Analytic Functions

By M.A. Evgrafov

This highly regarded text is directed toward advanced undergraduates and graduate students in mathematics who are interested in developing a firm foundation in the theory of functions of a complex variable. The treatment departs from traditional presentations in its early development of a rigorous discussion of the theory of multiple-valued analytic functions on the basis of analytic continuation. Thus it offers an early introduction of Riemann surfaces, conformal mapping, and the applications of residue theory. M. A. Evgrafov focuses on aspects of the theory that relate to modern research and assumes an acquaintance with the basics of mathematical analysis derived from a year of advanced calculus.
Starting with an introductory chapter containing the fundamental results concerning limits, continuity, and integrals, the book addresses analytic functions and their properties, multiple-valued analytic functions, singular points and expansion in series, the Laplace transform, harmonic and subharmonic functions, extremal problems and distribution of values, and other subjects. Chapters are largely self-contained, making this volume equally suitable for the classroom or independent study.

• Language: English
• Publisher: Dover Publications
• Release date: Sep 18, 2019
• ISBN: 9780486843667

Book preview

ANALYTIC

FUNCTIONS

ANALYTIC

FUNCTIONS

M. A. EVGRAFOV

Translated by

Scripta Technica, Inc.

Edited by

Bernard R. Gelbaum

DOVER PUBLICATIONS, INC.

MINEOLA, NEW YORK

Copyright

Copyright © 1966 by Scripta Technica, Inc.

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2019, is an unabridged republication of the English translation originally printed by W. B. Saunders Company, Ltd., Philadelphia, in 1966. The work was first published in Russian by Izdatel’stvo Nauka, Moscow, in 1965, under the title Analiticheskiye Funktsyi.

Library of Congress Cataloging-in-Publication Data

Names: Evgrafov, Marat Andreevich, author.

Title: Analytic functions / M.A. Evgrafov.

Other titles: Analiticheskie funkëtìsii. English

Description: Dover edition [2019 edition]. | Mineola, New York : Dover

Publications, Inc., 2019. | English translation originally published:

Philadelphia : W.B. Saunders Company, 1966. Reprinted: New York : Dover

Publications, 1978.

Identifiers: LCCN 2019012114| ISBN 9780486837604 | ISBN 0486837602

Subjects: LCSH: Analytic functions.

