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Preface This book is the fruit of many years of teaching complex variables to students in applied mathematics by the first author and research by the third author with the close collaboration of the second author, who translatedapreliminaryRussianversionofthetextandcollectedandsolved all the exercises. It is an extended course in complex analysis and its applications, written in a style that is particulary well-suited for students in applied mathematics, science and engineering, and for users of complex analysis in the applications. The first half of the book is a clear and rigorous introduction to the theory of functions of one complex variable. The second half contains the evaluation of many new integration formulae and the summation of new infiniteseriesbythecalculusofresidue. Thelastchapterisconcernedwith theFatou–Juliatheoryformeromorphicfunctionsforfindingselectiveroots of some transcendental equations as found in the applications. Chapter 1 reviews the representation of complex numbers and intro- ducesanalytic(holomorphic)functions. InChapter2,bothtraditionaland non-traditional problems in conformal mapping are solved in great detail. Chapter 2 depends only on Chapter 1 and is independent of the other chapters;thusitcanbetakenanytime, afterthe studyofthefirstchapter. Chapters 3, 4, 5, 6 and 9 can be covered in that order. Chapters 7, 8, 10 and 11 cover more specialized topics and are beyond a usual introduction to analytic functions. The shortbibliographylists commonreferences in Englishand in Rus- sian and a few research papers. The exercises are elementary and aim at the understanding of the the- oryof analyticfunctions. Some of them canbe easily solvedwith symbolic software on computers. Answers to almost all odd-numbered exercises are found at the end of the book. The text benefitted from the remarksmade by generationsof students at the Riga Technical University and at the University of Ottawa. Miss Ellen Yanqing Zheng has read a preliminary version of the book in the winter of 1994 and made many corrections. vii viii PREFACE The authors express their thanks to Dr. Thierry Giordano, who has used the manuscript as lecture notes since 1995. He has made many valu- able suggestions for improving the first part of the book. Mr.AndréMontpetitoftheCentrederecherchesmathématiquesofthe Université de Montréal has been generous in offering invaluable assistance for the composition of the text in A S LaTeX. This book benefitted from theMsupp−orts of the Natural Sciences and Engineering Council of Canada, the University of Ottawa, Riga Technical University, and the Centre de recherches mathématiques of the Université de Montréal. The authors express their warmest thanks to the dynamic and collab- orative editorial and production team of Academic Press Inc. M. Ya. Antimirov A. A. Kolyshkin Rémi Vaillancourt Riga, Ottawa, 24 November 1997 CHAPTER 1 Functions of a Complex Variable 1.1. Complex numbers 1.1.1. Algebraic operations on complex numbers. Definition 1.1.1. A complex number z is an ordered pair, (x,y), of realnumbers, x and y, where x is called the real part of z, written x= z, < andy iscalledtheimaginarypartofz,writteny = z. Thesetofcomplex = numbers is denoted by C. For clarity, the expressions z-plane and w-plane will be used to mean z C and w C, respectively, when referring to different copies of C. ∈ ∈ Two complex numbers, z = (x ,y ) and z = (x ,y ), are equal, 1 1 1 2 2 2 writtenz =z ,ifandonlyiftheirrealandimaginarypartsareequal;that 1 2 is, if and only if x =x and y =y . 1 2 1 2 Definition 1.1.2. Thesumoftwocomplexnumbers,z =(x ,y )and 1 1 1 z =(x ,y ), is defined to be the complex number 2 2 2 z =z +z =(x +x ,y +y ). 