − |y − x|

Appendix B

Automorphic Functions

Theorem B.1 If A is a point in the region R and C \ R has an unbounded component, then there is only one analytic relation f such that 1. f has domain R and range the unit-disc U , 2. f −1 is an analytic function with positive slope at (0, A), and 3. if K is a path from A in R, there is a path K0 from 0 such that K = f −1 /K0 . For the upcoming series of lemmas, use the following context. Suppose that R is a region such that C \ R has an unbounded component, and let A ∈ R. There is a one-to-one analytic function, with domain R and bounded range, which has slope 1 at (A, 0). Let H be the collection to which h belongs only in case 1. h is an analytic relation from R onto a bounded set, 2. h−1 is an analytic function with slope 1 at (0, A), and 3. if K is a path from A in R, there is a path K0 from 0 such that K = h−1 /K0 . If S is a subset of the domain of the relation F ⊆ C × C and F (S) is bounded, let |F |S denote the least upper bound of {|F (s)| | s ∈ S}. Lemma B.2 (1) If h ∈ H , and (B, C) ∈ h, and K is a path from B in R, there is a path K0 from C such that K = h−1 /K0 . Lemma B.3 (2) If h ∈ H , and (B1 , C1 ) ∈ h, and B2 ∈ R is different from B1 , then (B1 , C1 ) belongs to an analytic function which is a subset of h and has B2 in its domain. Lemma B.4 (3) If each of h, k ∈ H , and h(R) is the disc {z ∈ C | |z| < |h|R } and |k|R ≤ |h|R , then k = h. Lemma B.5 (4) If h ∈ H and h(R) is not the disc {z ∈ C | |z| < |h|R }, then there is an f ∈ H such that |f |R ≤ |h|R . Lemma B.6 (5) If b is the greatest lower bound of {|h|R | h ∈ H }, then b > 0 and there is only one member of H with range Db (0). © University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7

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Theorem B.7 Suppose A is a point in the region R, f is an analytic relation having all the properties (1)–(3) enumerated in the statement of Theorem B.1, and φ = f −1 1. If z, w ∈ U such that φ(z) = φ(w), u = φ  (z)/|φ  (z)| and v = φ  (w)/|φ  (w)|, and t is the linear-fractional transformation which contains all ordered pairs y−w x−z (x, y) ∈ C × C such that u 1−zx = v 1−wy , then t (U ) = U and φ ◦ t = φ. 2. If G is the collection of linear-fractional transformations such that s ∈ G implies s(U ) = U and φ ◦ s = φ, then (G, ◦) is a group (as defined earlier in Theorem A.8) sometimes called a transformation-group. 3. If s ∈ G and there is an x ∈ U such that s(x) = x, then s is the linear-fractional transformation which extends the identity function I . Remark B.8 The transformation-group described in Theorem B.7(2) is sometimes called the fundamental group of the region R with respect to the point A. If G is a transformation-group and φ is an analytic function such that if t ∈ G then φ ◦ t = φ, the function φ is said to be automorphic with respect to the group G. Theorem B.9 Suppose t is a linear-fractional transformation, t (U ) = U , and t (x) = x for all x ∈ U . One of the following two cases occurs: 1. t (y) = y holds for only one y ∈ C: in this case there exist a linear-fractional transformation s and b ∈ C such that |b − 12 | = 12 , b2 = b, and for each u ∈ U ,

u−b . If z is a sequence such that z0 ∈ U and, for each s −1 (t (s(u))) = 1−b 1−bu 1−b n ∈ N, zn+1 = t (zn ) then z is a one-to-one sequence which has the limit s(1). 2. t (y) = y holds for only two elements y ∈ C: in this case there exist a linearfractional transformation t and a real number c such that 0 < c < 1 and for u+c each u ∈ U , s −1 (t (s(u))) = 1+cu . If z is a sequence such that z0 ∈ U and, for each integer n ≥ 0, zn+1 = t (zn ) then z is a one-to-one sequence with the limit s(1).