Classification: LCC QA331 .E843 2019 | DDC 515/.73—dc23

LC record available at https://lccn.loc.gov/2019012114

Manufactured in the United States by LSC Communications

83760201

www.doverpublications.com

2019

Contents

Foreword

Preface

I. INTRODUCTION

1.Complex Numbers

2.Sets, Functions and Curves

3.Limits and Series

4.Continuous Functions

5.Line Integrals

6.Integrals Depending on a Parameter

II. ANALYTIC FUNCTIONS AND THEIR PROPERTIES

1.Differentiable and Analytic Functions

2.Cauchy’s Theorem

3.Cauchy’s Integral Formula

4.Criteria for Analyticity

5.A Uniqueness Theorem

6.The Behavior of the Basic Elementary Functions

III. MULTIPLE-VALUED ANALYTIC FUNCTIONS

1.The Concept of a Complete Analytic Function

2.The Analytic Function In z

3.Monodromy Theorem

4.Riemann Surfaces

5.Examples of the Construction of Riemann Surfaces

IV. SINGULAR POINTS AND EXPANSION IN SERIES

1.The Notation of a Singular Point of Regular Function

2.Removal of Singularities

3.Isolated Singular Points

4.Residues and Laurent Series

5.Expansion of Meromorphic Functions in Series of Partial Fractions

6.The Argument Principle and Rouché’s Theorem

7.Implicit Functions and Inverse Functions

V. CONFORMAL MAPPINGS

1.General Information About Mappings

2.Linear Fractional Transformations

3.Conformal Mapping of the Elementary Functions

4.The Riemann-Schwarz Symmetry Principle

5.The Schwarz-Christoffel Integral

6.Approximation of Conformal Mappings Near the Boundary

VI. THE THEORY OF RESIDUES

1.Generalized Contour Integrals

2.Analytic Continuation of Contour Integrals

3.Evaluation of Definite Integrals

4.Asymptotic Formulas for Integrals

5.The Summation of Series

6.Basic Formulas Relating to Euler’s Gamma Function

VII. THE LAPLACE TRANSFORM

1.The Inversion Formula for the Laplace Transform

2.The Convolution Theorem and Other Formulas

3.Examples of the Application of the Method

4.The Generalized Laplace Transform

5.The Use of Analytic Continuation

6.The Mellin Transform

VIII. HARMONIC AND SUBHARMONIC FUNCTIONS

1.Basic Properties of Harmonic Functions

2.Subharmonic Functions

3.The Dirichlet Problem and Poisson’s Integral

4.Harmonic Measure

5.Uniqueness Theorems for Bounded Functions

6.The Phragmen-Lindelöf Theorems

IX. CONFORMAL MAPPINGS OF MULTIPLY CONNECTED DOMAINS

1.The Existence of Conformal Mappings

2.Corresponding Boundaries Under Conformal Mappings

3.The Automorphism of a Conformal Mapping

4.The Dirichlet Problem and Mapping Onto Canonical Domains

5.Mapping of the Plane with Deleted Points

6.Automorphic and Elliptic Functions

X. EXTREMAL PROBLEMS AND DISTRIBUTION OF VALUES

1.The Principle of the Hyperbolic Metric

2.The Symmetrization Principle

3.Bounds for Functions Univalent in the Mean

4.The Principle of Length and Area

5.Distribution of Values of Entire and Meromorphic Functions

6.Nevanlinna’s Theorem of Defect

INDEX

Foreword

This book represents a healthy departure from traditional presentations of analytic function theory. Among the interesting features are the early introduction of Riemann surfaces, conformal mapping, and the applications of residue theory. The orientation of the author is modern in that he dwells at much greater length on those aspects of the theory that lead into the modern parts of research, rather than on some of the classical (albeit elegant) topics, whose relevance to further research is becoming less and less clear.

The book should serve the needs of all graduate students who wish to get a firm foundation in the theory of functions of a complex variable and who wish at the same time to be made aware of the most important lines of its modern development.

Bernard R. Gelbaum

Preface

The present text is Preface designed for students and other readers acquainted with the fundamentals of mathematical analysis as presented in a year of advanced calculus.

The order of presentation of material in this textbook is significantly different from that of other texts in analytical function theory. We shall give a rigorous discussion of the theory of multiple-valued analytic functions presented on the basis of analytic continuation. This theory is presented toward the very end in many textbooks in use today, but it enters in the present book much nearer the beginning (Chap. III). A strong argument can be made for such arrangement of the material. First of all, from the point of view of logic, analytic continuation plays no less a role in the theory of complex variables than that of the theory of limits in analysis. Second, it is appropriate from the purely practical point of view, since the earlier utilization of analytic continuation allows great economy in space and time in the presentation to follow. The usual objection made against this arrangement is based on the opinion that analytic continuation is difficult to understand. However, the difficulty is greatly exaggerated. Moreover, whatever the difficulties, they must be overcome in any event upon the introduction of the elementary multiple-valued functions, and by more artificial (and therefore less intelligible) means.

In any case, my lecturing experience in the theory of analytic functions at the Moscow Physical-Technical Institute has convinced me that two or three difficult (but completely accessible) lectures were fully justified by the better understanding of the material that followed. The exercises could be gone over much more easily, since the problem of isolating analytic branches is laborious and difficult to understand.

It was also valuable in that the student developed from the beginning a correct and accurate point of view of the subject studied. In the writing of the book I tried to make the separate chapters as independent as possible of one another. The object of this was to make the book available for a variety of courses with considerable variation in content. The volume of material presented in the text considerably exceeds the content of the courses usually given at the Institute. It is worthwhile to stress that all the chapters are written on a level completely accessible to third year students.

