# FUNCTIONS OF ONE COMPLEX VARIABLE PDF

functions of one complex variable for students who are mathematically mature enough to understand and execute e - d arguments. The actual pre- requisites for . The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for. Germany, the. Graduate Texts in Mathematics Functions of One Complex Variable watermarked, DRM-free; Included format: PDF; ebooks can be used on all reading devices.

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Functions of one complex variable U / John B. Conway. p. cm. — (Graduate texts in mathematics ; ). Includes bibliographical references (p. —) and index. FACULTY OF. MATHEMATICS. Functions of one complex variable. Syllabus. Course code: Number of ECTS credits: 6. Semester: 1st (September- January). Conway, John B - Functions of One Complex Variables I (1).pdf - Ebook download as PDF File .pdf), Text File .txt) or read book online.

John B. Editorial Board S. Mathematics Subject Classification Functions of one complex variable I I John B. Functions of one complex variable. Includes bibliographical references p.

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Functions of complex variables. All rights reserved. NY , USA , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks. Solkover reprint of the hardcover 15t Edition 15 14 13 springeronline.

I wish to thank publicly Earl Berkson. John Dixon's treatment of Cauchy's Theorem has been included. Richard Varga. Andrew Lenard. Springer-Verlag is making a contribution to our discipline by increasing its efforts to disseminate the recent developments in mathematics.

John Mairhuber. John Plaster. This appendix contains some bibliographical material and a guide for further reading. There are also minor changes that have been made.

The present proof is due to Sandy Grabiner and does not use "pole pushing". Hans Sagan. Springer-Verlag asked me to prepare a list of corrections for a third printing. There are four major differences between the present edition and its predecessor. Robert Olin. I have a strong attachment to the homotopic version that appeared in the first edition and have proved this form of Cauchy's Theorem as it was done there. Louis Brickman. In a sense the "pole pushing" is buried in the concept of uniform approximation and some ideas from Banach algebras.

When I mentioned that r had some ideas for more substantial revisions. I wish to thank the staff at Springer-Verlag New York not only for their treatment of my book. Donald Perlis. Jeffrey Nunemacher. James P.

This version is very geometric and quite easy to apply. Donald C. James Deddens. David Stegenga. This has the advantage of providing a quick proof of the theorem in its full generality. When it was apparent that the second printing was nearly sold out. Gerard Keough. Glenn Schober. Several colleagues in the mathematical community have helped me greatly by providing constructive criticism and pointing out typographical errors.

Conway VI. In the present time of shrinking graduate enrollments and the consequent reluctance of so many publishers to print advanced texts and monographs. The exercises are varied in their degree of difficulty.

The other guiding principle followed is that all definitions. Section two applies Runge's Theorem to obtain a more general form of Cauchy's Theorem. PREFACE This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E.

Chapter VII initiates the student in the consideration of functions as points in a metric space. The topics from advanced calculus that are used e. Leibniz's rule for differentiating under the integral sign are proved in detail.

The proof presented in section VIII. The last two sections of Chapter VII are not needed in the rest of the book although they are a part of classical mathematics which no one should completely disregard. Chapter IX studies analytic continuation and introduces the reader to analytic manifolds and covering spaces.

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Sections four and five need no defense. The actual prerequisites for reading this book are quite minimal.

## Solutions Manual

Sections one through three can be considered as a unit and will give the reader a knowledge of analytic vii. Proofs are given with the student in mind. Except for the material at the beginning of Section VI.

The remaining chapters are independent topics and may be covered in any order desired. The main results of sections three and four should be read by everyone. In addition to having applications to other parts of analysis..

Section six is an application of the factorization theorem. Some are meant to fix the ideas of the section in the reader's mind and some extend the theory or give applications to other parts of mathematics. This view of Complex Analysis as "An Introduction to Mathematics" has influenced the writing and selection of subject matter for this book. Most are presented in detail and when this is not the case the reader is told precisely what is missing and asked to fill in the gap as an exercise.

