34258750 Complex Variables Introduction and Applications

The study of complex variables is important for students in engineering and thephysical sciences and is a central subject in mathematics. In addition to beingmathematically elegant, complex variables provide a powerful tool for solvingproblems that are either very difcult or virtually impossible to solve in anyother way.

Part I of this text provides an introduction to the subject, including analyticfunctions, integration, series, and residue calculus. It also includes transformmethods, ordinary differential equations in the complex plane, numerical meth-ods, and more. Part II contains conformal mappings, asymptotic expansions,and the study of RiemannHilbert problems. The authors also provide an ex-tensive array of applications, illustrative examples, and homework exercises.

This new edition has been improved throughout and is ideal for use in intro-ductory undergraduate and graduate level courses in complex variables.

Complex VariablesIntroduction and Applications

Second Edition

Cambridge Texts in Applied Mathematics

FOUNDING EDITORProfessor D.G. Crighton

EDITORIAL BOARDProfessor M.J. Ablowitz, Department of Applied Mathematics,

University of Colorado, Boulder, USA.Professor J.-L. Lions, College de France, France.

Professor A. Majda, Department of Mathematics, New York University, USA.Dr. J. Ockendon, Centre for Industrial and Applied Mathematics, University of Oxford, UK.

Professor E.B. Saff, Department of Mathematics, University of South Florida, USA.

Maximum and Minimum PrinciplesM.J. Sewell

SolitonsP.G. Drazin and R.S. Johnson

The Kinematics of MixingJ.M. Ottino

Introduction to Numerical Linear Algebra and OptimisationPhillippe G. CiarletIntegral Equations

David Porter and David S.G. StirlingPerturbation Methods

E.J. HinchThe Thermomechanics of Plasticity and Fracture

Gerard A. MauginBoundary Integral and Singularity Methods for Linearized Viscous Flow

C. PozrikidisNonlinear Systems

P.G. DrazinStability, Instability and Chaos

Paul GlendinningApplied Analysis of the Navier-Stokes Equations

C.R. Doering and J.D. GibbonViscous Flow

H. Ockendon and J.R. OckendonSimilarity, Self-similarity and Intermediate Asymptotics

G.I. BarenblattA First Course in the Numerical Analysis of Differential Equations

A. IserlesComplex Variables: Introduction and Applications

Mark J. Ablowitz and Athanssios S. Fokas

Complex VariablesIntroduction and Applications

Second Edition

MARK J. ABLOWITZUniversity of Colorado, Boulder

ATHANASSIOS S. FOKASUniversity of Cambridge

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo

Cambridge University PressThe Edinburgh Building, Cambridge , United Kingdom

First published in print format

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Contents

Sections denoted with an asterisk (*) can be either omitted or readindependently.

Preface page xi

Part I Fundamentals and Techniques of Complex FunctionTheory 1

1 Complex Numbers and Elementary Functions 31.1 Complex Numbers and Their Properties 31.2 Elementary Functions and Stereographic Projections 8

1.2.1 Elementary Functions 81.2.2 Stereographic Projections 15

1.3 Limits, Continuity, and Complex Differentiation 201.4 Elementary Applications to Ordinary Differential Equations 26

2 Analytic Functions and Integration 322.1 Analytic Functions 32

2.1.1 The CauchyRiemann Equations 322.1.2 Ideal Fluid Flow 40

2.2 Multivalued Functions 462.3 More Complicated Multivalued Functions and Riemann

Surfaces 612.4 Complex Integration 702.5 Cauchys Theorem 812.6 Cauchys Integral Formula, Its Generalization and

Consequences 91

vii

viii Contents

2.6.1 Cauchys Integral Formula and Its Derivatives 912.6.2 Liouville, Morera, and Maximum-Modulus

Theorems 952.6.3 Generalized Cauchy Formula and Derivatives 98

2.7 Theoretical Developments 105

3 Sequences, Series, and Singularities of Complex Functions 1093.1 Denitions and Basic Properties of Complex Sequences,

Series 1093.2 Taylor Series 1143.3 Laurent Series 1273.4 Theoretical Results for Sequences and Series 1373.5 Singularities of Complex Functions 144

3.5.1 Analytic Continuation and Natural Barriers 1523.6 Innite Products and MittagLefer Expansions 1583.7 Differential Equations in the Complex Plane: Painleve

Equations 1743.8 Computational Methods 196

3.8.1 Laurent Series 1963.8.2 Differential Equations 198

4 Residue Calculus and Applications of Contour Integration 2064.1 Cauchy Residue Theorem 2064.2 Evaluation of Certain Denite Integrals 2174.3 Principal Value Integrals and Integrals with Branch

Points 2374.3.1 Principal Value Integrals 2374.3.2 Integrals with Branch Points 245

4.4 The Argument Principle, Rouches Theorem 2594.5 Fourier and Laplace Transforms 2674.6 Applications of Transforms to Differential Equations 285

Part II Applications of Complex Function Theory 309

5 Conformal Mappings and Applications 3115.1 Introduction 3115.2 Conformal Transformations 3125.3 Critical Points and Inverse Mappings 3175.4 Physical Applications 3225.5 Theoretical Considerations Mapping Theorems 341

Contents ix

5.6 The SchwarzChristoffel Transformation 3455.7 Bilinear Transformations 3665.8 Mappings Involving Circular Arcs 3825.9 Other Considerations 400

5.9.1 Rational Functions of the Second Degree 4005.9.2 The Modulus of a Quadrilateral 4055.9.3 Computational Issues 408

6 Asymptotic Evaluation of Integrals 4116.1 Introduction 411

6.1.1 Fundamental Concepts 4126.1.2 Elementary Examples 418

6.2 Laplace Type Integrals 4226.2.1 Integration by Parts 4236.2.2 Watsons Lemma 4266.2.3 Laplaces Method 430

6.3 Fourier Type Integrals 4396.3.1 Integration by Parts 4406.3.2 Analog of Watsons Lemma 4416.3.3 The Stationary Phase Method 443

6.4 The Method of Steepest Descent 4486.4.1 Laplaces Method for Complex Contours 453

6.5 Applications 4746.6 The Stokes Phenomenon 488

6.6.1 Smoothing of Stokes Discontinuities 4946.7 Related Techniques 500

6.7.1 WKB Method 5006.7.2 The Mellin Transform Method 504

7 RiemannHilbert Problems 5147.1 Introduction 5147.2 Cauchy Type Integrals 5187.3 Scalar RiemannHilbert Problems 527

7.3.1 Closed Contours 5297.3.2 Open Contours 5337.3.3 Singular Integral Equations 538

7.4 Applications of Scalar RiemannHilbert Problems 5467.4.1 RiemannHilbert Problems on the Real Axis 5587.4.2 The Fourier Transform 5667.4.3 The Radon Transform 567

x Contents

7.5 Matrix RiemannHilbert Problems 5797.5.1 The RiemannHilbert Problem for Rational

Matrices 5847.5.2 Inhomogeneous RiemannHilbert Problems and

Singular Equations 5867.5.3 The RiemannHilbert Problem for Triangular

Matrices 5877.5.4 Some Results on Zero Indices 589

7.6 The DBAR Problem 5987.6.1 Generalized Analytic Functions 601

7.7 Applications of Matrix RiemannHilbert Problems and Problems 604

Appendix A Answers to Odd-Numbered Exercises 627

Bibliography 637

Index 640

Preface

The study of complex variables is beautiful from a purely mathematical pointof view and provides a powerful tool for solving a wide array of problemsarising in applications. It is perhaps surprising that to explain real phenomena,mathematicians, scientists, and engineers often resort to the complex plane.In fact, using complex variables one can solve many problems that are eithervery difcult or virtually impossible to solve by other means. The text providesa broad treatment of both the fundamentals and the applications of this subject.

This text can be used in an introductory one- or two-semester undergraduatecourse. Alternatively, it can be used in a beginning graduate level course andas a reference. Indeed, Part I provides an introduction to the study of complexvariables. It also contains a number of applications, which include evaluationof integrals, methods of solution to certain ordinary and partial differentialequations, and the study of ideal uid ow. In addition, Part I develops asuitable foundation for the more advanced material in Part II. Part II containsthe study of conformal mappings, asymptotic evaluation of integrals, the so-called RiemannHilbert and DBAR problems, and many of their applications.In fact, applications are discussed throughout the book. Our point of view isthat students are motivated and enjoy learning the material when they can relateit to applications.

To aid the instructor, we have denoted with an asterisk certain sections that aremore advanced. These sections can be read independently or can be skipped.We also note that each of the chapters in Part II can be read independently.Every effort has been made to make this book self-contained. Thus advancedstudents using this text will have the basic material at their disposal withoutdependence on other references.

We realize that many of the topics presented in this book are not usu-ally covered in complex variables texts. This includes the study of ordinary

xi

xii Preface

differential equations in the complex plane, the solution of linear partial differ-ential equations by integral transforms, asymptotic evaluation of integrals, andRiemannHilbert problems. Actually some of these topics, when studied atall, are only included in advanced graduate level courses. However, we believethat these topics arise so frequently in applications that early exposure is vital.It is fortunate that it is indeed possible to present this material in such a waythat it can be understood with only the foundation presented in the introductorychapters of this book.

We are indebted to our families, who have endured all too many hours ofour absence. We are thankful to B. Fast and C. Smith for an outstanding jobof word processing the manuscript and to B. Fast, who has so capably usedmathematical software to verify many formulae and produce gures.

Several colleagues helped us with the preparation of this book. B. Herbstmade many suggestions and was instrumental in the development of the com-putational section. C. Schober, L. Luo, and L. Glasser worked with us on manyof the exercises. J. Meiss and C. Schober taught from early versions of themanuscript and made valuable suggestions.

David Benney encouraged us to write this book and we extend our deepappreciation to him. We would like to take this opportunity to thank thoseagencies who have, over the years, consistently supported our research efforts.Actually, this research led us to several of the applications presented in this book.We thank the Air Force Ofce of Scientic Research, the National ScienceFoundation, and the Ofce of Naval Research. In particular we thank ArjeNachman, Program Director, Air Force Ofce of Scientic Research (AFOSR),for his continual support.

Since the rst edition appeared we are pleased with the many positive anduseful comments made to us by colleagues and students. All necessary changes,small additions, and modications have been made in this second edition. Ad-ditional information can be found from www.cup.org/titles/catalogue.