I will now point out the relations among the chapters. Chapter I need only be employed for purposes of reference. Chapters II–IV are essential for all that follows. Chapters VI and VII have no connection with V and VIII–X. Chapter VIII relies to a considerable extent on Chapter V, and itself serves as the base for the two that follow. Chapters IX and X have only slight connections with one another.

A subject and name index is included at the end of the book.

In conclusion I should like to thank all those who contributed to my work on this book. First of all, I thank my teachers: Corresponding Member of the Academy of Sciences of the U. S. S. R. A. O. Gel’fond and Academician M. V. Keldysh. My views took shape under the influence of my discussions with them, and this influence is more noticeable in this book than in other of my works. My students helped me very much: Candidates in the Physical-Mathematical Sciences I. S. Arshon and G. M. Mordasova, who carefully read all the variants of the manuscript as they made their appearance. V. V. Zarutskaya, the editor of this book, helped me very much in finishing the final version of the manuscript. I am very thankful to professors V. B. Lidskiy and B. V. Shabat, and to docents O. V. Lokutsievskiy and M. B. Fedoryuk for their helpful observations.

M. A. Evgrafov

ANALYTIC

FUNCTIONS

CHAPTER I

Introduction

The study of the theory of analytic functions requires that the student have mastered a full course of mathematical analysis. It is therefore quite natural to assume that all the results needed for the presentation of the theory are already known from analysis. Unfortunately, all the questions treated in a course of analysis are presented in terms of real functions of real variables. For this reason, before we can present the theory of analytic functions, we must introduce at least the statements of the fundamental results concerning limits, continuity and integrals. Since the presentation of the proofs of a small number of theorems does not replace a full course in analysis, and since the reader who has studied analysis can readily prove the theorems stated himself, we have given only the statements of the theorems in the majority of cases. Exception to this rule is made only in a comparatively small number of cases where the theorem in question is not characteristic of analysis.

1. COMPLEX NUMBERS

By a complex number z we mean an ordered pair of real numbers (a, b).

We say that two complex numbers z = (a, b) and ζ = (c, d) are equal, if a = c and b = d.

The operations of addition and multiplication are defined for complex numbers by means of the following rules:

By using the definition, verify that the operations on the complex numbers have the following properties:

1. Associativity, i.e.,

2. Commutativity, i.e.,

3. Distributivity, i.e.,

We shall define multiplication of the complex number z = (a, b) by the real number c by means of the equation

Then any complex number can be written in the form

The number e1 behaves like an identity under multiplication, since ze1 = z for all z. It is therefore reasonable to regard e1 as the identity or 1. It is customary to use the notation e2 = i for the number e2 and to call it the imaginary unit. It is easily verified that i² = -1. Thus, complex numbers can be written in the form

The complex number a + 0i is identified with the real number a, and the complex number 0 + ib is called an imaginary number.

The notion of complex numbers may be considered as an extension of the concept of real numbers. The same basic axioms hold for complex numbers as for real numbers, with the exception of the order axioms and in particular the Archimedean property concerning order. The concepts of greater than and less than have no meaning for complex numbers.

The real numbers are represented by the points of a line, while complex numbers are represented in a natural way by the points of a plane. Specifically, the complex number a + bi is represented by the point of the (coordinate) plane with abscissa a and ordinate b. Complex numbers can likewise be represented by vectors, particularly since they may be added like vectors. However, the analogy between complex numbers and vectors should not be carried much further. Neither the scalar nor the vector product have any relation to the multiplication of complex numbers.

We next introduce a series of traditional names and notations for the complex numbers.

The plane in which we represent the complex numbers is called the complex plane.

The axis of abscissas in the complex plane is called the real axis, and the axis of ordinates the imaginary axis.

Let z = a + bi. We shall employ the following names and notations:

a = Re z — the real part of z;

b = Im z — the imaginary part of z;

a—bi = 2 — the complex conjugate of z;

the modulus of z (absolute value of z);

arg z—the argument of z; this is the number φ defined (modulo 2π) by the equations

All the quantities introduced have simple geometrical interpretations in the complex plane. Thus, the modulus of z is the distance from the origin of coordinates to the point z, the argument of z is the angle between the positive real axis and the vector issuing from the origin in the direction of z.