The results of the first three sections of this chapter are used repeatedly in the remainder of the book.

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Complex Variables is a subject which has something for all mathematicians.. Runge's Theorem is the inspiration for much of the theory of Function Algebras. It is possible to cover this material in a single semester only if a number of proofs are omitted. Some consideration was given to including chapters on some or all of the following: There are other topics that could have been covered.

I would like to thank the Department of Mathematics for making its resources available to me during its preparation. Chapter X studies harmonic functions including a solution of the Dirichlet Problem and the introduction of Green's Function. Her encouragement was the most valuable assistance I received. This book evolved from classes taught at Indiana University.

But the line had to be drawn somewhere and these topics were the victims. If this can be called applied mathematics it is part of applied mathematics that everyone should know. I would especially like to thank the students in my classes. When a function other than a path is being discussed. For those readers who would like to explore this material or to further investigate the topics covered in this book.

Most of the notation used is standard. With regard to Picard's Theorem it should be mentioned that another proof is available. I must thank my wife to whom this book is dedicated. The proof presented here uses only elementary arguments while the proof found in most other books uses the modular function. Although they are independent. Latin letters are used for the domain and Greek letters are used for the range. The word "iff" is used in place of the phrase "if and only if".

I must also thank Ceil Sheehan for typing the final draft of the manuscript under unusual circumstances. Classification of singularities ix Volume II 1. Sequences and completeness Compactness Continuity Uniform convergence 11 14 17 20 24 28 Power series Analytic functions Analytic functions as mapping. Counting zeros. The Riemann Mapping Theorem. The space of continuous functions C G. Spaces of merom orphic functions. The Riemann zeta function. Table of Contents XI.

The Area Theorem Disk mappings: An application: The Green function. Wiener's criterion for regular points References List of Symbols Index xiii. Logarithmic capacity: Regular points for the Dirichlet problem The Dirichlet principle and Sobolev spaces Szeg6's Theorem In particular we assume a knowledge of the ordering of IR. We will write a for the complex number a.

Although it has been traditional to study functions of several real variables before studying analytic function theory. It is also assumed that every reader is familiar with sequential convergence in IR and with infinite series. From this point on we abandon the ordered pair notation for complex numbers.

C satisfies the associative. The field of complex numbers We define C. In fact. It is assumed that each reader is acquainted with the real number system and all its properties.

More generally. The real numbers We denote the set of all real numbers by IR. That is. There will not be any occasion when the deep results of this an: We conclude this section by introducing two operations on IC which are not field operations. Exercises 1. Note that 2. Find the real and imaginary parts of each of the following: Find the absolute value and conjugate of each of the following: This is called the triangle inequality because.

With this in mind the last equation of Exercise 4 in the preceding section states the parallelogram law: The sum of the squares of the lengths of the sides of a parallelogram equals the sum of the squares of the lengths of its diagonals. On encounter Let R z be a rational function of z. If z and ware complex numbers. These form two sides of a parallelogram with 0.

The addition of complex numbers is exactly the addition law of the vector space [R2. Note also that Iz-wl is exactly the distance between z and w. To show this first observe that for any z in C. The complex plane From the definition of complex numbers it is dear that each z in C can be identified with the unique point Re z. Z3' By using easy to see that we need only show 3.

The complex plane 3 3. A fundamental property of a distance function is that it satisfies the triangle inequality see the next chapter. Notice that 8 plus any multiple of " can be substituted for 8 in the above equations.

Prove Oifw O. From looking at a triangle and considering the geometrical significal'lce of 3. Z is the point obtained by reflecting z across the x-axis i. We introduce the notation 4.

It is clear that equality will occur when the two points are colinear with the origin. Now let a be any complex number and. This is also easy Because of the ambiguity of e. Show that equality occurs in In fact Iz-wl Now that we have given a geometric interpretation of the absolute value let us see what taking a complex conjugate does to a point in the plane. This point has polar coordinates Cr.