Part IFundamentals and Techniques of Complex

Function Theory

The rst portion of this text aims to introduce the reader to the basic notions andmethods in complex analysis. The standard properties of real numbers and thecalculus of real variables are assumed. When necessary, a rigorous axiomaticdevelopment will be sacriced in place of a logical development based uponsuitable assumptions. This will allow us to concentrate more on examples andapplications that our experience has demonstrated to be useful for the studentrst introduced to the subject. However, the important theorems are stated andproved.

1

1Complex Numbers and Elementary Functions

This chapter introduces complex numbers, elementary complex functions, andtheir basic properties. It will be seen that complex numbers have a simple two-dimensional character that submits to a straightforward geometric description.While many results of real variable calculus carry over, some very importantnovel and useful notions appear in the calculus of complex functions. Appli-cations to differential equations are briey discussed in this chapter.

1.1 Complex Numbers and Their PropertiesIn this text we use Eulers notation for the imaginary unit number:

i2 = 1 (1.1.1)A complex number is an expression of the form

z = x + iy (1.1.2)Here x is the real part of z, Re(z); and y is the imaginary part of z, Im(z).

If y = 0, we say that z is real; and if x = 0, we say that z is pure imaginary.We often denote z, an element of the complex numbers as z C; where x , anelement of the real numbers is denoted by x R. Geometrically, we representEq. (1.1.2) in a two-dimensional coordinate system called the complex plane(see Figure 1.1.1).

The real numbers lie on the horizontal axis and pure imaginary numbers onthe vertical axis. The analogy with two-dimensional vectors is immediate. Acomplex number z = x + iy can be interpreted as a two-dimensional vector(x, y).

It is useful to introduce another representation of complex numbers, namelypolar coordinates (r, ):

x = r cos y = r sin (r 0) (1.1.3)

3

4 1 Complex Numbers and Elementary Functions

r

x

y

z = x+iy

Fig. 1.1.1. The complex plane (z plane)

Hence the complex number z can be written in the alternative polar form:

z = x + iy = r(cos + i sin ) (1.1.4)The radius r is denoted by

r =

x2 + y2 |z| (1.1.5a)(note: denotes equivalence) and naturally gives us a notion of the absolutevalue of z, denoted by |z|, that is, it is the length of the vector associated withz. The value |z| is often referred to as the modulus of z. The angle is calledthe argument of z and is denoted by arg z. When z = 0, the values of can befound from Eq. (1.1.3) via standard trigonometry:

tan = y/x (1.1.5b)where the quadrant in which x , y lie is understood as given. We note that arg z is multivalued because tan is a periodic function of with period . Given z = x + iy, z = 0 we identify to have one value in the interval0 < 0 + 2 , where 0 is an arbitrary number; others differ by integermultiples of 2 . We shall take 0 = 0. For example, if z = 1 + i , then|z| = r = 2 and = 34 + 2n , n = 0,1,2, . . . . The previous remarksapply equally well if we use the polar representation about a point z0 = 0. Thisjust means that we translate the origin from z = 0 to z = z0.

At this point it is convenient to introduce a special exponential function. Thepolar exponential is dened by

cos + i sin = ei (1.1.6)Hence Eq. (1.1.4) implies that z can be written in the form

z = rei (1.1.4)This exponential function has all of the standard properties we are familiar

with in elementary calculus and is a special case of the complex exponential

1.1 Complex Numbers and Their Properties 5

function to be introduced later in this chapter. For example, using well-knowntrigonometric identities, Eq. (1.1.6) implies

e2 i = 1 e i = 1 e i2 = i e 3 i2 = iei1 ei2 = ei(1+2) (ei )m = eim (ei )1/n = ei/n

With these properties in hand, one can solve an equation of the form

zn = a = |a|ei = |a|(cos + i sin), n = 1, 2, . . .

Using the periodicity of cos and sin, we have

zn = a = |a|ei(+2m) m = 0, 1, . . . , n 1

and nd the n roots

z = |a|1/nei(+2m)/n m = 0, 1, . . . , n 1.

For m n the roots repeat.If a = 1, these are called the n roots of unity: 1, , 2, . . . , n1, where

= e2 i/n . So if n = 2, a = 1, we see that the solutions of z2 = 1 = eiare z = {ei/2, e3i/2}, or z = i . In the context of real numbers there are nosolutions to z2 = 1, but in the context of complex numbers this equation hastwo solutions. Later in this book we shall show that an nth-order polynomialequation, zn + an1zn1 + + a0 = 0, where the coefcients {a j }n1j=0 arecomplex numbers, has n and only n solutions (roots), counting multiplicities(for example, we say that (z 1)2 = 0 has two solutions, and that z = 1 is asolution of multiplicity two).

The complex conjugate of z is dened as

z = x iy = rei (1.1.7)

Two complex numbers are said to be equal if and only if their real andimaginary parts are respectively equal; namely, calling zk = xk + iyk , for k =1, 2, then

z1 = z2 x1 + iy1 = x2 + iy2 x1 = x2, y1 = y2Thus z = 0 implies x = y = 0.

Addition, subtraction, multiplication, and division of complex numbers fol-low from the rules governing real numbers. Thus, noting i2 = 1, we have

z1 z2 = (x1 x2)+ i(y1 y2) (1.1.8a)


and

z1z2 = (x1 + iy1)(x2 + iy2) = (x1x2 y1 y2)+ i(x1 y2 + x2 y1) (1.1.8b)

In fact, we note that from Eq. (1.1.5a)

zz = zz = (x + iy)(x iy) = x2 + y2 = |z|2 (1.1.8c)

This fact is useful for division of complex numbers,

z1z2= x1 + iy1

x2 + iy2 =(x1 + iy1)(x2 iy2)(x2 + iy2)(x2 iy2)

= (x1x2 + y1 y2)+ i(x2 y1 x1 y2)x22 + y22

= x1x2 + y1 y2x22 + y22

+ i(x2 y1 x1 y2)x22 + y22

(1.1.8d)

It is easily shown that the commutative, associative, and distributive laws ofaddition and multiplication hold.

Geometrically speaking, addition of two complex numbers is equivalent tothat of the parallelogram law of vectors (see Figure 1.1.2).

The useful analytical statement

||z1| |z2|| |z1 + z2| |z1| + |z2| (1.1.9)

has the geometrical meaning that no side of a triangle is greater in length thanthe sum of the other two sides hence the term for inequality Eq. (1.1.9) is thetriangle inequality.

Equation (1.1.9) can be proven as follows.

|z1 + z2|2 = (z1 + z2)(z1 + z2) = z1z1 + z2z2 + z1z2 + z1z2= |z1|2 + |z2|2 + 2 Re(z1z2)

iy

x

Fig. 1.1.2. Addition of vectors

1.1 Complex Numbers and Their Properties 7

Hence

|z1 + z2|2 (|z1| + |z2|)2 = 2(Re(z1z2) |z1||z2|) 0 (1.1.10)where the inequality follows from the fact that

x = Re z |z| =

x2 + y2

and |z1z2| = |z1||z2|.Equation (1.1.10) implies the right-hand inequality of Eq. (1.1.9) after taking

a square root. The left-hand inequality follows by redening terms. Let

W1 = z1 + z2 W2 = z2Then the right-hand side of Eq. (1.1.9) (just proven) implies that

|W1| |W1 + W2| + | W2|or |W1| |W2| |W1 + W2|

which then proves the left-hand side of Eq. (1.1.9) if we assume that|W1| |W2|; otherwise, we can interchange W1 and W2 in the above discussionand obtain

||W1| |W2|| = (|W1| |W2|) |W1 + W2|Similarly, note the immediate generalization of Eq. (1.1.9)

nj=1

z j

n

j=1|z j |

Problems for Section 1.1

1. Express each of the following complex numbers in polar exponential form:

(a) 1 (b) i (c) 1+ i

(d) 12+

32

i (e) 12

32

i

2. Express each of the following in the form a + bi , where a and b are real:

(a) e2+i/2 (b) 11+ i (c) (1+ i)

3 (d) |3+ 4i |

(e) Dene cos(z) = (eiz + ei z)/(2), and ez = ex eiy .Evaluate cos(i/4+ c), where c is real


3. Solve for the roots of the following equations:

(a) z3 = 4 (b) z4 = 1(c) (az + b)3 = c,where a, b, c > 0 (d) z4 + 2z2 + 2 = 0

4. Estabilish the following results:

(a) z + w = z + w (b) |z w| |z| + |w| (c) z z = 2iIm z(d) Rez |z| (e) |wz + wz| 2|wz| (f) |z1z2| = |z1||z2|

5. There is a partial correspondence between complex numbers and vec-tors in the plane. Denote a complex number z = a+ bi and a vectorv = ae1+ be2, where e1 and e2 are unit vectors in the horizontal and ver-tical directions. Show that the laws of addition z1 z2 and v1 v2 yieldequivalent results as do the magnitudes |z|2, |v|2 = v v. (Here v v is theusual vector dot product.) Explain why there is no general correspondencefor laws of multiplication or division.

1.2 Elementary Functions and Stereographic Projections

1.2.1 Elementary FunctionsAs a prelude to the notion of a function we present some standard denitionsand concepts. A circle with center z0 and radius r is denoted by |z z0| = r .A neighborhood of a point z0 is the set of points z for which

|z z0| < (1.2.1)

where is some (small) positive number. Hence a neighborhood of the pointz0 is all the points inside the circle of radius , not including its boundary.An annulus r1 < |z z0| < r2 has center z0, with inner radius r1 and outerradius r2. A point z0 of a set of points S is called an interior point of S ifthere is a neighborhood of z0 entirely contained within S. The set S is said tobe an open set if all the points of S are interior points. A point z0 is said to bea boundary point of S if every neighborhood of z = z0 contains at least onepoint in S and at least one point not in S.

A set consisting of all points of an open set and none, some or all of itsboundary points is referred to as a region. An open region is said to be boundedif there is a constant M > 0 such that all points z of the region satisfy |z| M ,that is, they lie within this circle. A region is said to be closed if it containsall of its boundary points. A region that is both closed and bounded is called

1.2 Elementary Functions, Stereographic Projections 9

iy

x

Fig. 1.2.1. Half plane

compact. Thus the region |z| 1 is compact because it is both closed andbounded. The region |z| < 1 is open and bounded. The half plane Re z > 0(see Figure 1.2.1) is open and unbounded.