Let us note two important inequalities satisfied by the modulus.

Theorem 1.1.For any complex numbers

Proof. Let us consider the triangle with vertices 0, z1, z1 + z2. The lengths of its sides are: |z1| (from 0 to z1), |z2| (from z1 to z1 + z2 and | z1 + z2 | (from z1 + z2 to 0), since the distance between two points z and ζ equals |z—ζ|. We know that the length of a side of a triangle is not greater than the sum of the lengths of the other two sides and not less than the absolute value of their difference. Applying this statement to the side from z1 + z2, to 0, we obtain the required inequality. [Ed.: A strictly analytic proof may be based on the Schwarz inequality for real numbers.]

It is now easy to describe various regions or lines in the complex plane with the aid of the notation introduced. For example:

The points z satisfying the inequality | z—z0| < R lie in a circle of radius R with center at the point z0.

The points z satisfying the inequality Im z > 0 lie in the upper half-plane, i.e., above the real axis.

The points z satisfying the inequality |arg z—θ| η lie within an angle of 2η with vertex at the origin and with a bisector making an angle θ with the positive real axis. The points of the bisector are described by the equality arg z = θ.

Complex numbers are often represented in the so-called exponential or trigonometric form:

(here eiφ is to be understood according to Euler’s formula as cos φ + i sin φ). It is not difficult to see that

We can easily deduce the following result from the definition of the product of complex numbers:

The modulus of the product of complex numbers is the product of the moduli, and the argument of the product is the sum (modulo 2π) of the arguments of the moduli.

We shall usually be dealing with the so-called extended complex plane, the complex plane supplemented by the point at infinity, which corresponds to the conventional complex number ∞. The extended complex plane is also called the complex sphere or Riemann sphere. This name is justified in terms of the following geometric interpretation (stereographic projection).

Imagine a plane in the three-dimensional space and a sphere of radius 1/2 placed above this plane and tangent to it at the origin of coordinates. We denote the origin of coordinates by O and the opposite pole of the sphere by P. We now make each point z of the plane correspond to a point A (z) on the sphere where the sphere with the line joining the points z and P intersect. Moreover, the set of points of the sphere with the exception of P are found to be in a one-one correspondence with the points of the plane. We readily observe that as | z | → ∞, the point A(z) approaches the point P. Therefore it is natural to say that the point P of the sphere corresponds to the point at infinity of the extended plane.

The formula expressing the coordinates of the point A (z) of the Riemann sphere in terms of the coordinates of the point of the plane, sometimes finds application.

Theorem 12.The point z = x + iy under stereographic projection corresponds to the point A (z) of the sphere

with coordinates

Proof. Since the projection of the point A lies on the line Oz, then, ξ = λx, η = λy, where λ is some real constant. We shall find ζ in terms of |z|. Let us consider the cross section of the sphere cut by the plane passing through the points 0, P and z (Fig. 1). The right triangles OPz and OAz are similar. The altitude of the triangle OAz is equal to ζ and its hypotenuse is |z|. The segment OA is a leg of the triangle OAz and the altitutde of the triangle OPz. From the similarity of the triangles OPz and OAz we have

Fig. 1.

and after that ζ and η.

Corollary. We denote the distance between the points A (z) and A (w) by k(w, z). We can easily obtain

with the aid of the formula of Theorem 1.2.

The quantity k (w, z) is called the chordal distance between the points w and z.

2. SETS, FUNCTIONS AND CURVES

We shall need to deal with a variety of sets in the extended complex plane, and to avoid ambiguity we shall define here the terms to be employed.

We shall usually write the formula z ∈ E in place of the words "the point z belongs to the set E," and the formula z E in place of the words "the point z does not belong to the set E."

By the intersection of the sets E1 and E2 we mean the set E consisting of all points belonging to both E1 and E2.

By the distance between the sets E1 and E2 we mean the quantity

We shall call the circle |z — z0| < r, where r is any positive number, a neighborhood of the point z0.