1 2 1 2 1 2 The commutativity and the associativity of the addition, z +z =z +z , 1 2 2 1 z +(z +z )=(z +z )+z , 1 2 3 1 2 3 follow from Definition 1.1.2. The complex number zero, 0 = (0,0), such that z +0 = z for all z C, is introduced in the same way as the real ∈ number 0 in the set of real numbers. Definition 1.1.3. Theproduct oftwocomplexnumbers,z =(x ,y ) 1 1 1 and z =(x ,y ), is defined to be the complex number 2 2 2 z =z z =(x x y y ,x y +x y ). 1 2 1 2 1 2 1 2 2 1 − 1 2 1. FUNCTIONS OF A COMPLEX VARIABLE Thecommutativity,theassociativity andthedistributivity ofthemul- tiplication, z z =z z , 1 2 2 1 z (z z )=(z z )z , 1 2 3 1 2 3 (z +z )z =z z +z z , 1 2 3 1 3 2 3 follow from Definition 1.1.3. The set R of real numbers becomes a subset of the set C of complex numbers if a R is identified with a = (a,0) C. It then follows, from ∈ ∈ Definitions1.1.2and1.1.3ofadditionandmultiplication,respectively,that alltheknownpropertiesoftheadditionandthemultiplicationofrealnum- bers are also valid for complex numbers. Therefore the set C of complex numbers can be considered as an extension of the set R of real numbers. Note that (a,0) (x,y)=(ax,ay). × The complex numbers are not ordered. Hence the order relations < and > cannot be applied to complex numbers; that is, given two distinct nonreal complex numbers, z and z , it is impossible to write z > z or 1 2 1 2 z <z , without violating some properties of the real numbers. 1 2 Definition 1.1.4. A complex number of the form (0,y) is said to be a pure imaginary number. The complex number (0,1)is called the imaginaryunit and is denoted by the symbol i: i = (0,1). The number (0,y) can be considered as the product of the real number y =(y,0) and the imaginary unit (0,1), (y,0) (0,1)=(y 0 0 1,y 1+0 0)=(0,y). × × − × × × Therefore we can write (0,y)=iy. Squaring the imaginary unit, we have i i=(0,1) (0,1)=(0 0 1 1,0 1+1 0)=( 1,0), × × × − × × × − that is, i2 = 1. (1.1.1) − 1.1.2. Algebraic form of complex numbers. The previous rela- tion(1.1.1)allowsonetogiveadirectcomputationallyconvenientalgebraic meaning to complex numbers. Definition 1.1.5. The algebraic form of the complex number z =(x,y)=(x,0)+(0,y) is z =x+iy. (1.1.2) 1.1. COMPLEX NUMBERS 3 Notation 1.1.1. Complex numbers in the algebraic form are usually denoted by z =x+iy, ζ =ξ+iη, w =u+iv, and a=α+iβ. The letters c and d are also used. Toperformadditionandmultiplicationofcomplexnumbers,onesimply uses the usual rules of the algebra of polynomials plus the rules i2 = 1, − i3 = i and i4 =1. − Definition1.1.6. Thecomplexnumberz¯=x iyiscalledthecomplex − conjugate of z =x+iy. The subtraction of complex numbers is defined as the inverse of the addition. Giventwo complex numbers z =x +iy andz =x +iy , the 1 1 1 2 2 2 difference, z z , is the complex number z such that z +z =z . Thus 2 1 1 2 − z =z z =x x +i(y y ). 2 1 2 1 2 1 − − − The division of complex numbers is defined as the inverse of the mul- tiplication. If z =x +iy and z =x +iy =0, then z =z /z if 1 1 1 2 2 2 1 2 6 z z =z . (1.1.3) 2 1 Lettingz =x+iyin(1.1.3),performingthemultiplicationandequating the real and imaginary parts on the right- and left-hand sides of (1.1.3), respectively, we obtain a system of equations for x = z and y = z. < = Solving this system, we get x +iy x x +y y x y x y 1 1 1 2 1 2 2 1 1 2 x+iy = = +i − . (1.1.4) x +iy x2+y2 x2+y2 2 2 2 2 2 2 It is easy to check that the same result can be found by multiplying the numerator and the denominator of the fraction z /z by z¯ =x iy . 