Theorem B.10 Suppose A is a point in the region R, f is an analytic relation having all the properties (1)–(3) enumerated in the statement of Theorem B.1, φ = f −1 , and h is an analytic function with domain the simple region D and range lying in R. If (p, q) ∈ h and (z, q) ∈ φ then 1. there is an analytic function g from D into U such that (p, z) ∈ g and h = φ ◦ g, and 2. if r > 0 and Dr (0) ⊆ D then r|h (p)| ≤ (1 − |z|2 )|φ  (z)|. Remark B.11 In the preceding theorem we see that if D = C then h is constant: hence, R = C and C \ R contains at least two points. Exercise B.1 1. There exists an analytic function φ with domain U such that (a) 0 ∈ / φ  (U ), (b) φ has positive slope at (0, φ(0)), and

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(c) there is a path K from φ(0) in φ(U ) such that if K0 is a path from 0 in U then K = φ/K0 . 2. There exist regions R1 , R2 and points Ai ∈ Ri for i = 1, 2, such that R1 is not simple and the fundamental group of R1 with respect to A1 is the fundamental group of R2 with respect to A2 . 3. If t is a linear-fractional transformation and t (U ) = U then, for any x, y ∈ U , hyp(t (x), t (y)) = hyp(x, y). If f is an analytic function from U into U and f is not the restriction to U of a linear-fractional transformation t such that t (U ) = U then, for any distinct x, y ∈ U , hyp(f (x), f (y)) < hyp(x, y).

Appendix C

Excepted Values and Uniformization

The following notation will be used throughout this appendix. • Let ω = exp(2π i/3), so that ω¯ = ω2 and 1 + ω + ω2 = 0. • Let t be the linear-fractional transformation which extends the function

z+iω , so z+iω2 2 and t −1 extends the function iω2 z−ω 1−z . if n ∈ {1, 2, 3} then z belongs to Cn only

that t maps the right half-plane onto U • Let C1 , C2 and C3 be circles such that if |z + 2wn |2 = 3. • Let T1 , T2 , and T3 be the inversions of C∞ in the circles C1 , C2 , and C3 , respectively.

Exercise C.1 The notation and ideas in this series of exercises is cumulative and may be referenced in later items.   i 1. Show that T1 ◦ t = t ◦ J , T2 ◦ t = t ◦ (J + 2i), and T3 ◦ t = t ◦ J 1/4 + +i/2 2 . 2. Let v and w be the linear-fractional transformations extending the functions   2 z+1 1 − 2ω 2ω+z and ωz respectively. Show that v◦t = t ◦(I +i) and w◦t = t ◦ i + I ; moreover, v(C1 ) = C2 , w(C1 ) = C2 , and w(C2 ) = C3 , so that v ◦T1 = T2 ◦v, w ◦ T1 = T2 ◦ w, and w ◦ T2 = T3 ◦ w. 3. Let E1 denote the restriction of − exp(−π t −1 (z)) to U . Establish the following: E1 (U ) is {z ∈ C | 0 < |z| < 1}; E1 maps C1 ∩ U onto the real number interval (−1, 0), and maps C2 ∩ U onto the real number interval (0, 1); E1 ◦ T1 = E1 ◦ T2 = J ◦ E1 ; E1 ◦ v = −E1 . 4. Let V be a sequence such that V1 is 

   z ∈ C  |z| < 1, |z + 2ω2 |2 > 3, 

© University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7

 2    z − 2 1 − ω  > 1 ,  3  3

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and if n is a positive integer then V2n is V2n−1 ∪ v(V2n−1 ), and V2n+1 is V2n ∪ T1 (V2n ). Show that if n is a positive integer, T1 (V2n−1 ) = V2n−1 and z ∈ t −1 (V2n−1 ) only in case Re z > 0 and −n < Im z < n. 5. There is a sequence g such that g1 is the restriction of E1 to V1 and if n is a positive integer then (a) g2n is a function with domain V2n such that if z ∈ V2n−1 then g2n (z) = g2n−1 (z) and g2n (v(z)) = −g2n−1 (z), and (b) g2n+1 is a function with domain V2n+1 such that if z ∈ V2n then g2n+1 (z) = g2n (z) and g2n+1 (T1 (z)) = J (g2n (z)). 6. If n is a positive integer and  z ∈ c(V2n−1 ) then T2 (z) ∈ v(V2n−1 ) and g2n (T2 (z)) = J (g2n (z)) ; E1 is ∞ i=1 gn . 7. Let W be a sequence such that W1 is the set of complex numbers z satisfying all of (a) |z| < 1 (b) |z + 2ω2 |2 > 3 (c) |z + 2|2 > 3 2  2   (d) z − 2 ω 3−ω  >    1  (e) z − 2 1−ω 3 > 3