We shall call the set |z| > R for any R (the exterior of a circle)* a neighborhood of the point at infinity.

We shall call the point z a limit point of the set E, if an infinite number of points of the set E are in any neighborhood of the point z.

The point z is called an interior point of the set E, if it has a neighborhood consisting entirely of points of E.

The point z is called an exterior point of the set E if it has a neighborhood consisting only of points not belonging to E.

The point z is called a boundary point of the set E, if in every neighborhood of the point z are found points belonging and points not belonging to E.

The union of all the boundary points of the set E is called its boundary.

A set is said to be closed, if it contains its boundary.

It is possible to show that the boundary of a set is always a closed set.

The set obtained by taking the union of E and its boundary is called the closure of E

A set is said to be open, if all its points are interior points.

We shall define the notion of connectedness for open and closed sets. The definitions are essentially different.

An open set is called connected if any two of its points can be joined by a polygonal line all of whose points belong to the set.

A closed set is called connected, if it is impossible to decompose it into two subsets which are a positive distance apart.*

A connected open set is called a domain.

The closure of a domain is called a closed domain.

A domain is said to be n-connected if its boundary consists of n-connected subsets.

In particular, a 1-connected (simply connected) domain has a boundary consisting of one connected set, i.e., roughly speaking, of one curve or one point. A multiply connected domain can be imagined as a simply connected domain in which holes have been made (n—1 holes in an n-connected domain).

Suppose we are given a set E in the complex plane and a rule by which a complex number f(z) is made to correspond to each point z of the set E. Then we will say that f (z) is a function of the complex variable z defined on the set E. The set E is called the domain of definition of the function f(z).

If we use the notation z = x + iy, f(z) = u + iv, then we can study the function f (z) of the complex variable z as a pair of functions u(x, y) and v(x, y) of two real variables x and y.

Functions of a complex variable can be studied geometrically as mappings of one complex plane (or, more exactly, a part of it) into another. The mapping given by the function of a complex variable w = f(z) is equivalent to the real mapping

in which

The mapping (2.1) is called nondegenerate at the point (x0, y0) if the Jacobian D(x0, y0) of the transformation is different from zero. We recall that the Jacobian of a transformation is the quantity

The linear transformation

where

is called the principal linear part of the transformation (2.1) at the point (x0, y0).

The transformation (2.1) is called one-one in the domain D, if the transformation takes distinct points of the domain D into distinct points of the plane.

In the study of topology it is shown that a one-one transformation defined by a pair of continuous functions u(x, y) and v(x, y) must take an n-connected domain into an n-connected domain.

We shall discuss in somewhat greater detail domains and the curves bounding them.

Let us begin with the general concept of a continuous curve.

Let x(t) and y(t) be continuous functions of the parameter t on the segment [a, b], and z(t) = x(t) + iy(t).

We shall call the equation z = z(t), a ≤ t ≤ b, the parametric equation of a curve. Moreover, we shall assume that two functions

define the same curve only if there exists a monotonically increasing and continuous function φ (t) on the segment [a1, b1] such that

The definition given says that we assume in general that a curve has a parametric representation. It is evident that this representation involves first, a set of points (of the plane) presented in the form z = z (t), a ≤ t ≤ b (it is easy to convince oneself that this set does not vary under the transition from one parametric equation to another); second, the order in which the points of this set are traversed (the order of the points is likewise maintained under the transition from one equation to another). Clearly, we have described a very special situation.

We call a curve defined as above a continuous curve.

Let us suppose that at least one of the parametric equations of a curve is such that the functions x(t) and y (t) are continuously differentiable on the segment [a, b] with the exception of at most a finite number of points, at which the left- and right-hand limits of the derivatives exist. Then we call the curve piecewise smooth. (The exceptional points will be called corners.)

Since the points of the curve are ordered, we may speak of the beginning (initial point) and the end (terminal point) of the curve. These are the points Z (a) and Z (b) respectively.

The curve is called closed, if its initial and terminal points coincide.