Let z1, z2, . . . , zn be points in the plane. The n 1 line segments z1z2, z2z3,. . . , zn1zn taken in sequence form a broken line. An open region is said to beconnected if any two of its points can be joined by a broken line that is containedin the region. (There are more detailed denitions of connectedness, but thissimple one will sufce for our purposes.) For an example of a connected regionsee Figure 1.2.2.)

A disconnected region is exemplied by all the points interior to |z| = 1 andexterior to |z| = 2: S = {z : |z| < 1, |z| > 2}.

A connected open region is called a domain. For example the set (seeFigure 1.2.3)

S = {z = rei : 0 < arg z < 0 + }

is a domain that is unbounded.

z

z

z

zz

z2

1

3

5

6

4

Fig. 1.2.2. Connected region


0

0+

Fig. 1.2.3. Domain a sector

Because a domain is an open set, we note that no boundary point of thedomain can lie in the domain. Notationally, we shall refer to a region asR; theclosed region containingR and all of its boundary points is sometimes referredto as R. If R is closed, then R = R. The notation z R means z is a pointcontained in R. Usually we denote a domain by D.

If for each z R there is a unique complex number w(z) then we say w(z)is a function of the complex variable z, frequently written as

w = f (z) (1.2.2)

in order to denote the function f . Often we simply write w = w(z), or just w.The totality of values f (z) corresponding to z R constitutes the range off (z). In this context the setR is often referred to as the domain of denitionof the function f . While the domain of denition of a function is frequently adomain, as dened earlier for a set of points, it does not need to be so.

By the above denition of a function we disallow multivaluedness; no morethan one value of f (z) may correspond to any point z R. In Sections 2.2and 2.3 we will deal explicitly with the notion of multivaluedness and its ram-ications.

The simplest function is the power function:

f (z) = zn, n = 0, 1, 2, . . . (1.2.3)

Each successive power is obtained by multiplication zm+1 = zm z, m =0, 1, 2, . . . A polynomial is dened as a linear combination of powers

Pn(z) =n

j=0a j z j = a0 + a1z + a2z2 + + anzn (1.2.4)


where the a j are complex numbers (i.e.,1 a j C). Note that the domain ofdenition of Pn(z) is the entire z plane simply written as z C. A rationalfunction is a ratio of two polynomials Pn(z) and Qm(z), where Qm(z) =m

j=0 b j z j

R(z) = Pn(z)Qm(z) (1.2.5)

and the domain of denition of R(z) is the z plane, excluding the points whereQm(z) = 0. For example, the function w = 1/(1+ z2) is dened in the z planeexcluding z = i . This is written as z C \ {i,i}.

In general, the function f (z) is complex and when z = x + iy, f (z) can bewritten in the complex form:

w = f (z) = u(x, y)+ i v(x, y) (1.2.6)

The function f (z) is said to have the real part u, u = Re f , and the imaginarypart v, v = Im f . For example,

w = z2 = (x + iy)2 = x2 y2 + 2i xy

which implies

u(x, y) = x2 y2 and v = 2xy.

As is the case with real variables we have the standard operations on func-tions. Given two functions f (z) and g(z), we dene addition, f (z) + g(z),multiplication f (z)g(z), and composition f [g(z)] of complex functions.

It is convenient to dene some of the more common functions of a complexvariable which, as with polynomials and rational functions, will be familiarto the reader.

Motivated by real variables, ea+b = eaeb, we dene the exponential function

ez = ex+iy = ex eiy

Noting the polar exponential denition (used already in section 1.1, Eq. (1.1.6))

eiy = cos y + i sin y

we see that

ez = ex (cos y + i sin y) (1.2.7)

1 Hereafter these abbreviations will frequently be used: i.e. = that is; e.g. = for example.


Equation (1.2.7) and standard trigonometric identities yield the properties

ez1+z2 = ez1 ez2 and (ez)n = enz, n = 1, 2 . . . (1.2.8)

We also note

|ez| = |ex || cos y + i sin y| = ex

cos2 y + sin2 y = ex

and

(ez) = ez = exiy = ex (cos y i sin y)

The trigonometric functions sin z and cos z are dened as

sin z = eiz ei z

2i(1.2.9)

cos z = eiz + ei z

2(1.2.10)

and the usual denitions of the other trigonometric functions are taken:

tan z = sin zcos z

, cot z = cos zsin z

, sec z = 1cos z

, csc z = 1sin z

(1.2.11)

All of the usual trigonometric properties such as

sin(z1 + z2) = sin z1 cos z2 + cos z1 sin z2,sin2 z + cos2 z = 1, . . . (1.2.12)

follow from the above denitions.The hyperbolic functions are dened analogously

sinh z = ez ez

2(1.2.13)

cosh z = ez + ez

2(1.2.14)

tanh z = sinh zcosh z

, coth z = cosh zsinh z

, sechz = 1cosh z

, cschz = 1sinh z

Similarly, the usual identities follow, such as

cosh2 z sinh2 z = 1 (1.2.15)


From these denitions we see that as functions of a complex variable, sinh zand sin z (cosh z and cos z) are simply related

sinh i z = i sin z, sin i z = i sinh zcosh i z = cos z, cos i z = cosh z (1.2.16)

By now it is abundantly clear that the elementary functions dened in thissection are natural generalizations of the conventional ones we are familiar within real variables. Indeed, the analogy is so close that it provides an alternativeand systematic way of dening functions, which is entirely consistent withthe above and allows the denition of a much wider class of functions. Thisinvolves introducing the concept of power series. In Chapter 3 we shall lookmore carefully at series and sequences. However, because power series ofreal variables are already familiar to the reader, it is useful to introduce thenotion here.

A power series of f (z) about the point z = z0 is dened as

f (z) = limn

nj=0

a j (z z0) j =j=0

a j (z z0) j (1.2.17)

where a j , z0 are constants.Convergence is of course crucial. For simplicity we shall state (motivated

by real variables but without proof at this juncture) that Eq. (1.2.17) converges,via the ratio test, whenever

limn

an+1an|z z0| < 1 (1.2.18)

That is, it converges inside the circle |z z0| = R, where

R = limn

anan+1

when this limit exists (see also Section 3.2). If R = , we say the series con-verges for all nite z; if R = 0, we say the series converges only for z = z0.R is referred to as the radius of convergence.

The elementary functions discussed above have the following power seriesrepresentations:

ez =j=0

z j

j! , sin z =j=0

(1) j z2 j+1(2 j + 1)! , cos z =

j=0

(1) j z2 j(2 j)!

sinh z =j=0

z2 j+1

(2 j + 1)! , cosh z =j=0

z2 j

(2 j)!

(1.2.19)


where j! = j ( j 1)( j 2) 3 2 1 for j 1, and 0! 1. The ratio testshows that these series converge for all nite z.

Complex functions arise frequently in applications. For example, in theinvestigation of stability of physical systems we derive equations for small de-viations from rest or equilibrium states. The solutions of the perturbed equationoften have the form ezt , where t is real (e.g. time) and z is a complex numbersatisfying an algebraic equation (or a more complicated transcendental sys-tem). We say that the system is unstable if there are any solutions with Re z > 0because |ezt | as t . We say the system ismarginally stable if thereare no values of z with Re z > 0, but some with Re z = 0. (The correspondingexponential solution is bounded for all t .) The system is said to be stable anddamped if all values of z satisfy Re z < 0 because |ezt | 0 as t .

A function w = f (z) can be regarded as a mapping or transformation of thepoints in the z plane (z = x + iy) to the points of the w plane (w = u + iv).In real variables in one dimension, this notion amounts to understanding thegraph y = f (x), that is, the mapping of the points x to y = f (x). In complexvariables the situation is more difcult owing to the fact that we really have fourdimensions hence a graphical depiction such as in the real one-dimensionalcase is not feasible. Rather, one considers the two complex planes, z and w,separately and asks how the region in the z plane transforms or maps to acorresponding region or image in the w plane. Some examples follow.

Example 1.2.1 The function w = z2 maps the upper half z-plane including thereal axis, Im z 0, to the entire w-plane (see Figure 1.2.4). This is particularlyclear when we use the polar representation z = rei . In the z-plane, liesinside 0 < , whereas in the w-plane, w = r2e2i = Rei , R = r2, = 2 and lies in 0 < 2 .

z

iy

r

x w

iv R = r

u

w = z

-plane

-plane

2

2

= 2

Fig. 1.2.4. Map of z w = z2


z

iy

x- iy

w

x

w = z

-plane-plane

Fig. 1.2.5. Conjugate mapping

Example 1.2.2 The function w = z maps the upper half z-plane Im z > 0 intothe lower halfw-plane (see Figure 1.2.5). Namely, z = x+ iy and y > 0 implythat w = z = x iy. Thus w = u + iv u = x , v = y.

The study and understanding of complex mappings is very important, and wewill see that there are many applications. In subsequent sections and chapterswe shall more carefully investigate the concept of mappings; we shall not gointo any more detail or complication at this juncture.

It is often useful to add the point at innity (usually denoted by or z)to our, so far open, complex plane. As opposed to a nite point where theneighborhood of z0, say, is dened by Eq. (1.2.1), here the neighborhood of zis dened by those points satisfying |z| > 1/ for all (sufciently small) > 0.One convenient way to dene the point at innity is to let z = 1/t and then tosay that t = 0 corresponds to the point z. An unbounded region R containsthe point z. Similarly, we say a function has values at innity if it is denedin a neighborhood of z. The complex plane with the point z included isreferred to as the extended complex plane.

1.2.2 Stereographic ProjectionConsider a unit sphere sitting on top of the complex plane with the south poleof the sphere located at the origin of the z plane (see Figure 1.2.6). In thissubsection we show how the extended complex plane can be mapped onto thesurface of a sphere whose south pole corresponds to the origin and whose northpole to the point z. All other points of the complex plane can be mapped ina one-to-one fashion to points on the surface of the sphere by using the follow-ing construction. Connect the point z in the plane with the north pole using astraight line. This line intersects the sphere at the point P . In this way eachpoint z(= x+iy) on the complex plane corresponds uniquely to a point P on thesurface of the sphere. This construction is called the stereographic projectionand is diagrammatically illustrated in Figure 1.2.6. The extended complex plane


P

N

S

C(X,Y,Z)

(0,0,0)

(0,0,2) iy

z=x+iy

z

x

-plane

Fig. 1.2.6. Stereographic projection

is sometimes referred to as the compactied (closed) complex plane. It is oftenuseful to view the complex plane in this way, and knowledge of the constructionof the stereographic projection is valuable in certain advanced treatments.