1 2 2 2 2 − 1.1.3. Geometricrepresentationofcomplexnumbers. Weshall represent the complex number z = x + iy by the point A in the plane with coordinates (x,y) referred to the Cartesian coordinate system x0y. Such a plane is called the complex plane, the x-axis being called the real axis and the y-axis being called the imaginary axis. There is a one-to-one correspondence between the points of the complex plane and the set of complexnumbers. Thereforeinthe sequelweshallnotdistinguishbetween acomplexnumberanditscorrespondingpointinthecomplexplane,sothat we shallsay,for example,the “point3+2i,”the “trianglewith verticesz , 1 z and z ,” etc. 2 3 InFig1.1,thevectorO−→A=(x,y)isidentifiedwiththecomplexnumber z =x+iy. The angleθ formedby O−→A and the positivex-axis is calledthe argument of z and is denoted by argz: y θ =argz, if tan(argz)= . (1.1.5) x 4 1. FUNCTIONS OF A COMPLEX VARIABLE y A y z = x + iy r θ = Arg z x x 0 Figure 1.1. The vector O−→A = (x,y) identified with the complex number z =x+iy. ThelengthrofthevectorO−→Aiscalledthemodulus ofthecomplexnumber z and is denoted by z , | | z = O−→A =r= x2+y2 0. (1.1.6) | | | | ≥ The angle argz is usually taken in onepof the half-open intervals, (2k 1)π <argz (2k+1)π, k =0, 1, 2,..., (1.1.7) − ≤ ± ± or 2kπ argz <2(k+1)π, k =0, 1, 2,.... (1.1.8) ≤ ± ± The principal value of the argument of z is defined to be the angle Argz such that y tan(Argz)= , π<Argz π, (1.1.9) x − ≤ by taking k =0 in (1.1.7), or y tan(Argz)= , 0 Argz <2π, (1.1.10) x ≤ by taking k =0 in (1.1.8). In this book, the choice of (1.1.9) or (1.1.10) will be dictated by each problem in hand and should be clear from the context. Generally, (1.1.9) is used in Chapters 1, 3, 4 and 5, and (1.1.10) is used in Chapters 2, 6, 7 and 8. Most computers use the the principal value given by (1.1.9). Withthechoice(1.1.9),therearethreecasestobeconsideredforArgz: y (a) If x>0 (see Fig 1.2), Argz =Arctan . x y (b) If x<0 and y >0 (see Fig 1.3), Argz =Arctan +π. x y (c) If x<0 and y <0 (see Fig 1.4), Argz =Arctan π. x − 1.1. COMPLEX NUMBERS 5 y y z = x + iy 0 Arg z = Arctan _xy x _y Arg z = Arctan x z = x + iy x 0 Figure 1.2. The principal value of arg z for x>0. y z = x + iy Arg z = Arctan _y + π x 0 Arctan _y x x Figure 1.3. The principal value of arg z for x<0, y >0. y Arctan _y x x 0 Arg z = Arctan _ y – π x z = x + iy Figure 1.4. The principal value of arg z for x<0, y <0. Hence, for (1.1.9), Arctan y , x>0, x Argz = Arctan(cid:0)xy(cid:1)+π, x<0, y >0, (1.1.11)  Arctan(cid:0)xy(cid:1)−π, x<0, y <0. In any case, one sees that(cid:0)ar(cid:1)gz =Argz+2kπ for k ∈Z, that is, arg is periodic of period 2π. 6 1. FUNCTIONS OF A COMPLEX VARIABLE y A z + z A 1 2 1 z 1 z – z A 2 1 2 z 2 x 0 B z – z 2 1 Figure 1.5. Geometric representation of the sum, O−→A, and difference, O−→B =A−→A , of two complex numbers. 1 2 Note 1.1.1. The definitions (1.1.9) or (1.1.10) of Argz mean that a cutismadealongthenegativeorpositiverealaxis,respectively. Ingeneral terms, a cut is a double line that is not allowed to be crossed when angles are measured. Therefore, with (1.1.9) Argz = π on the upper part of the cut and Argz = π on the lower part of the cut. Such a cut can be taken − along an arbitrary direction, but formula (1.1.11) differs from cut to cut. Most computers and calculators take the cut along the negative real axis so that the principal value, Argz, of the argument of z is given by (1.1.9) so that (1.1.11) holds. In the Russian mathematical literature, the roles of arg and Arg are interchanged. Letusconsiderthe geometricmeaningofthe sumanddifferenceofthe two complex numbers z =x +iy and z =x +iy . 1 1 1 2 2 2 In Fig 1.5 the vectors O−A→ = (x ,y ) and O−A→ = (x ,y ) correspond 1 1 1 2 2 2 to z and z , respectively. 