1 3

and for all other positive integers n, W3n−1 is W3n−2 ∪ w(W3n−2 ), W3n is W3n−1 ∪ w(W3n−1 ), and W3n+1 is W3n ∪ T1 (W3n ). Show that W1 is a simple region and t −1 (W1 ) is the region of complex numbers z satisfying all of (a) (b) (c) (d)

Re z > 0 −1  1 2 2

8. If z ∈ U then there is an integer n > 0 such that z ∈ Wn . 9. Let f1 be the one-to-one analytic function with domain t −1 (W1 ) and range the right half-plane, which has positive slope at (1, 1). Show f1 ◦ J = J ◦ f1 and, if z ∈ t −1 (W1 ) and Im z = 0, Im(z) · Im(f1 (z)) > 0. The restriction of f1 to the set of all positive numbers is an increasing function with range the set of all positive numbers. 10. Let W0 be {z ∈ C | z ∈ W1 and |z + 2ω|2 > 3}, and f2 be a function with domain W0 such that if z ∈ W0 then f2 (z) =

f1 (t −1 (z))2 − f1 (ω/ i)2 . f1 (t −1 (z))2 − f1 (i/ω)2

Show f2 is a one-to-one analytic function with range U , and f2 ◦ w = w ◦ f2 , and f1 (ω/ i) = ωf1 (i/ω).

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11. Let f3 =

(f1 ◦ t −1 )2 . (f1 ◦ t −1 )2 + f1 (ω/ i)f1 (i/ω)

Show the following: f3 ◦ T1 = J ◦ f3 . If z ∈ W0 then f3 (z) =

1 − ω f2 (z) − ω2 1 and f3 (ωz) = . 2 1 − f3 (z) 1 − ω f2 (z) − ω

f3 is a one-to-one analytic function and z belongs to the range of f3 only if either Im z = 0 or 0 < Re z < 1. f3 maps C1 ∩ U onto (0, 1). 12. There is a sequence h such that h1 = f3 and, for each integer n > 0, (a) h3n−1 is a function with domain W3n−1 such that if z ∈ W3n−2 then 1 h3n−1 (z) = h3n−2 (z) and h3n−1 (ωz) = 1−h3n−2 (z) , (b) h3n is a function with domain W3n such that if z ∈ W3n−1 then h3n (z) = 1 h3n−1 (z) and h3n (ωz) = 1−h3n (z) , and 1 (c) h3n+1 is a function with domain W3n+1 such that if z ∈ W3n then h3n+1 (z) = h3 n(z) and h3n+1 (T1 (z)) = J (h3n (z)).  Definition C.1 The modular function E2 is defined to be the function ∞ i=1 hn . Theorem C.2 E2 is an analytic function with domain U such that 1. E2 (U ) = {z ∈ C | z(1 − z) = 0}, 2. E2 maps C1 ∩ U onto the segment (0, 1), 3. E2 ◦ T1 = E2 ◦ T2 = E2 ◦ T3 = J ◦ E2 and E2 ◦ w =

1 1−E2 ,

4. E2 (0) = 1 + ω and E2−1 is an analytic relation, and 5. if K is a path from 1 + ω in E2 (U ) then there is a path K0 from 0 in U such that K = E2 /K0 . Theorem C.3 If f is an entire function and there exist two points neither of which belongs to the range of f , then f is constant: Theorem C.4 If A is a point in the region R and C \ R contains two points, then there is only one analytic relation f such that 1. f has domain R and range the unit-disc U , 2. f −1 is an analytic function with positive slope at (0, A), and 3. if K is a path from A in R there is a path K0 from 0 such that K = f −1 /K0 . Exercise C.2 (*) If A1 is a point in the region R1 , A2 is a point in the region R2 , and each of C \ R1 and C \ R2 each have more than one element, then there is a oneto-one analytic function f from R1 onto R2 which has positive slope at (A1 , A2 ) only in case . . ., in which case f consists of all ordered pairs of the form . . .. Theorem C.5 If R is a region and f is a sequence of analytic functions with domain R, each of which does not have either 0 or 1 in its range, then either