So, more concretely, the point P : (X, Y, Z) on the sphere is put into corre-spondence with the point z = x + iy in the complex plane by nding on the sur-face of the sphere, (X, Y, Z), the point of intersection of the line from the northpole of the sphere, N : (0, 0, 2), to the point z = x + iy on the plane. The con-struction is as follows. We consider three points in the three-dimensional setup:

N = (0, 0, 2): north poleP = (X, Y, Z): point on the sphereC = (x, y, 0): point in the complex plane

These points must lie on a straight line, hence the difference of the points PNmust be a real scalar multiple of the difference C N , namely

(X, Y, Z 2) = s(x, y,2) (1.2.20)

where s is a real number (s = 0). The equation of the sphere is given by

X2 + Y 2 + (Z 1)2 = 1 (1.2.21)

Equation (1.2.20) implies

X = sx, Y = sy, Z = 2 2s (1.2.22)

Inserting Eq. (1.2.22) into Eq. (1.2.21) yields, after a bit of manipulation

s2(x2 + y2 + 4) 4s = 0 (1.2.23)


This equation has as its only nonvanishing solution

s = 4|z|2 + 4 (1.2.24)

where |z|2 = x2 + y2. Thus given a point z = x + iy in the plane, we have onthe sphere the unique correspondence:

X = 4x|z|2 + 4 , Y =4y

|z|2 + 4 , Z =2|z|2|z|2 + 4 (1.2.25)

We see that under this mapping, the origin in the complex plane z = 0 yieldsthe south pole of the sphere (0, 0, 0), and all points at |z| = yield the northpole (0, 0, 2). (The latter fact is seen via the limit |z| with x = |z| cos ,y = |z| sin .) On the other hand, given a point P = (X, Y, Z) we can nd itsunique image in the complex plane. Namely, from Eq. (1.2.22)

s = 2 Z2

(1.2.26)

and

x = 2X2 Z , y =

2Y2 Z (1.2.27)

The stereographic projection maps any locus of points in the complex planeonto a corresponding locus of points on the sphere and vice versa. For example,the image of an arbitrary circle in the plane, is a circle on the sphere that doesnot pass through the north pole. Similarly, a straight line corresponds to a circlepassing through the north pole (see Figure 1.2.7). Here a circle on the spherecorresponds to the locus of points denoting the intersection of the sphere withsome plane: AX + BY +C Z = D, A, B,C, D constant. Hence on the spherethe images of straight lines and of circles are not really geometrically different

N

S

C

image:straight

line

circle

image:circle

(0,0,2)

(0,0,0)

-plane

i y

z

Fig. 1.2.7. Circles and lines in stereographic projection


from one another. Moreover, the images on the sphere of two nonparallelstraight lines in the plane intersect at two points on the sphere one of which isthe point at innity. In this framework, parallel lines are circles that touch oneanother at the point at innity (north pole). We lose Euclidean geometry on asphere.


1. Sketch the regions associated with the following inequalities. Determineif the region is open, closed, bounded, or compact.

(a) |z| 1 (b) |2z + 1+ i | < 4 (c) Re z 4(d) |z| |z + 1| (e) 0 < |2z 1| 2

2. Sketch the following regions. Determine if they are connected, and whatthe closure of the region is if they are not closed.

(a) 0 < arg z (b) 0 arg z < 2(c) Re z > 0 and Im z > 0

(d) Re (z z0)> 0 and Re (z z1)< 0 for two complex numbers z0, z1(e) |z| < 12 and |2z 4| 2

3. Use Eulers formula for the exponential and the well-known series expan-sions of the real functions ex , sin y, and cos y to show that

ez =j=0

z j

j!

Hint: Use

(x + iy)k =k

j=0

k!j!(k j)! x

j (iy)k j

4. Use the series representation

ez =j=0

z j

j! , |z|


Use these results to deduce where the power series for sin2 z and sech zwould converge. What can be said about tan z?

5. Use any method to determine series expansions for the following func-tions:

(a) sin zz

(b) cosh z 1z2

(c) ez 1 z

z

6. Let z1 = x1 and z2 = x2, with x1, x2 real, and the relationship

ei(x1+x2) = eix1 eix2

to deduce the known trigonometric formulae

sin(x1 + x2) = sin x1 cos x2 + cos x1 sin x2cos(x1 + x2) = cos x1 cos x2 sin x1 sin x2

and therefore show

sin 2x = 2 sin x cos xcos 2x = cos2 x sin2 x

7. Discuss the following transformations (mappings) from the z plane to thew plane; here z is the entire nite complex plane.

(a) w = z3 (b) w = 1/z

8. Consider the transformation

w = z + 1/z z = x + iy w = u + iv

Show that the image of the points in the upper half z plane (y > 0) thatare exterior to the circle |z| = 1 corresponds to the entire upper half planev > 0.

9. Consider the following transformation

w = az + bcz + d , = ad bc = 0

(a) Show that the map can be inverted to nd a unique (single-valued) zas a function of w everywhere.


(b) Verify that the mapping can be considered as the result of three suc-cessive maps:

z = cz + d, z = 1/z, w = c

z + ac

where c = 0 and is of the form

w = ad

z + bd

when c = 0.The following problems relate to the subsection on stereographic projec-tion.

10. To what curves on the sphere do the lines Re z = x = 0 and Im z = y = 0correspond?

11. Describe the curves on the sphere to which any straight lines on the zplane correspond.

12. Show that a circle in the z plane corresponds to a circle on the sphere.(Note the remark following the reference to Figure 1.2.7 in Section 1.2.2)

1.3 Limits, Continuity, and Complex DifferentiationThe concepts of limits and continuity are similar to that of real variables. In thissense our discussion can serve as a brief review of many previously understoodnotions. Consider a function w = f (z) dened at all points in some neighbor-hood of z = z0, except possibly for z0 itself. We say f (z) has the limit w0 if asz approaches z0, f (z), approaches w0 (z0, w0 nite). Mathematically, we say

limzz0

f (z) = w0 (1.3.1)

if for every (sufciently small) > 0 there is a > 0 such that

| f (z) w0| < whenever 0 < |z z0| < (1.3.2)where the absolute value is dened in section 1.1 (see, e.g. Eqs. 1.1.4 and1.1.5a).

This denition is clear when z0 is an interior point of a region R in whichf (z) is dened. If z0 is a boundary point of R, then we require Eq. (1.3.2) tohold only for those z R.

Figure 1.3.1 illustrates these ideas. Under the mapping w = f (z), all pointsinterior to the circle |z z0| = with z0 deleted are mapped to points interior tothe circle |w w0| = . The limit will exist only in the case when z approachesz0 (that is, z z0) in an arbitrary direction; then this implies that w w0.

1.3 Limits, Continuity, and Complex Differentiation 21

z

z

w = f(z)

ww

z w-plane-plane

o o

Fig. 1.3.1. Mapping of a neighborhood

This limit denition is standard. Let us consider the following examples.

Example 1.3.1 Show that

limzi

2(

z2 + i z + 2z i

)= 6i. (1.3.3)

We must show that given > 0, there is a > 0 such that2( z2 + i z + 2z i) 6i

= 2( (z i)(z + 2i)(z i)) 6i

< (1.3.4)whenever

0 < |z i | < (1.3.5)Since z = i , inequality (1.3.4) implies that 2|z i | < . Thus if = /2,

Eq. (1.3.5) ensures that Eq. (1.3.4) is satised. Therefore Eq. (1.3.3) is demon-strated.

This limit denition can also be applied to the point z = . We say thatlim

z f (z) = w0 (1.3.6)

(w0 nite) if for every (sufciently small) > 0 there is a > 0 such that

| f (z) w0| < whenever |z| > 1

(1.3.7)

We assert that the following properties are true. (The proof is an exercise ofthe limit denition and follows that of real variables.) If for z R we havetwo functions w = f (z) and s = g(z) such that

limzz0

f (z) = w0, limzz0

g(z) = s0


then

limzz0

( f (z)+ g(z)) = w0 + s0limzz0

( f (z)g(z)) = w0s0and

limzz0

f (z)g(z)

= w0s0

(s0 = 0)

Similar conclusions hold for sums and products of a nite number of functions.As mentioned in Section 1.2, the point z = z = is often dealt with via thetransformation

t = 1z

The neighborhood of z = z corresponds to the neighborhood of t = 0. Sothe function f (z) = 1/z2 near z = z behaves like f (1/t) = t2 near zero;that is, t2 0 as t 0, or 1/z2 0 as z .

In analogy to real analysis, a function f (z) is said to be continuous atz = z0 if

limzz0

f (z) = f (z0) (1.3.8)

(z0, f (z0) nite). Equation (1.3.8) implies that f (z) exists in a neighborhoodof z = z0 and that the limit, as z approaches z0, of f (z) is f (z0) itself. In termsof , notation, given > 0, there is a > 0 such that | f (z) f (z0)| < whenever |z z0| < . The notion of continuity at innity can be ascertainedin a similar fashion. Namely, if limz f (z) = w, and f () = w, thenthe denition for continuity at innity, limz f (z) = f (), is the following:Given > 0 there is a > 0 such that | f (z) w| < whenever |z| > 1/.

The theorems on limits of sums and products of functions can be used toestablish that sums and products of continuous functions are continuous. Itshould also be pointed out that since | f (z) f (z0)| = | f (z) f (z0)|, thecontinuity of f (z) at z0 implies the continuity of the complex conjugate f (z)at z = z0. (Recall the denition of the complex conjugate, Eq. (1.1.7)). Thusif f (z) is continuous at z = z0, then

Re f (z) = ( f (z)+ f (z))/2Im f (z) = ( f (z) f (z))/2i

and | f (z)|2 = ( f (z) f (z))are all continuous at z = z0.


We shall say a function f (z) is continuous in a region if it is continuousat every point of the region. Usually, we simply say that f (z) is continuouswhen the associated region is understood. Considering continuity in a regionR generally requires that = (, z0); that is, depends on both and thepoint z0 R. Function f (z) is said to be uniformly continuous in a regionR if = (); that is, is independent of the point z = z0.