Example Calculate the nth roots of unity. By means of 4.

As a special case of 4. How many such z can you find? In light of 4. Moreover if z. Find the sixth roots of unity. We are now in a position to consider the following problem: For a given complex number a -O and an integer n By the formulas for the sine and cosine of the sum of two angles we get 4. What is the smallest value of k? What can be said if a and bare nonprimitive roots of unity?

Lines and half planes in the complex plane Let L denote a straight line in C. A primilive nth rool oj unity is a complex number a such that l. Show that if a and bare primitive nth and mth roots of unity. Calculate the following: L is determined by a point in L and a direction vector.

From elementary analytic geometry. Ha is the half plane lying to the left of L. As a first step in answering this question. For the moment. That is 5. Hence Ho is the half plane lying to the left of the line L if we are "walking along L in the direction of b. Ha is the translation of Ho by a. Exercise 1. We will find equations expressing Xl' x 2. Thus 1C00 is represented as the sphere S.

## Differentiation of the Martinelli‐Bochner Integrals and the Notion of Hyperderivability

This intersects. Clearly Z approaches N. Let us explore this representation. Now for each point z in IC consider the straight line in 1R3 through z and N. To accomplish this and to give a concrete picture of 1C00 we represent 1C00 as the unit sphere in 1R 3. N is the north pole on S.

We also wish to introduce a distance function on 1C00 in order to discuss continuity properties of functions assuming the value infinity. The extended plane and its spherical representation Often in complex analysis we will be concerned with functions that become infinite as the variable approaches a given point. I-t y.

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X3 EIR3: Since t: X2' Xl then 6. The extended plane and its spherical representation which this line intersects S. Find the coordinates of Win terms of the coordinates of Z and Z'. The Complex Number System 10 Exercises 1. A projects onto a circle in C. Let A be a circle lying in S. Let Z and Z' be points on S corresponding to z and z' respectively. Which subsets of S correspond to the real and imaginary axes in C? Give the details in the derivation of 6. For each of the following points in C.

## Functions of One Complex Variable I

Use this information to show that if A contains the point N then its stereo graphic projection on C is a straight line.

Examples 1. Then C. This makes both IR. Yn in IR n define d x. If the reader has never encountered the concept of a metric space before this. This metric space does not appear in the study of analytic function theory. Definition and examples of metric spaces A metric space is a pair X. To show that the function d satisfies the triangle inequality one merely considers all possibilities of equality among x.

E consists only of the point x if E Notice here that B x. Gn are open sets in X then so is c If n n Gk. E is not contained in this set no matter how small we choose E. The most common error made upon learning of open and closed sets is to interpret the definition of closed set to mean that if a set is not open it is. The proof of c is left as an exercise for the reader.

J any indexing set.. Let X. We denote the empty set. B S consists of all complex valued functions whose range is contained inside some disk of finite radius. The following proposition is the complement of Proposition 1. Fn are closed sets in X then so is U Fk. J any indexing set. These are the sets which contain all their "edge".

A set F c X is closed if its complement. For a metric space X. The proof. That is.. In fact if J. The proofs of a - e are left to the reader. A subset A of a metric space X is dense if A. Notice that int A may be empty and A. Let A and B be subsets of a metric space X. Then Xo E int X -A and. Definition and examples of metric spaces 13 closed. Prove that a set G c X is open if and only if X. Example 1. By part e.

Let A be a subset of X. By Propositions 1. Suppose G c X is open. Which of the following subsets of C are open and which are closed: To prove f assume xoEA.

The closure of A. Then the interior of A. Give the details of the proof of 1. G is closed. Show that IC"" d where d is given by I. Prove Proposition 1. Show that each of the examples of metric spaces given in 1.

But a An equivalent formulation of connectedness is to say that X is not connected if there are disjoint open sets A and B in X.