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there is a subsequence of f which is continuously convergent on R or there is a subsequence g of f such that g1 is continuously convergent on R with limit having only the value 0. Lemma C.6 If 0 < r < 1 then there is a real number b in the interval (0, 1) such that if z ∈ C and |z − b| < 1 − b then |E2 (z) − 1| < r. Lemma C.7 If 0 < r < 1 then there is a real number b in the interval (0, 1) such that if z ∈ U ∩ W1 and |z − l| < b, then |z − |z|| < r(1 − |z|). Lemma C.8 There is an analytic function from a simple region onto R. Theorem C.9 If f is an analytic function from E1 (U ) into E2 (U ) then 0 is not an essential singularity of f . Corollary C.10 If the entire function g is not constant and is not a polynomial then there is a P ∈ C such that each complex number different from P is the second term of each of infinitely many ordered pairs in g. Theorem C.11 (Uniformization Theorem) If R is a region then there exist a simple region D and an analytic function g from D onto R having the following properties: 1. g −1 is an analytic relation with domain R such that no element in R is a g −1 boundary-point of any disc lying in R, and 2. if F is an analytic relation with domain R such that no element in R is an F boundary-point of any disc lying in R, then F ◦ g contains an analytic function h such that F = {(g(z), h(z)) | z ∈ D}.

Appendix D

Well-ordering

Definition D.1 A well-ordering of the nonempty set V is a subset R of V ×V such that 1. 2. 3. 4.

if (x, y) ∈ V × V then either (x, y) or (y, x) is in R, if (x, y) ∈ R and (y, x) ∈ R then y = x, if (x, y) ∈ R and (y, z) ∈ R then (x, z) ∈ R, and if ∅ = W ⊆ V then there is an x ∈ W such that for all y ∈ W , we have (x, y) ∈ R. If R is a well-ordering of the set V = ∅ then

1. a subset T of V is called an initial R-segment of V if T = ∅, and either T = V , or x ∈ T and y ∈ V \ T implies (x, y) ∈ R. 2. if ∅ = S ⊆ V then the R-first member of S is that member x ∈ S such that for all y ∈ S, we have (x, y) ∈ R. Theorem D.2 Suppose the set M has more than one element, and G is a function from the collection of all nonempty subsets of M such that if ∅ = K ⊆ M then G(K) is a member of K. There is a well-ordering R of M such that 1. G(M) is the R-first member of M, and 2. if H is an initial R-segment of M different from M then G(M \ H ) is the R-first member of M \ H . Suggestion For an approach to a proof, with M and G as stated in the theorem: Definition D.3 If K ⊆ M then K is said to be G-normal provided K =  ∅ and there exists a well-ordering P of K such that 1. G(M) is the P -first member of K and, 2. if H is an initial P -segment of K different from K, then G(M \ H ) is the P -first member of K \ H . In this case, the set K is said to be G-normal with respect to P . © University of Houston and Trinity College 2020 J. S. Mac Nerney, An Introduction to Analytic Functions, https://doi.org/10.1007/978-3-030-42085-7