As in real analysis, a function that is continuous in a compact (closed andbounded) region R is uniformly continuous and bounded; that is, there is aC > 0 such that | f (z)| < C . (The proofs of these statements follow from theanalogous statements of real analysis.) Moreover, in a compact region, themodulus | f (z)| actually attains both its maximum and minimum values on R;this follows from the continuity of the real function | f (z)|.Example 1.3.2 Show that the continuity of the real and imaginary parts of acomplex function f (z) implies that f (z) is continuous.

f (z) = u(x, y)+ iv(x, y)We know that

limzz0

f (z) = limxx0yy0

(u(x, y)+ iv(x, y))

= u(x0, y0)+ iv(x0, y0) = f (z0)which completes the proof. It also illustrates that we can appeal to real analysisfor many of the results in this section.

Conversely, we have

|u(x, y) u(x0, y0)| | f (z) f (z0)||v(x, y) v(x0, y0)| | f (z) f (z0)|

(because | f |2 = |u|2+|v|2) in which case continuity of f (z) implies continuityof the real and imaginary parts of f (z). Namely, this follows from the fact thatgiven > 0, there is a > 0 such that | f (z) f (z0)| < whenever |zz0| < (and note that |x x0| < |z z0| < , |y y0| < |z z0| < ).

Let f (z) be dened in some region R containing the neighborhood of apoint z0. The derivative of f (z) at z = z0, denoted by f (z0) or d fdz (z0), isdened by

f (z0) = limz0

( f (z0 +z) f (z0)z

)(1.3.9)

provided this limit exists. We sometimes say that f is differentiable at z0.


Alternatively, letting z = z z0, Eq. (1.3.9) has another standard form

f (z0) = limzz0

( f (z) f (z0)z z0

)(1.3.10)

If f (z0) exists for all points z0 R, then we say f (z) is differentiable inR or just differentiable, if R is understood. If f (z0) exists, then f (z) iscontinuous at z = z0. This follows from

limzz0

( f (z) f (z0)) = limzz0

( f (z) f (z0)z z0

)limzz0

(z z0)

= f (z0) limzz0

(z z0) = 0

A continuous function is not necessarily differentiable. Indeed it turns outthat differentiable functions possess many special properties.

On the other hand, because we are now dealing with complex functions thathave a two-dimensional character, there can be new kinds of complications notfound in functions of one real variable. A prototypical example follows.

Consider the function

f (z) = z (1.3.11)Even though this function is continuous, as discussed earlier, we now show thatit does not possess a derivative. Consider the difference quotient:

limz0

(z0 +z) z0z

= limz0

z

z q0 (1.3.12)

This limit does not exist because a unique value of q0 cannot be found; indeed itdepends on how z approaches zero. Writing z = rei , q0 = limz0 e2i .So ifz 0 along the positive real axis ( = 0), then q0 = 1. Ifz 0 alongthe positive imaginary axis, then q0 = 1 (because = /2, e2i = 1), etc.Thus we nd the surprising result that the function f (z) = z is not differentiableanywhere (i.e., for any z = z0) even though it is continuous everywhere! Infact, this situation will be seen to be the case for general complex functionsunless the real and imaginary parts of our complex function satisfy certaincompatibility conditions (see Section 2.1). Differentiable complex functions,often called analytic functions, are special and important.

Despite the fact that the formula for a derivative is identical in form to that ofthe derivative of a real-valued function, f (z), a signicant point to note is thatf (z) follows from a two-dimensional limit (z = x + iy or z = rei ). Thus forf (z) to exist, the relevant limit must exist independent of the direction from


which z approaches the limit point z0. For a function of one real variable weonly have two directions: x < x0 and x > x0.

If f and g have derivatives, then it follows by similar proofs to those of realvariables that

( f + g) = f + g

( f g) = f g + f g( fg

)= ( f g f g)/g2 (g = 0)

and if f (g(z)) and g(z) exist, then

[ f (g(z))] = f (g(z))g(z)

In order to differentiate polynomials, we need the derivative of the elementaryfunction f (z) = zn , n is a positive integer

ddz

(zn) = nzn1 (1.3.13)

This follows from

(z +z)n znz

= nzn1 + a1zn2z + a2zn3z2 + . . .+zn nzn1

as z 0, where a1, a2, . . ., are the appropriate binomial coefcients of(a + b)n .

Thus we have as corollaries to this result

ddz

(c) = 0, c = constant (1.3.15a)

ddz

(a0 + a1z + a2z2 + + am zm) = a1 + 2a2z + 3a3z2 + + mam zm1

(1.3.15b)

Moreover, with regard to the (purely formal at this point) powerseries expan-sions discussed earlier, we will nd that

ddz

( n=0

anzn

)=

n=0

nanzn1 (1.3.15)

inside the radius of convergence of the series.


We also note that the derivatives of the usual elementary functions behave inthe same way as in real variables. Namely

ddz

ez = ez, ddz

sin z = cos z, ddz

cos z = sin zddz

sinh z = cosh z, ddz

cosh z = sinh z(1.3.16)

etc. The proofs can be obtained from the fundamental denitions. For example,

ddz

ez = limz0

ez+z ezz

= ez limz0

(ez 1z

)= ez (1.3.17)

where we note that

limz0

ez 1z

= limx0y0

((ex cosy 1)+ iex siny

(x + iy))= 1 (1.3.18)

One can put Eq. (1.3.18) in real/imaginary form and use polar coordinatesfor x , y. This calculation is also discussed in the problems given for thissection. Later we shall establish the validity of the power series formulaefor ez (see Eq. (1.2.19)), from which Eq. (1.3.18) follows immediately (sinceez = 1+ z + z2/2+ ) without need for the double limit. The other formulaein Eq. (1.3.16) can also be deduced using the relationships (1.2.9), (1.2.10),(1.2.13), (1.2.14).

1.3.1 Elementary Applications to Ordinary Differential EquationsAn important topic in the application of complex variables is the study ofdifferential equations. Later in this text we discuss differential equations in thecomplex plane in some detail, but in fact we are already in a position to seewhy the ideas already presented can be useful. Many readers will have had acourse in differential equations, but it is not really necessary for what we shalldiscuss. Linear homogeneous differential equations with constant coefcientstake the following form:

Lnw = dnw

dtn+ an1 d

n1wdtn1

+ a1 dwdt + a0w = 0 (1.3.19)

where {a j }n1j=0 are all constant, n is called the order of the equation, and (forour present purposes) t is real. We could (and do, later in section 3.7) allow t


to be complex, in which case the study of such differential equations becomesintimately connected with many of the topics studied later in this text, but fornow we keep t real. Solutions to Eq. (1.3.19) can be sought in the form

w(t) = cezt (1.3.20)

where c is a nonzero constant. Substitution of Eq. (1.3.20) into Eq. (1.3.19), andfactoring cezt from each term (note ezt does not vanish), yields the followingalgebraic equation:

zn + an1zn1 + + a1z + a0 = 0 (1.3.21)

There are various subcases to consider, but we shall only discuss the proto-typical one where there are n distinct solutions of Eq. (1.3.22), which we call{z1, z2, . . . , zn}. Each of these values, say z j , yields a solution to Eq. (1.3.19)w j = c j ez j t , where c j is an arbitrary constant. Because Eq. (1.3.19) is a linearequation, we have the more general solution

w(t) =n

j=1w j =

nj=1

c j ez j t (1.3.22)

In differential equation texts it is proven that Eq. (1.3.22) is, in fact, the mostgeneral solution. In applications, the differential equations (Eq. (1.3.19)) fre-quently have real coefcients {a j }n1j=0. The study of algebraic equations of theform (Eq. (1.3.21)), discussed later in this text, shows that there are at most nsolutions precisely n solutions if we count multiplicity of solutions. In fact,when the coefcients are real, then the solutions are either real or come in com-plex conjugate pairs. Corresponding to complex conjugate pairs, a real solutionw(t) is found by taking complex conjugate constants c j and c j correspondingto each pair of complex conjugate roots z j and z j . For example, consider onesuch real solution, call it wp, corresponding to the pair z, z:

wp(t) = cezt + cezt (1.3.23)

We can rewrite this in terms of trigonometric functions and real exponentials.Let z = x + iy:

wp(t) = ce(x+iy)t + ce(xiy)t

= ext [c(cos yt + i sin yt)+ c(cos yt i sin yt)]= (c + c)ext cos yt + i(c c)ext sin yt (1.3.24)


Because c + c = A, i(c c) = B are real, we nd that this pair of solutionsmay be put in the real form

wc(t) = Aext cos yt + Bext sin yt (1.3.25)Two examples of these ideas are simple harmonic motion (SHM) and vibrationsof beams:

d2wdt2

+ 02w = 0 (SHM) (1.3.26a)d4wdt4

+ k4w = 0 (1.3.26b)

where 02 and k4 are real nonzero constants, depending on the parameters inthe physical model. Looking for solutions of the form of Eq. (1.3.20) leads tothe equations

z2 + 02 = 0 (1.3.27a)z4 + k4 = 0 (1.3.27b)

which have solutions (see also Section 1.1)z1 = i0, z2 = i0 (1.3.28a)

z1 = kei/4 = k2 (1+ i)

z2 = ke3i/4 = k2 (1+ i)

z3 = ke5i/4 = k2 (1 i)

z4 = ke7i/4 = k2 (1 i)

(1.3.28b)

It follows from the above discussion that the corresponding real solutionsw(t) have the form

w = A cos0t + B sin0t (1.3.29a)

w = e kt2[

A1 coskt

2+ B1 sin kt2

]+ e kt2

[A2 cos

kt2+ B2 sin kt2

](1.3.29b)

where A, B, A1, A2, B1, and B2 are arbitrary constants.In this chapter we have introduced and summarized the basic properties of

complex numbers and elementary functions. We have seen that the theory offunctions of a single real variable have so far motivated many of the notions of


complex variables; though the two-dimensional character of complex numbershas already led to some signicant differences. In subsequent chapters a numberof entirely new and surprising results will be obtained, and the departure fromreal variables will become more apparent.