But the definition of B a. A is open because if a E A then B a. We will show that A cannot also be closed-and hence. This is an example of a non-connected space. It is closed because its complement in X. Similarly B is also both open and closed in X. The proof that other types of intervals are connected is similar and it will be left as an exercise.

A metric space X. X must be connected. A set X c IR is connected iff X is an interval. B is also closed. Connectedness Let us start this section by giving an example. But this gives [a. Do Exercise 8 with "closed" in place of "open. Now a moment's thought will show that one of the segments making up P will have one end point in A and another in B.

This shows that B b. Z2"'" zn' b] be a polygon from a to b with PeG. However it can be shown that both Sand T are open Exercise 2. G-A is open so that A is closed. One could obtain a. Thus we must have that B z. The plan is to show that A is simultaneously open and closed in G.

But we don't have to perform such a construction. Now suppose that G is connected and fix a point a in G. To show how to construct a polygon lying in G!

Proof There are two ways of proving this corollary. An open set Gee is connected iff for any two points a. Connectedness 15 The proof of the converse is Exercise 1. E then. If Gee is open and connected and a and b are points in G then there is a polygon P in G from a to b which is made up of line segments parallel to either the real or imaginary axis.

E c G-A. G must be connected. From the definition. Proof Suppose that G satisfies this condition and let us assume that G is not connected. We will obtain a contradiction. Since G is open this was not needed in the first half. But C must be a component.

The remainder of the proof will be valid with one exception.

## Solutions Manual

Thus C is maximal and part a is proved. Again the lemma says that C 1 U C z is connected. If the reader examines the example at the beginning of this section he will notice that both A and B are components and. Proof a Let! III 2. Define the set A as in the proof of 2. Another proof can be obtained by modifying the proof of Theorem 2.

III It will now be shown that any set S in a metric space can be expressed. Also notice that the hypotheses of the preceding lemma apply to the collection g. A subset D of a metric space X is a component of. Note that part a says that X is the union of its components. D is connected and there is no connected subset of X that properly contains D. Proof Let A be a subset of the metric space CD. Then clearly every component of X is a point and each point is a component.

Show that if F c X is closed and connected then for every pair of points a. Their central role in calculus is duplicated in the study of metric spaces and complex analysis. Let G be open in IC. I s the hypothesis that F be closed needed? If F is a set which satisfies this property then F is not necessarily connected.

Sequences and completeness One of the most useful concepts in a metric space is that of a convergent sequence. II Exercises 1. Show that the sets Sand T in the proof of Theorem 2. Give an example to illustrate this.

The proof is left as an exercise. Prove the following generalization of Lemma 2. By Lemma 2. Let C be a component of G and let Xo c C. Which of the following su bsets X of IC are connected. The purpose of this exercise is to show that a connected subset of IR is an interval. That is B xo. Then S is countable and each component of G contains a point of S.

Many concepts in the theory of metric spaces can be phrased in terms of sequences. IJ is a limit point of A but i is not. The following is an example. Now suppose F is not closed.

We do not wish to call a point such as i a limit point. Let z'" z be points in C. One of their attributes is that you know the limit will exist even though you can't produce it. Such sequences are called Cauchy sequences. By Proposition 1. C is complete.. III Consider Coo with its metric d 1. Since X is complete. In spite of this. Zentralblatt MATH: Behnke and K.

Stein, 1. Stein, 2. Behnke and P. Thullen, 1. Theorie der Funktionen mehrerer komplexer Veraenderlichen, Ergebnisse der Mathematik, vol. Bergman, 1. Jahrbuch database Zbl : Bergman, 2. Sbornik N.

Bergman, 3. Bergman, 4. About this Textbook This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments.

Show all. Pages Complex Integration Conway, John B. Singularities Conway, John B. Harmonic Functions Conway, John B. Entire Functions Conway, John B.Behnke and P. Equation 3. For each m: To see this.. The plan is to show that A is simultaneously open and closed in G.

Iff' is differentiable then f is twice differentiable. Moreover if z.

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