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In the following lemmas, let ∅ = K ⊆ M: Lemma D.4 If K is G-normal with respect to P , and if H is a nonempty set, then the following two statements are equivalent: 1. H is an initial P -segment of K different from K. 2. There is a member y ∈ K different from G(M) such that x ∈ H implies (x, y) ∈ P and x = y. Lemma D.5 If K is G-normal with respect to P and to Q, then P = Q. Lemma D.6 If K is G-normal with respect to P and if H is an initial P -segment of K, then H is G-normal with respect to P ∩ (H × H ). Lemma D.7 Suppose K is G-normal with respect to P , the subset L of M is Gnormal with respect to Q, and K ⊆ L. Then K is an initial Q-segment of L and P is Q ∩ (K × K). Lemma D.8 If K and L are G-normal subsets of M then either K ⊆ L or L ⊆ K. Lemma D.9 Let V be the union of all G-normal subsets of M, and let R be the union of all well-orderings of M for which there exists K ⊆ M such that K is G-normal. Then V is G-normal with respect to R, and V = M. Definition D.10 A monotonic collection is a collection C, each member of which is a set, such that if X, Y ∈ C then either X ⊆ Y or Y ⊆ X. Theorem D.11 Suppose the collection S ofsets has the property that, for each nonempty monotonic subcollection C of S, C ∈ S. If X ∈ S then there is a member of S which contains X and which is not contained in any other member of S. Corollary D.12 If f is an analytic function then there is an analytic function which contains f and which is not contained in any other analytic function. Corollary D.13 If the simple region D lies in the region R then there is a simple region which contains D, which lies in R, and which is not contained in any other simple region lying in R. Theorem D.14 If M is a set with more than one element, then there is a wellordering R of M which is most economical in the sense that if H is an initial R-segment of M different from M then there does not exist a one-to-one function from M onto H . Exercise D.1 If D ⊆ R ⊆ C, then D is said to be dense in R if each point of R is either a point of D or is a limit-point of D. Show that for any region R, there is a simple region D which is dense R.

Appendix E

Analytic Surfaces

Let Ma denote the relation to which (p, f ) belongs only in case p is a complex number and f is an analytic function with domain D such that either D = C or D is a disc with center p and f is not a subset of any other analytic function with domain a disc with center p. Let Ra denote the relation to which (X, h) belongs only in case 1. there is a (p, f ) ∈ Ma and a disc d, having center p and contained in the domain of f , such that X is the subset of Ma to which (q, g) belongs only in case q ∈ d and f ∪ g is a function, and 2. h is the function to which (x, z) belongs only in case x ∈ X and z is the first term of x. Exercise E.1 (1) Ra is an analytic structure for Ma an in the sense that Ra is a relation with domain Q such that  1. each member of Q is a subset of Ma and Q is Ma , 2. if (X, h) is in Ra then b is a one-to-one function from X onto a region, and 3. if each of (X, h) and (Y, k) is in Ra and V = X ∩ Y then k ◦ h−1 is analytic in each component of h(V ). Definition E.1 If M is a set and R is an analytic structure for M, the member P ∈ M is said to be a limit-point relative to R of the subset S of M provided there is an (X, h) ∈ R such that P ∈ X and if D is a disc having center h(P ) and contained in h(X) then h−1 (D) contains a member of S different from P . (Implicitly, meaning has now been assigned to the phrases connected relative to R, open relative to R, continuous relative to R, etc.) If each of R1 and R2 is an analytic structure for the set M, R1 is analytically equivalent to R2 provided that R1 ∪ R2 is an analytic structure for M.

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Exercise E.2 (2) If f is an analytic function with domain D0 and  is the relation to which (p, x) belongs only in case p ∈ D0 and x is a member (p, f ) of Ma such that there is an analytic function which has p in its domain and which is a subset of f0 and of f , then  is a function from D0 into Ma which is continuous relative to Ra . Exercise E.3 (3) Let Z be the relation to which (F, C) belongs only in case there is a member (p, f ) of Ma such that 1. F is the analytic relation which contains f and which is not contained in any other analytic relation, and 2. C is the component of Ma relative to Ra to which (p, f ) belongs. Show Z is a one-to-one function. Definition E.2 An analytic surface is an ordered pair (S, R) such that S is a set, R is an analytic structure for S, S is connected relative to R, and if P1 and P2 are members of S then there are disjoint subsets T1 and T2 of S which are open relative to R and contain P1 and P2 , respectively. Remark E.3 If F is an analytic relation which is not a subset of any other analytic relation and R = {(X, h) ∈ Ra | X ∈ Z(F )}, then (Z(F ), R) is an analytic surface—the Riemann surface of F . Exercise E.4 (4) If S is a region in C and R is the relation to which (X, h) belongs only in case X is S and h is the restriction of the identity function I to S, then (S, R) is an analytic surface. Exercise E.5 (5) If (S, R) is an analytic surface and x0 , x1 ∈ S, there exist an integer n > 0 and a sequence {Vp }np=0 , each value of which is in the domain of R, such that x0 ∈ V0 and x1 ∈ Vn and, for each integer p ∈ [1, n], the set S ∩ Vp−1 ∩ Vp = ∅. Exercise E.6 (6) The ordered pair (S, R), such that S is the unit sphere and R is the relation to which (X, h) belongs only in case either 1. X is the set of all elements of S different from (0, 0, 1) and if (a, b, c) is in X then h(a, b, c) = a+ib 1−c , or 2. X is the set of all elements of S different from (0, 0, −1) and if (a, b, c) is in X then h(a, b, c) = a−ib 1+c , is an analytic surface—the Riemann sphere—and S is compact relative to R in the sense that each infinite subset of S has a limit-point relative to R. Exercise E.7 (7) If (S, R1 ) is an analytic surface, there is an analytic surface (S, R2 ) such that R2 is analytically equivalent to R1 and if (Y, k) ∈ R2 then k(Y ) is simple. Definition E.4 If each of (S1 , R1 ) and (S2 , R2 ) is an analytic surface, the function G from S1 into S2 is analytic relative to (R1 , R2 ) provided that if (z, w) ∈ G and