Problems for Section 1.31. Evaluate the following limits:

(a) limzi (z + 1/z) (b) limzz0 1/zm, m integer

(c) limzi sinh z (d) limz0 sin zz

(e) limz sin zz

(f) limz z2

(3z + 1)2 (g) limzz

z2 + 1

2. Establish a special case of lHopitals rule. Suppose that f (z) and g(z)have formal power series about z = a, and

f (a) = f (a) = f (a) = = f (k)(a) = 0g(a) = g(a) = g(a) = = g(k)(a) = 0

If f (k+1)(a) and g(k+1)(a) are not simultaneously zero, show that

limza

f (z)g(z)

= f(k+1)(a)

g(k+1)(a)

What happens if g(k+1)(a) = 0?

3. If |g(z)| M , M > 0 for all z in a neighborhood of z = z0, show that iflimzz0 f (z) = 0, then

limzz0

f (z)g(z) = 0

4. Where are the following functions differentiable?

(a) sin z (b) tan z (c) z 1z2 + 1 (d) e

1/z (e) 2z

5. Show that the functions Re z and Imz are nowhere differentiable.

6. Let f (z) be a continuous function for all z. Show that if f (z0) = 0, thenthere must be a neighborhood of z0 in which f (z) = 0.


7. Let f (z) be a continuous function where limz0 f (z) = 0. Show thatlimz0(e f (z) 1) = 0. What can be said about limz0((e f (z) 1)/z)?

8. Let two polynomials f (z) = a0 + a1z + + anzn and g(z) = b0 + b1z+ + bm zm be equal at all points z in a region R. Use the concept of alimit to show that m = n and that all the coefcients {a j }nj=0 and {b j }nj=0must be equal. Hint: Consider limz0( f (z) g(z)), limz0( f (z) g(z))/(z), etc.

9. (a) Use the real Taylor series formulae

ex = 1+ x + O(x2), cos x = 1+ O(x2),sin x = x(1+ O(x2))

where O(x2) means we are omitting terms proportional to power x2(i.e, lim

x0(O(x2))/(x2) = C , where C is a constant), to establish the

following:

limz0

(ez (1+ z)) = limr0

(er cos eir sin (1+ r(cos + i sin ))) = 0

(b) Use the above Taylor expansions to show that (c.f. Eq. (1.3.18))

limz0

(ez 1z

)= lim

r0

{(er cos cos(r sin ) 1)+ ier cos sin(r sin )

r(cos + i sin )}

= 1

10. Let z = x be real. Use the relationship (d/dx)eix = ieix to nd thestandard derivative formulae for trigonometric functions:

ddx

sin x = cos x

ddx

cos x = sin x

11. Suppose we are given the following differentialequations:

(a) d3w

dt3 k3w = 0

(b) d6w

dt6 k6w = 0


where t is real and k is a real constant. Find the general real solution ofthe above equations. Write the solution in terms of real functions.

12. Consider the following differential equation:

x2d2wdx2

+ x dwdx

+ w = 0

where x is real.

(a) Show that the transformation x = et implies that

xd

dx= d

dt,

x2d2

dx2= d

2

dt2 d

dt

(b) Use these results to nd that w also satises the differential equationd2wdt2

+ w = 0

(c) Use these results to establish that w has the real solutionw = Cei(log x) + Cei(log x)

or

w = A cos(log x)+ B sin(log x)

13. Use the ideas of Problem 12 to nd the real solution of the followingequations (x is real and k is a real constant):

(a) x2 d2w

dx2+ k2w = 0, 4k2 > 1

(b) x3 d3w

dx3+ 3x2 d

2w

dx2+ x dw

dx+ k3w = 0

2Analytic Functions and Integration

In this chapter we study the notion of analytic functions and their properties. Itwill be shown that a complex function is differentiable if and only if there is animportant compatibility relationship between its real and imaginary parts. Theconcepts of multivalued functions and complex integration are considered insome detail. The technique of integration in the complex plane is discussed andtwo very important results of complex analysis are derived: Cauchys theoremand a corollary Cauchys integral formula.

2.1 Analytic Functions

2.1.1 The CauchyRiemann EquationsIn Section 1.3 we dened the notion of complex differentiation. For conve-nience, we remind the reader of this denition here. The derivative of f (z),denoted by f (z), is dened by the following limit:

f (z) = limz0

f (z +z) f (z)z

(2.1.1)

We write the real and imaginary parts of f (z), f (z) = u(x, y) + iv(x, y),and compute Eq. (2.1.1) for (a)z=x real and (b)z= iy pure imaginary(i.e., we take the limit along the real and then along the imaginary axis). Then,for case (a)

f (z) = limx0

(u(x +x, y) u(x, y)

x+ i v(x +x, y) v(x, y)

x

)= ux (x, y)+ ivx (x, y) (2.1.2)

We use the subscript notation for partial derivatives, that is, ux = u/x and

32

2.1 Analytic Functions 33

vx = v/x . For case (b)

f (z) = limy0

u(x, y +y) u(x, y)iy

+ i (v(x, y +y) v(x, y))iy

= iuy(x, y)+ vy(x, y) (2.1.3)Setting Eqs. (2.1.2) and (2.1.3) equal yields

ux = vy, vx = uy (2.1.4)Equations (2.1.4) are called the CauchyRiemann conditions.

Equations (2.1.4) are a system of partial differential equations that are neces-sarily satised if f (z) has a derivative at the point z. This is in stark contrast toreal analysis where differentiability of a function f (x) is only a mild smoothnesscondition on the function. We also note that if u, v have second derivatives, thenwe will show that they satisfy the equations uxx + uyy = 0 and vxx + vyy = 0(c.f. Eqs. (2.1.11a,b)).

Equation (2.1.4) is a necessary condition that must hold if f (z) is dif-ferentiable. On the other hand, it turns out that if the partial derivatives ofu(x, y), v(x, y) exist, satisfy Eq. (2.1.4), and are continuous, then f (z) =u(x, y)+ iv(x, y) must exist and be differentiable at the point z = x + iy; thatis, Eq. (2.1.4) is a sufcient condition as well. Namely, if Eq. (2.1.4) holds,then f (z) exists and is given by Eqs. (2.1.12.1.2).

We discuss the latter point next. We use a well-known result of real analysisof two variables, namely, if ux , uy and vx , vy are continuous at the point (x, y),then

u = uxx + uyy + 1|z|v = vxx + vyy + 2|z| (2.1.5)

where |z| =x2 +y2, limz0 1 = limz0 2 = 0, and

u = u(x +x, y +y) u(x, y)v = v(x +x, y +y) v(x, y)

Calling f = u + iv, we have fz

= uz

+ i vz

=(

uxx

z+ uy y

z

)+ i(vxx

z+ vy y

z

)+ (1 + i2) |z|

z, |z| = 0 (2.1.6)

34 2 Analytic Functions and Integration

Then, letting z|z| = ei and using Eq. (2.1.4), Eq. (2.1.6) yields

fz

= (ux + ivx )x + iyz

+ (1 + i2)ei

= f (z)+ (1 + i2)ei (2.1.7)after noting Eq. (2.1.2) and manipulating. Taking the limit of z approachingzero yields the desired result.

We state both of the above results as a theorem.

Theorem 2.1.1 The function f (z) = u(x, y) + iv(x, y) is differentiable at apoint z = x + iy of a region in the complex plane if and only if the partialderivatives ux , uy , vx , vy are continuous and satisfy the CauchyRiemannconditions (Eq. (2.1.4)) at z = x + iy.

A consequence of the CauchyRiemann conditions is that the level curvesof u, that is, the curves u(x, y) = c1 for constant c1, are orthogonal to the levelcurves of v, where v(x, y) = c2 for constant c2, at all points where f (z) existsand is nonzero. From Eqs. (2.1.2) and (2.1.4) we have

| f (z)|2 =(u

x

)2+(v

x

)2=(u

x

)2+(u

y

)2=(v

x

)2+(v

y

)2hence the two-dimensional vector gradientsu = ( u

x, uy

)andv = ( v

x, vy

)are nonzero. We know from vector calculus that the gradient is orthogonal toits level curve (i.e., du = u ds = 0, where ds points in the direction ofthe tangent to the level curve), and from the CauchyRiemann condition (Eq.(2.1.4)) we see that the gradients u, v are orthogonal because their vectordot product vanishes:

u v = ux

v

x+ uy

v

y

= ux

u

y+ uy

u

x= 0

Consequently, the two-dimensional level curves u(x, y) = c1 and v(x, y) = c2are orthogonal.

The CauchyRiemann conditions can be written in other coordinate sys-tems, and it is frequently valuable to do so. Here we quote the result in polarcoordinates:

u

r= 1

r

v

v

r= 1

r

u

(2.1.8)


Equation (2.1.8) can be derived in the same manner as Eq. (2.1.4). Analternative derivation uses the differential relationships

x= cos

r sin

r

y= sin

r+ cos

r

(2.1.9)

which are derived from x = r cos and y = r sin , r2 = x2+ y2, tan = y/x .Employing Eq. (2.1.9) in Eq. (2.1.4) yields

cos u

r sin

r

u

= sin v

r+ cos

r

v

sin u

r+ cos

r

u

= cos v

r+ sin

r

v

Multiplying the rst of these equations by cos , the second by sin , andadding yields the rst of Eqs. (2.1.8). Similarly, multiplying the rst by sin ,the second by cos , and adding yields the second of Eqs. (2.1.8).

Similarly, using the rst relation of Eq. (2.1.9) in f (z) = u/x + iv/xyields

f (z) = cos ur

sin r

u

+ i cos v

r i sin

r

v

= (cos i sin )(u

r+ i v

r

)hence,

f (z) = ei(u

r+ i v

r

)(2.1.10)

Example 2.1.1 Let f (z) = ez = ex+iy = ex eiy = ex (cos y + i sin y). VerifyEq. (2.1.4) for all x and y, and then show that f (z) = ez .

u = ex cos y, v = ex sin yu

x= ex cos y = v

y

u

y= ex sin y = v

x

f (z) = ux

+ i vx

= ex (cos y + i sin y)

= ex eiy = ex+iy = ez


We have therefore established the fact that f (z) = ez is differentiable forall nite values of z. Consequently, standard functions like sin z and cos z,which are linear combinations of the exponential function eiz (see Eqs. (1.2.91.2.10)) are also seen to be differentiable functions of z for all nite values of z.It should be noted that these functions do not behave like their real counterparts.For example, the function sin x oscillates and is bounded between 1 for allreal x . However, we have

sin z = sin(x + iy) = sin x cos iy + cos x sin iy= sin x cosh y + i cos x sinh y

Because |sinh y| and |cosh y| tend to innity as y tends to innity, we see thatthe real and imaginary parts of sin z grow without bound.