An arbitrary curve may intersect itself any number of times. We can imagine it as a tangled thread lying on the table. It is clear that an arbitrary curve need not be the boundary of some domain. The following much narrower class of curves has significance.

Curves that do not intersect themselves, i.e., such that the function z(t) takes on different values for different values of t, a < t < b, are called simple curves (it is not considered an intersection if the initial and terminal points coincide).

The following result is well known:

Jordan’s theorem. A simple, closed curve divides the plane into two domains and is their common boundary.

It follows from Jordan’s theorem that the following expression has meaning for simple curves: motion to the left (or to the right) along the curve.

We shall define still some classes of curves that occur as the boundaries of domains.

Suppose that a given piecewise-smooth curve can be divided into a finite number of pieces, each of which is a simple curve. Suppose, furthermore, that there exists a division of the curve into simple parts such that two distinct parts either have only end points in common or completely coincide, but have opposite directions. We call such a curve a piecewise-smooth curve with folds (folds are the parts that coincide and have opposite direction).

Let C be a closed piecewise-smooth curve with folds, and let the set of points lying on C coincide with the boundary of a domain D. If the domain D remains on the left when we traverse any simple part of the curve C, then we shall say that C is the boundary curve of the domain D.

Folds of the curve inside the domain are called cross cuts.

Example 1. We shall study the domain described by the inequality

This domain is obtained by removing the radius (0, 1) from the circle | z | < 1. It is easy to make a model of this domain by cutting a circle out of paper and then cutting it along a radius. The boundary curve of this domain is a piecewise-smooth curve with a fold (the slit in the paper). The equation of the curve is easy to find, say, by expressing the points as a function of time as we move along the curve with constant velocity. One of the possible equations is z = z (t), 0 ≤ t < 3π, where

Another class of curves, for which it is possible to define a concept of length, has great importance in questions connected with integration.

Suppose we are given an arbitrary continuous curve. We choose an arbitrary number of points on it (the first is the initial point of the curve, the last is its terminal point). We obtain a polygonal line by joining these points by lines in the order of their succession along the curve. If the set of lengths of the polygonal lines thus obtained is bounded, then we say that the curve is rectifiable.

The least upper bound of the lengths of these polygonal lines is called the length of the curve.

3. LIMITS AND SERIES

Since it is customary in analysis to study only the limits of real functions, we shall briefly present the basic facts about limits of functions of a complex variable.

Let the function f(z) be defined on the set Ε, ζ be a limit point of the set E, and suppose that there exists a number A satisfying the condition: For every ε > 0 we can find a δ > 0 such that for all z, 0 < | z—ζ |< δ, the inequality | f(z) —A|< ε is satisfied.

Then we say that the function f(z) has a limit A as z → ζ in the set E. This fact will be denoted in one of the two following ways:

If some neighborhood of the point ζ is contained in the set E, then we can omit the requirement z ∈ E in these formulas.

The formulation is easily modified in the cases that ζ = ∞ or A = ∞ (or both at once). For ζ = ∞ we must write | z |> R, z ∈ E, in place of 0 < | z —ζ |< δ, z ∈ E, and for A = ∞ we must write | f(z)| > R in place of | f(z) —A|< ε.

The limit of a sequence is the special case of a limit of a function which occurs when E coincides with the set of positive integers.

We shall state some properties of limits, the proofs of which will be left to the reader. (Here the discussion is only about finite limits.)

If the limit of each of a finite number of terms exists, then the sum of the limits is equal to the limit of the sum of these terms.

If the limit of each of a finite number of factors exists, then the product of the limits is equal to the limit of the product of these factors.

If the limits of the numerator and the denominator exist and if the denominator and the limit of the denominator are different from zero, then the limit of the quotient exists and is equal to the quotient of the limits.

In order that the limit of a complex variable exist, it is necessary and sufficient that the limits of the real and the imaginary parts exist.

In order that the limit of f (z) exist as z approaches ζ in the set E, it is necessary and sufficient that the following condition be satisfied: for any ε > ο we can find α δ > 0 such that for any z' ∈ Ε, z' ∈ Ε, | z—ζ |< δ, |z' —ζ |< δ, the inequality

holds (Cauchy’s criterion).