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(X, h) ∈ R1 and (Y, k) ∈ R2 and z ∈ X and w ∈ Y then k ◦ G ◦ h−1 is analytic in each component of its domain. Exercise E.8 (8) Reformulate Theorems 5.17, 9.19, and C.3 as statements about analytic surfaces. Exercise E.9 (9) Supposing that F is a maximal analytic relation and (Z(F ), R) is the Riemann surface of F , consider the relation G to which (z, w) belongs only in case z is a member (p, f ) of Z(F ) such that (p, w) ∈ f . Exercise E.10 (10) Supposing that (S, R) is an analytic surface and A ∈ S, ˆ R) ˆ and the function : consider the ordered pair (S, 1. Sˆ is the relation to which (B, C) belongs only in case B is a member of S and C is a set of which there is a member x such that (a) x is a function from [0, 1] into S, continuous relative to R, such that x(0) = A and x(1) = B, and (b) y ∈ C only if there is a function F from [0, 1] × [0, 1] into S, continuous relative to R, such that F [I, 0] = x and F [I, 1] = y and, for each t ∈ [0, 1], F (0, t) = A and F (1, t) = B.1 2. Rˆ is the relation to which (Y, k) belongs only in case there is a member (X, h) of R, a simple region D lying in h(X), and a member (B, C) of Sˆ such that B is in h−1 (D) and (a) Y is the subset of Sˆ to which (B1 , C1 ) belongs only in case there is a member z of C and a function z1 from [0, 1] into h−1 (D), continuous relative to R, such that z1 (0) is B and z1 (1) is B1 and C1 contains the function x1 from [0, 1] such that if t is in [0, 1/2] then x1 (t) is z(2t) but if t is in [1/2, 1] then x1 (t) is z1 (2t − 1), and (b) k is a function with domain Y such that if (B1 , C1 ) ∈ Y then h(B1 ) is the second term of the ordered pair in k of which the first term is (B1 , C1 ). 3.  is the function to which (u, v) belongs only in case u belongs to Sˆ and v is the first term of u.

1 This

is the composite of F with pairs of relations as defined on page 23.

Selected Works by Mac Nerney

1. Mac Nerney, J.S.: Continued fractions in which the elements are operators in a linear space. Ph.D. thesis, University of Texas at Austin (1951) 2. Mac Nerney, J.S.: Hellinger integrals and Stieltjes integral equations. In: Bulletin of the American Mathematical Society, vol. 60, pp. 50–51 (1954) 3. Mac Nerney, J.S.: Continuous products in linear spaces. Journal of the Elisha Mitchell Scientific Society 71(2), 185–200 (1955) 4. Mac Nerney, J.S.: Stieltjes integrals in linear spaces. Annals of Mathematics pp. 354–367 (1955) 5. Mac Nerney, J.S.: Hellinger integrals in inner product spaces. Journal of the Elisha Mitchell Scientific Society 76(2), 252–273 (1960) 6. Mac Nerney, J.S.: Hermitian moment sequences. Transactions of the American Mathematical Society pp. 45–81 (1962) 7. Mac Nerney, J.S.: Integral equations and semigroups. Illinois Journal of Mathematics 7(1), 148–173 (1963) 8. Mac Nerney, J.S.: A linear initial-value problem. Bulletin of the American Mathematical Society 69(3), 314–329 (1963) 9. Mac Nerney, J.S.: A nonlinear integral operation. Illinois Journal of Mathematics 8(4), 621–638 (1964) 10. Mac Nerney, J.S.: Finitely additive set functions. Houston J. Math 6 (1980)