1 2 Since z +z =(x +x )+i(y +y ), then the vector 1 2 1 2 1 2 O−→A=(x +x ,y +y ) 1 2 1 2 corresponds to the complex number z +z . Thus, the sum of the vectors 1 2 O−A→ and O−A→ , 1 2 O−→A=O−A→ +O−A→ , (1.1.12) 1 2 corresponds to the sum of the complex numbers z and z . Similarly, the 1 2 vector O−→A=O−A→ +O−A→ + +O−A→ (1.1.13) 1 2 n ··· correspondstothesumz +z + +z ofthecomplexnumbersz ,z , ,z 1 2 n 1 2 n ··· ··· represented by the vectors O−A→ ,O−A→ ,...,O−A→ , respectively. The vector 1 2 n 1.1. COMPLEX NUMBERS 7 O−→A joins the beginning andthe endofthe polygonalline OA A A . It 1 2 n ··· follows from Fig 1.5 and formulae (1.1.12) and (1.1.13) that O−→A O−A→ + O−A→ , O−→A O−A→ + O−A→ + + O−A→ , 1 2 1 2 n ≤ ≤ ··· that(cid:12)is, w(cid:12)e h(cid:12)ave t(cid:12)he t(cid:12)riang(cid:12)le ineq(cid:12)uali(cid:12)ty, (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) z +z z + z , (1.1.14) 1 2 1 2 | |≤| | | | and its generalizationto n numbers, z +z + +z z + z + + z . 1 2 n 1 2 n | ··· |≤| | | | ··· | | These inequalities can be written in the short form n n z z . (1.1.15) k k ≤ | | (cid:12)k=1 (cid:12) k=1 (cid:12)X (cid:12) X Equality in (1.1.14)and(cid:12)(1.1.15(cid:12))holds only if allthe complex numbers (cid:12) (cid:12) z lie on the same straight line in the complex plane. k Inequality(1.1.15)isbasicforestimatingthemoduliofsumsofcomplex numbers and integrals of functions of a complex variable. On the other hand, since z z = (x x )+i(y y ), then the 2 1 2 1 2 1 − − − vectorO−→B =(x x ,y y )correspondsto the complexnumber z z . 2 1 2 1 2 1 − − − In this case, O−→B =A−→A =O−A→ O−A→ , (1.1.16) 1 2 2 1 − that is, the vector O−→B corresponds to the difference of the given complex numbers and is represented by a difference of the vectors O−A→ and O−A→ . 2 1 It follows from Fig 1.5 and formula (1.1.16) that z z = A−→A = (x x )2+(y y )2, (1.1.17) 2 1 1 2 2 1 2 1 | − | | | − − that is, the modulus of the diffperence, z2 z1 of two complex numbers is − equal to the distance between the points z and z in the complex plane. 1 2 Sincethe distanceinCandR2 isgivenbythe sameformula,itwillbe seen inthenextsubsectionsthatthedefinitionofaneighborhoodofapoint,the set of interior or exterior points of a disk in C, etc., will be the same as in R2. Hence CandR2 havethe samenotionsofcontinuityandlimit, thatis, the same topology. For example, if z = x +iy = constant and ρ = constant > 0, then 0 0 0 the formula z z =ρ (1.1.18) 0 | − | represents the geometric locus of all the points z which are at distance ρ from the point z . Thus (1.1.18) is the equation of a circle centered at z 0 0 and of radius ρ (see Fig 1.6). If z = x+iy and z = x +iy , it follows 0 0 0 8 1. FUNCTIONS OF A COMPLEX VARIABLE y |z – z |> ρ 0 ρ z 0 |z – z |< ρ 0 x 0 Figure 1.6. (Shaded)interiorand(unshaded)exteriorof a disk. from (1.1.17) and (1.1.18) that (x x )2+(y y )2 =ρ2, (1.1.19) 0 0 − − whichistheCartesianequationofthecircleofradiusρ,centeredat(x ,y ). 0 0 In Fig 1.6, the inequality z z <ρ represents the (shaded) set of points 0 | − | insidethediskwhereastheinequality z z >ρrepresentsthe(unshaded) 0 | − | set of points outside the same disk. 1.1.4. Trigonometric form of complex numbers. Oneeasilysees fromFig1.1thatifz =x+iy,thenx=rcosθ andy =rsinθ withr = z . | | Thus, we have the following definition. Definition 1.1.7. Thetrigonometricformofthecomplexnumberz = x+iy is z =r(cosθ+isinθ), (1.1.20) where x=rcosθ, y =rsinθ and r= z . | | Thetrigonometricform(1.1.20)ofcomplexnumbersallowsonetogive a simple geometric meaning to the product and quotient of two complex numbers. Given z =r (cosθ +isinθ ), z =r (cosθ +isinθ ), 1 1 1 1 2 2 2 2 by the usual rules of algebra the product of z and z is 1 2 z z =r r cosθ cosθ sinθ sinθ 1 2 1 2 1 2 1 2 − (cid:2) +i(cosθ1sinθ2+sinθ1cosθ2) , (1.1.21) which, upon using trigonometric identities for sums of angles, r(cid:3)educes to z z =r r cos(θ +θ )+isin(θ +θ ) . (1.1.22) 1 2 1 2 1 2 1 2 It follows from (1.1.22) that(cid:2) (cid:3) z z = z z , arg(z z )=argz +argz , (1.1.23) 1 2 1 2 1 2 1 2 | | | || |