Example 2.1.2 Let f (z) = z = x iy, so that u(x, y) = x and v(x, y) = y.Since u/x = 1 while v/y = 1, condition (2.1.4) implies f (z) does notexist anywhere (see also section 1.3).

Example 2.1.3 Let f (z) = zn = rnein = rn(cos n + i sin n), for integern, so that u(r, ) = rn cos n and v(r, ) = rn sin n . Verify Eq. (2.1.8) andshow that f (z) = nzn1 (z = 0 if n < 0). By differentiation, we have

u

r= nrn1 cos n = 1

r

v

v

r= nrn1 sin n = 1

r

u

From Eq. (2.1.10),f (z) = ei (nrn1)(cos n + i sin n)

= nrn1eiein = nrn1ei(n1)

= nzn1

Example 2.1.4 If a function is differentiable and has constant modulus, showthat the function itself is constant. We may write f in terms of real, imaginary,or complex forms where

f = u + iv = Rei

R2 = u2 + v2, tan = vu

R = constant


From Eq. (2.1.8) we have

uu

r+ v v

r= 1

r

(uv

v u

)= u

2

r

(v

u

)so

r

(u2 + v2) = 2u2

r

(v

u

)Thus (v/u)/ = 0 because R2 = u2 + v2 = constant.

Similarly, using Eq. (2.1.8),

u2

r

(v

u

)=(

uv

r v u

r

)

= 1r

(uu

+ v v

)= 1

2r

(u2 + v2) = 0

Thus v/u = constant, which implies is constant, and hence so is f .

We have observed that the system of partial differential equations (PDEs),Eq. (2.1.4), that is, the CauchyRiemann equations, must hold at every pointwhere f (z) exists. However, PDEs are really of interest when they hold notonly at one point, but rather in a region containing the point. Hence we givethe following denition.

Denition 2.1.1 A function f (z) is said to be analytic at a point z0 if f (z) isdifferentiable in a neighborhood of z0. The function f (z) is said to be analyticin a region if it is analytic at every point in the region.

Of the previous examples, f (z) = ez is analytic in the entire nite z plane,whereas f (z) = z is analytic nowhere. The function f (z) = 1/z2 (Example2.1.3, n = 2) is analytic for all nite z = 0 (the punctured z plane).

Example 2.1.5 Determine where f (z) is analytic when f (z) = (x + y)2 +2i(x y) for real and constant.

u(x, y) = (x + y)2, v(x, y) = 2(x y)u

x= 2(x + y) v

y= 2

u

y= 2(x + y) v

x= 2


The CauchyRiemann equations are satised only if 2 = 1 and only on thelines x y = 1. Because the derivative f (z) exists only on these lines,f (z) is not analytic anywhere since it is not analytic in the neighborhood ofthese lines.

If we say that f (z) is analytic in a region, such as |z| R, we mean thatf (z) is analytic in a domain containing the circle because f (z) must exist ina neighborhood of every point on |z| = R. We also note that some authors usethe term holomorphic instead of analytic.

An entire function is a function that is analytic at each point in the entirenite plane. As mentioned above, f (z) = ez is entire, as is sin z and cos z. Sois f (z) = zn (integer n 0), and therefore, any polynomial.

A singular point z0 is a point where f fails to be analytic. Thus f (z) = 1/z2has z = 0 as a singular point. On the other hand, f (z) = z is analytic nowhereand has singular points everywhere in the complex plane. If any regionR existssuch that f (z) is analytic inR, we frequently speak of the function as being ananalytic function. A further and more detailed discussion of singular pointsappears in Section 3.5.

As we have seen from our examples and from Section 1.3, the standard dif-ferentiation formulae of real variables hold for functions of a complex variable.Namely, if two functions are analytic in a domain D, their sum, product, andquotient are analytic in D provided the denominator of the quotient does notvanish at any point in D. Similarly, the composition of two analytic functionsis also analytic.

We shall see, in a later section (2.6.1), that an analytic function has derivativesof all orders in the region of analyticity and that the real and imaginary partshave continuous derivatives of all orders as well. From Eq. (2.1.4), because2v/xy = 2v/yx ,we have

2u

x2=

2v

xy2v

yx=

2u

y2

hence

2u 2u

x2+

2u

y2= 0 (2.1.11a)

and similarly

2v 2v

x2+

2v

y2= 0 (2.1.11b)


Equations (2.1.11a,b) demonstrate that u and v satisfy certain uncoupledPDEs. The equation 2w = 0 is called Laplaces equation. It has wideapplicability and plays a central role in the study of classical partial differentialequations. The function w(x, y) satisfying Laplaces equation in a domain Dis called an harmonic function in D. The two functions u(x, y) and v(x, y),which are respectively the real and imaginary parts of an analytic function inD, both satisfy Laplaces equation in D. That is, they are harmonic functionsin D, and v is referred to as the harmonic conjugate of u (and vice versa).The function v may be obtained from u via the CauchyRiemann conditions. Itis clear from the derivation of Eqs. (2.1.11a,b) that f (z) = u(x, y)+ iv(x, y)is an analytic function if and only if u and v satisfy Eqs. (2.1.11a,b) and v isthe harmonic conjugate of u.

The following example illustrates how, given u(x, y), it is possible to obtainthe harmonic conjugate v(x, y) as well as the analytic function f (z).

Example 2.1.6 Suppose we are given u(x, y) = y2x2 in the entire z = x + iyplane. Find its harmonic conjugate as well as f (z).

u

x= 2x = v

y v = 2xy + (x)

u

y= 2y = v

x v = 2xy + (y)

where (x), (x) are arbitrary functions of x and y, respectively. Taking thedifference of both expressions for v implies (x)(y) = 0, which can onlybe satised by = = c = constant; thus

f (z) = y2 x2 2i xy + ic= (x2 y2 + 2i xy)+ ic = z2 + ic

It follows from the remark following Theorem 2.1.1, that the two level curvesu = y2 x2 = c1 and v = 2xy = c2 are orthogonal to each other at eachpoint (x, y). These are two orthogonal sets of hyperbolae.

Laplaces equation arises frequently in the study of physical phenomena.Applications include the study of two-dimensional ideal uid ow, steadystate heat conduction, electrostatics, and many others. In these applicationswe are usually interested in solving Laplaces equation 2w = 0 in a domainD with boundary conditions, typically of the form

w + wn

= on C (2.1.12)


where w/n denotes the outward normal derivative of w on the boundary ofD denoted by C; , , and are given functions on the boundary. We referto the solution of Laplaces equation when = 0 as the Dirichlet problem, andwhen = 0 the Neumann problem. The general case is usually called themixed problem.

2.1.2 Ideal Fluid FlowTwo-dimensional ideal uid ow is one of the prototypical examples ofLaplaces equations and complex variable techniques. The corresponding owcongurations are usually easy to conceptualize. Ideal uid motion refersto uid motion that is steady (time independent), nonviscous (zero friction;usually called inviscid), incompressible (in this case, constant density), andirrotational (no local rotations of uid particles). The two-dimensional equa-tions of motion reduce to a system of two PDEs (see also the discussion inSection 5.4, Example 5.4.1):

(a) incompressibility (divergence of the velocity vanishes)

v1

x+ v2

y= 0 (2.1.13a)

where v1 and v2 are the horizontal and vertical components of the two-dimen-sional vector v, that is, v = (v1, v2); and

(b) irrotationality (curl of the velocity vanishes)

v2

x v1

y= 0 (2.1.13b)

A simplication of these equations is found via the following substitutions:

v1 = x

= y

v2 = y

= x

(2.1.14)

In vector form: v = . We call the velocity potential, and the streamfunction. Equations (2.1.132.1.14) show that and satisfy Laplaces equa-tion. Because the CauchyRiemann conditions are satised for the functions and , we have, quite naturally, an associated complex velocity potential(z):

(z) = (x, y)+ i(x, y) (2.1.15)


The derivative of (z) is usually called the complex velocity

(z) = x

+ i x

= x

i y

= v1 iv2 (2.1.16)

The complex conjugate (z) = /x + i/y = v1 + iv2 is analogous tothe usual velocity vector in two dimensions.

The associated boundary conditions are as follows. The normal derivativeof (i.e., the normal velocity) must vanish on a rigid boundary of an idealuid. Because we have shown that the level sets (x, y) = constant and(x, y) = const. are mutually orthogonal at any point (x, y), we conclude thatthe level sets of the stream function follow the direction of the ow eld;namely, they follow the direction of the gradient of , which are themselvesorthogonal to the level sets of . The level curves (x, y) = const. are calledstreamlines of the ow. Consequently, boundary conditions in an ideal owproblem at a boundary can be specied by either giving vanishing conditionson the normal derivative of at a boundary (no ow through the boundary)or by specifying that (x, y) is constant on a boundary, thereby making theboundary a streamline. /n = n, n being the unit normal, implies that points in the direction of the tangent to the boundary. For problems withan innite domain, some type of boundary condition usually a boundednesscondition must be given at innity. We usually specify that the velocity isuniform (constant) at innity.

Briey in this section, and in subsequent sections and Chapter 5 (see Section5.4), we shall discuss examples of uid ows corresponding to various complexpotentials. Upon considering boundary conditions, functions (z) that areanalytic in suitable regions may frequently be associated with two-dimensionaluid ows, though we also need to be concerned with locations of nonanalyticityof (z). Some examples will clarify the situation.

Example 2.1.7 The simplest example is that of uniform ow

(z) = v0ei0 z = v0(cos 0 i sin 0)(x + iy), (2.1.17)where v0 and 0 are positive real constants. Using Eqs. (2.1.15, 2.1.16), thecorresponding velocity potential and velocity eld is given by

(x, y) = v0(cos 0x + sin 0 y) v1 = x

= v0 cos 0

v2 = y

= v0 sin 0

which is identied with uniform ow making an angle 0 with the x axis, as


y

x

0

Fig. 2.1.1. Uniform ow

in Figure 2.1.1. Alternatively, the stream function (x, y) = v0(cos 0 y sin 0x) = const. reveals the same ow eld.