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Mac Nerney’s theorem numbering in the original edition

Original Theorem 1 Theorem 2 Theorem 3 Theorem 4 Theorem 5 Theorem 6 Theorem 7 Theorem 8 Theorem 9 Theorem 10 Theorem 11 Theorem 12 Theorem 13 Theorem 14 Theorem 15 Theorem 16 Theorem 17 Theorem 18 Theorem 19 Theorem 20 Theorem 21 Theorem 22 Theorem 23 Theorem 24 Theorem 25 Theorem 26

New 2.4 2.6 2.7 2.9 2.10 2.13 2.15 3.5 3.9 3.19 3.22 4.2 4.5 4.8 4.9 4.11 4.17 5.2 5.7 5.8 5.10 5.13 5.14 5.15 5.17 5.18

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Original Theorem 27 Theorem 28 Theorem 29 Theorem 30 Theorem 31 Theorem 32 Theorem 33 Theorem 34 Theorem 35 Theorem 36 Theorem 37 Theorem 38 Theorem 39 Theorem 40 Theorem 41 Theorem 42 Theorem 43 Theorem 44 Theorem 45 Theorem 46 Theorem 47 Theorem 48 Theorem 49 Theorem 50 Theorem 51 Theorem 52 Theorem 53 Theorem 54 Theorem 55 Theorem 56 Theorem 57 Theorem 58 Theorem 59 Theorem 60 Theorem 61 Theorem 62 Theorem 63 Theorem 64 Theorem 65 Theorem 66 Theorem 67 Theorem 68

Mac Nerney’s theorem numbering in the original edition

New 5.20 5.21 6.1 6.4 6.9 6.12 6.14 7.2 7.3 7.4 7.6 7.8 7.9 7.11 7.12 7.13 7.14 7.16 8.1 8.2 8.3 8.4 8.5 8.7 8.8 8.9 8.11 8.12 8.13 8.15 8.16 9.5 9.8 9.15 9.16 9.19 10.2 10.4 10.6 10.8 10.10 10.12

Mac Nerney’s theorem numbering in the original edition

Theorem 1.1 Theorem 1.2 Theorem 1.3 Theorem 1.4 Theorem 2.1 Theorem 2.2 Theorem 2.3 Theorem 2.4 Theorem 3.1 Theorem 3.2 Theorem 3.3 Theorem 3.4 Theorem 3.5 Theorem 3.6 Theorem 4.1 Theorem 4.2 Theorem 4.3

A.1 A.5 A.7 A.8 B.1 B.7 B.9 B.10 C.2 C.3 C.4 C.5 C.9 C.11 D.2 D.11 D.14

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Index

A Analytic continuation, 60 Analytic function, 27 Analytic radius, 50 Analytic relation, 60 boundary-point of, 61 branch point of, 62 derivative of, 61 Riemann surface of, 80 Analytic structure, 79 compact relative to, 80 connected relative to, 79 continuous relative to, 79 equivalence, 79 limit-point relative to, 79 open relative to, 79 Analytic surface, 80 function analytic relative to, 80 Annulus, 42 ann(w; m, M), 42 Arg w, 28 B   C, 2 C, 1 Binomial sequence, 39 w , 39 p C C, 6 Cf (S, b), 14 C∞ , 54 Cr (w), 39 Cauchy integral, 21

chd(x, y), 57 Chordal distance, 57 Chordal metric, 57 properties of, 57 Circle, 56 inversion in a, 56 Collection, 1 Complement, 2 Complete normed algebra, 15 Complex function, 10 k’th derivative, 31 analytic, 27 of bounded variation, 18 Cauchy integral, 21 continuous, 13 derivative of, 22 entire, 27 essential singularity of, 45 modulus of, 14 order at ∞, 57 order at a point, 44 pole of, 45 residue, 43 slope at a point of, 22 total variation of, 18 zero of, 45 Complex matrix, 63 Complex number, 6 conjugate of, 6 imaginary part of, 6 modulus of, 6 principal argument of, 28 real part of, 6 Complex plane, 7 extended, 54