In what follows we shall need to use the symbols ~, o, O. The meaning of these symbols is as follows:

The formula

denotes that

The formula

denotes that

The formula

denotes that

(Roughly speaking, f(z) = O (φ (z)) (z ∈ E) is bounded on the set E.)

The concept of uniform convergence to a limit has great importance.

Suppose we are given a function f(z, w) depending on the parameter w and suppose that

for any fixed value of w ∈ E. We shall say that the convergence to the limit is uniform with respect to w ∈ Ε, if for any ε > 0 we can find a δ > 0, depending only on ε, but not on w, such that for z ∈ G, 0 <| z—ζ |< δ, and for all w ∈ Ε the inequality

is satisfied.

The concept of uniformity can also be applied to the symbols ο and ~. It means that the convergence to the limit occurring in the definition of the symbol is uniform with respect to the parameter in question. For the symbol 0, uniformity means that the constant C is uniformly bounded with respect to the parameter in question.)

We shall say that the series converges, has a limit as n → ∞. This limit is called the sum of the series.

is called absolutely convergent, converges. An absolutely convergent series converges.

We shall now introduce the basic facts about numerical series.

1. A necessary and sufficient condition for the convergence of the series is: for any ε > 0 we can find a positive integer Ν such that for any n > N and n' > N, the following inequality holds:

2. A necessary condition for the convergence of the series is that un → 0.

3. If the series is absolutely convergent and |vn| < |un|, then the series also converges absolutely.

also. The concept of uniform convergence has great importance for series of functions.

convergent for all z ∈ G, converges uniformly with respect to z ∈ G, if for any ε > 0 we can find a positive integer N, depending only on ε, but not on z, such that for n > N, n' > N, and for any z ∈ G

The following criterion (bearing the name: Weierstrass’ criterion) for uniform convergence of a series of functions is often applied:

If |un(z)| < un for all z ∈ G and the series converges, then the series converges uniformly with respect to z ∈ G.

In conclusion, we shall present the necessary information about power series, i.e., about series of the form

where z., a and cn are complex numbers.

The following statement is called Abel’s first theorem.

Theorem 3.1.If the series (3.1) converges for z = z1, then it converges absolutely and uniformly with respect to z in any circle |z —a| ≤ R, where R <| z1 —a |.

Proof. Since the series (3.1) converges for z = z1, then, according to property 2, cn (z1 —a)n→ 0. But a sequence converging to zero is bounded in modulus. This means that

Moreover, for any z from the circle | z—a | ≤R, R< | z1 —a|, we have the inequality

Therefore

converges absolutely for 0 ≤ θ < l. Applying

Weierstrass’ criterion, we obtain the statement of the theorem.

Abel’s first theorem leads to the following important conclusion:

There exists a number R possessing the property: for | z —a |< R the series (3.1) converges, and for | z—a| > R the series diverges. (The number R may be either zero or infinity.) This number R is called the radius of convergence of the series (3.1) and the circle | z—a | < R is called the circle of convergence of the series 3.1.)

We have the following formula giving the radius of convergence in terms of the coefficients of the series (3.1):

which bears the name of the Cauchy-Hadamard formula.

4. CONTINUOUS FUNCTIONS

The function f (z) defined on the set E is said to be continuous at the point ζ ∈ E, if for any ε > ο we can find a δ > 0, such that for z ∈ E, | z—ζ |< δ, we have | f (z) —f (ζ) |< ε.

If the function f (z) defined on the set Ε is continuous at each of the points of E, then it is said to be continuous on this set.

We shall list a series of properties of continuous functions:

The continuity of a function f(z) of a complex variable is equivalent to the continuity of the real functions u (x, y) and ν (x, y):

of two real variables x and y.

The sum and product of two continuous functions is continuous.

The quotient of two continuous functions is continuous at all points at which the denominator does not vanish.

If the values of the function f(z), continuous on the