Example 2.1.8 A somewhat more complicated ow conguration, ow arounda cylinder, corresponds to the complex velocity potential

(z) = v0(

z + a2

z

)(2.1.18)

where v0 and a are positive real constants and |z| > a. The correspondingvelocity potential and stream function are given by

= v0(

r + a2

r

)cos (2.1.19a)

= v0(

r a2

r

)sin (2.1.19b)


and for the complex velocity we have

(z) = v0(

1 a2

z2

)= v0

(1 a

2e2i

r2

)(2.1.20)

whereby from Eq. (2.1.16) the horizontal and vertical components of the ve-locity are given by

v1 = v0(

1 a2 cos 2

r2

)v2 = v0 a

2 sin 2r2

(2.1.21)

The circle r = a is a streamline ( = 0) as is = 0 and = . Asr , the limiting velocity is uniform in the x direction (v1 v0, v2 0).The corresponding ow eld is that of a uniform stream at large distancesmodied by a circular barrier, as in Figure 2.1.2, which may be viewed as owaround a cylinder with the same ow eld at all points perpendicular to the owdirection.

Note that the velocity vanishes at r = a, = 0, and = . These points arecalled stagnation points of the ow. On the circle r = a, which correspondsto the streamline = 0, the normal velocity is zero because the correspondingvelocity must be in the tangent direction to the circle. Another way to see thisis to compute the normal velocity from using the gradient in two-dimensionalpolar coordinates:

v = = r

ur + 1r

u

r=a

symmetric curves

= 0 = 0

= 0

=

=constant

vo constant

Fig. 2.1.2. Flow around a circular barrier


where ur and u are the unit normal and tangential vectors. Thus the velocity inthe radial direction is vr = r and the velocity in the circumferential directionis v = 1r . So the radial velocity at any point (r , ) is given by

r= v0

(1 a

2

r2

)cos

which vanishes when r = a. As mentioned earlier, as r the ow becomesuniform:

v0r cos = v0x v0r sin = v0 y

So for large r and correspondingly large y, the curves y = const are streamlinesas expected.


1. Which of the following satisfy the CauchyRiemann (C-R) equations? Ifthey satisfy the C-R equations, give the analytic function of z.

(a) f (x, y) = x iy + 1(b) f (x, y) = y3 3x2 y + i(x3 3xy2 + 2)

(c) f (x, y) = ey(cos x + i sin y)2. In the following we are given the real part of an analytic function of z. Find

the imaginary part and the function of z.

(a) 3x2 y y3 (b) 2x(c y), c = constant(c) y

x2 + y2 (d) cos x cosh y

3. Determine whether the following functions are analytic. Discuss whetherthey have any singular points or if they are entire.

(a) tan z (b) esin z (c) e1/(z1) (d) ez

(e) zz4 + 1 (f) cos x cosh y i sin x sinh y

4. Show that the real and imaginary parts of a twice-differentiable functionf (z) satisfy Laplaces equation. Show that f (z) is nowhere analytic unlessit is constant.


5. Let f (z) be analytic in some domain. Show that f (z) is necessarily aconstant if either the function f (z) is analytic or f (z) assumes only pureimaginary values in the domain.

6. Consider the following complex potential

(z) = k2

1z, k real,

referred to as a doublet. Calculate the corresponding velocity potential,stream function, and velocity eld. Sketch the stream function. The valueof k is usually called the strength of the doublet. See also Problem 4of Section 2.3, in which we obtain this complex potential via a limitingprocedure of two elementary ows, referred to as a source and a sink.

7. Consider the complex analytic function, (z) = (x, y)+ i(x, y), in adomain D. Let us transform from z to w using w = f (z), w = u + iv,where f (z) is analytic in D, with the corresponding domain in thew plane,D. Establish the following:

x= u

x

u+ vx

v

2

x2=

2u

x2

u

2u

xy

v+(u

x

)22

u2 2u

x

u

y2

uv

+(u

y

)22

v2

Also nd the corresponding formulae for /y and 2/y2. Recall thatf (z) = u

x i u

y , and u(x, y) satises Laplaces equation in the domainD. Show that

2x,y =2

x2+

2

y2= (u2x + u2y)(2u2 + 2v2

)= | f (z)|22u,v

Consequently, we nd that if satises Laplaces equation 2x,y = 0 inthe domain D, then so long as f (z) = 0 in D it also satises Laplacesequation 2u,v = 0 in domain D.


8. Given the complex analytic function (z) = z2, show that the real partof , (x, y) = Re(z), satises Laplaces equation, 2x,y = 0. Letz = (1 w)/(1+ w), where w = u + iv. Show that (u, v) = Re(w)satises Laplaces equation 2u,v = 0.

2.2 Multivalued FunctionsA single-valued function w = f (z) yields one value w for a given complexnumber z. A multivalued function admits more than one value w for a given z.Such a function is more complicated and frequently requires a great deal of care.Multivalued functions are naturally introduced as the inverse of single-valuedfunctions.

The simplest such function is the square root function. If we consider z = w2,the inverse is written as

w = z 12 (2.2.1)From real variables we already know that x1/2 has two values, often written as

x wherex 0. For the complex function (Eq. (2.2.1)) and from w2 = zwe can ascertain the multivaluedness by letting z = rei , and = p + 2n,where, say, 0 p < 2

w = r1/2eip/2en i (2.2.2)where r1/2 r 0 and n is an integer. (See also the discussion in Sec-tion 1.1.) For a given value z, the function w(z) takes two possible valuescorresponding to n even and n odd, namely

reip/2 and

reip/2ei = reip/2

An important consequence of the multivaluedness of w is that as z traversesa small circuit around z = 0, w does not return to its original value. Indeed,suppose we start at z = for real > 0. Let us see what happens to w aswe return to this point after going around a circle with radius . Let n = 0.When we start, p = 0 and w = ; when we return to z = , p = 2 andw = e 2i2 = . We note that the value can also be obtained fromp = 0 provided we take n = 1. In other words, we started with a value wcorresponding to n = 0 and ended up with a value w corresponding to n = 1!(Any even/odd values of n sufce for this argument.) The point z = 0 is calleda branch point. A point is a branch point if the multivalued function w(z) isdiscontinuous upon traversing a small circuit around this point. It should benoted that the point z = is also a branch point. This is seen by using thetransformation z = 1t , which maps z = to t = 0. Using arguments such

2.2 Multivalued Functions 47

C

xL R

Fig. 2.2.1. Closed circuit away from branch cut

z = re

z = re

x

0i

2 i

Fig. 2.2.2. Cut plane, z1/2

as that above, it follows that t = 0 is a branch point of the function t1/2, andhence z = is a branch point of the function z1/2. The points z = 0 andz = are the only branch points of the function z1/2. Indeed, if we take aclosed circuit C (see Figure 2.2.1) that does not enclose z = 0 or z = , thenz1/2 returns to its original value as z traverses C . Along C the phase will varycontinuously between = R and = L . So if we begin at zR = rReiR andfollow the curve C , the value z will return to exactly its previous value with nophase change. Hence z1/2 will not have a jump as the curve C is traversed.

The analytic study of multivalued functions usually is best effected by ex-pressing the multivalued function in terms of a single-valued function. Onemethod of doing this is to consider the multivalued function in a restrictedregion of the plane and choose a value at every point such that the resultingfunction is single-valued and continuous. A continuous function obtained froma multivalued function in this way is called a branch of the multivalued func-tion. For the functionw = z1/2 we can carry out this procedure by taking n = 0and restricting the region of z to be the open or cut plane in Figure 2.2.2. Forthis purpose the real positive axis in the z plane is cut out. The values of z = 0and z = are also deleted. The function w = z1/2 is now continuous in thecut plane that is an open region. The semiaxis Re z > 0 is referred to as abranch cut.

It should be noted that the location of the branch cut is arbitrary save thatit ends at branch points. If we restrict p to p < , n = 0 in thepolar representation of z = rei , = p + 2n , then the branch cut wouldnaturally be on the negative real axis. More complicated curves (e.g. spirals)could equally well be chosen as branch cuts but rarely do we do so becausea cut is chosen for convenience. The simplest choice (sometimes motivated


by physical application) is generally satisfactory. We reiterate that the mainpurpose of a branch cut is to articially create a region in which the function issingle-valued and continuous.

On the other hand, if we took a closed circuit that didnt enclose the branchpoint z = 0, then the function z1/2 would return to its same value. We depict, inFigure 2.2.1, a typical closed circuit C not enclosing the origin, with the choiceof branch cut (z = rei , 0 < 2 ) on the positive real axis.

Note that if we had chosenw = (zz0)1/2 as our prototype example, a (nite)branch point would have been at z = z0. Similarly, if we had investigatedw = (az + b)1/2, then a (nite) branch point would have been at b/a. (Ineither case, z = would be another branch point.) We could deduce thesefacts by translating to a new origin in our coordinate system and investigatingthe change upon a circuit around the branch point, namely, letting z = z0+rei ,0 < 2 . We shall see that multivalued functions can be considerably moreexotic than the ones described above.

A somewhat more complicated situation is illustrated by the inverse of theexponential function, that is, the logarithm (see Figure 2.2.3). Consider

z = ew (2.2.3)Let w = u + iv. We have, using the properties of the exponential function

z = eu+iv = eueiv = eu(cos v + i sin v) (2.2.4a)in polar coordinates z = reip for 0 p < 2 , so

r = eu

v = p + 2n, n integer (2.2.4b)From the properties of real variables

u = log rThus, in analogy with real variables, we write w = log z, which is

w = log z = log r + ip + 2n i (2.2.4c)where n = 0,1,2, . . . and where p takes on values in a particular range of2 . Here we take

0 p < 2When n = 0, Eq. (2.2.4) is frequently referred to as the principal branch of thelogarithm; the corresponding value of the function is referred to as the principalvalue. From (2.2.4) we see that, as opposed to the square root example, the

2.2 Multivalued Functions 49

function is innitely valued; that is, n takes on an innite number of integervalues. For example, if z = i , then |z| = r = 1, p = /2; hence

log i = log 1+ i(

2+ 2n

)n = 0,1,2, . . . (2.2.5)

Similarly, if z = x , a real positive quantity, |z| = r = |x |, then

log z = log |x | + 2n i n = 0,1,2, . . . (2.2.6)

The complex logarithm function differs from the real logarithm by additivemultiples of 2 i . If z is real and positive, we normally take n = 0 so that theprincipal branch of the complex logarithm function agrees with the u

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34258750 Complex Variables Introduction and Applications