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90 Component of a set, 26 Composite of a relation with a pair of relations, 23 Connected set, 26 Contour, 39 Convergence absolute convergence of a function sequence, 42 absolute convergence of a sequence, 42 Cauchy convergence of function sequence, 37 continuously convergent function sequence, 50 radius of, 42 totally convergent power-series, 42 totally divergent power-series, 42 Convex set, 26 cos, 40 Cosine function, 40 Curve, 21

D Dr (w), 30 Dense subset, 78 Distance, 7 Domain, 7

E Empty set, 1 ∅, 1 Entire function, 27 exp, 38 Exponential function, 38 Extended complex plane, 54 Extended real line, 56

F Family, 1 f (k) , 31 Function, 8 algebraic operations with, 9 analytic, 27 complex, see complex function composite, composition, 8 constant, 9 cosine, 40 exponential, 38 identity function, 10 meromorphic, 57 modular, 75

Index one-to-one, 9 principal logarithmic, 36 sine, 39 tangent, 40

G G-normal set, 77 Group, 66 fundamental group, 70 homotopy group, 66 isomorphism, 66 transformation-group, 70

H Half-plane, 7 Hyperbolic distance, 67 Hyperbolic metric, 67 hyp(x, y), 67

I I , 10 i, 6 Image, 7 Im(z), 6 ∈, 1 Induction, 3 54 ∞, b |dx|, 18 a b a y dx, 18 K f , 21 Intersection, 1 Inversion, 56

J J , 56

L Least upper bound, 12 Linear-fractional transformation, 55 Log, 36, 62 |F |S , 69 M Membership, 1 Meromorphic function, 57 Monotonic collection, 78 Mutually separated sets, 26

Index N N, 3 N+ , 4 Natural logarithmic relation, 62 Natural numbers, 3 Number-plane, see complex plane

O Open disc, 30 Open set, 26 Ordered pair, 2

P Path, 21 carrier of, 21 closed, 21 homotopically equivalent paths, 65 image of, 23 length, 21 linear, 26 polygonal, 26 sum of paths, 21 triangular, 30 winding number of, 39 f , 22 [g, h] , 23 π , 27 Point, 7 Point at infinity, 54 Point-function, see complex function Power-series, 37 binomial, 39 Principal circular contour, 39 Principal logarithmic function, 36 Principal square contour, 27

R R, 2 R∞ , 56 Radially convex set, 32 Range, 7 Real line, 7 extended, 56 Real numbers, 2 interval, 14 Region, 26 analytic radius of, 50 simple, 32 simply connected, 66

91 Relation, 7 algebraic curve, 63 analytic, see analytic relation composite, composition of two relations, 8 composite with a pair of relations, 23 equivalence, 8 extension of, 8 functional, 8 inverse of, 8 order, 8 partial-order, 8 restriction of, 8 well-ordering, 77 Residue, 43 Re(z), 6 Riemann sphere, 80 Riemann surface, 80

S S 2 , 54 Sr (w), 27 Sequence, 12 Cauchy, 12 cluster-point, 12 continuously convergent, 50 convergent, 12 Set, 1 disjoint, 1 Simple subdivision, 18 sin, 39 Sine function, 39 Singleton, 1 S + r, 28 Star-convex set, 32 Stieltjes integral, 18 Stieltjes subdivision, 18 norm of, 18  refinement of, 18 s |dx|, 20 s y dx, 18

T tan, 40 Tangent function, 40

U Union, 1 Unit sphere, 54

92 V Variable, 10 complex, 10 real, 10 |f |S , 14 W Well-ordering, 77

Index G-normal set, 77 Winding number, 39 W (K, w), 39

Z z1/2 , 9 zw , 39