Graduate Texts in Mathematics 103 Editorial Board F W Gehring P R Halmos (Managing Editor) C C Moore Graduate Texts in Mathematics I TAKEUTUZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHEs/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTUZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory 10 COHEN A Course in Simple Homotopy Theory 11 CONWAY Functions of One Complex Variable 2nd ed 12 BEALS Advanced Mathematical Analysis 13 ANDERSON/FuLLER Rings and Categories of Modules 14 GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities 15 BERBERIAN Lectures in Functional Analysis and Operator Theory 16 WINTER The Structure of Fields 17 ROSENBLATT Random Processes 2nd ed 18 HALMOS Measure Theory 19 HALMOS A Hilbert Space Problem Book 2nd ed., revised 20 HUSEMOLLER Fibre Bundles 2nd ed 21 HUMPHREYS Linear Algebraic Groups 22 BARNEs/MACK An Algebraic Introduction to Mathematical Logic 23 GREUB Linear Algebra 4th ed 24 HOLMES Geometric Functional Analysis and its Applications 2S' HEWITT/STROMBERG Real and Abstract Analysis 26 MANES Algebraic Theories 27 KELLEY General Topology 28 ZARISKUSAMUEL Commutative Algebra Vol I 29 ZARISKUSAMUEL Commutative Algebra VoL II 30 JACOBSON Lectures in Abstract Algebra I: Basic Concepts 31 JACOBSON Lectures in Abstract Algebra II: Linear Algebra 32 JACOBSON Lectures in Abstract Algebra III: Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Algebras and Several Complex Variables 2nd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory I 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and continued after Index Serge Lang Complex Analysis Second Edition With 132 Illustrations Springer Science+Business Media, LLC Serge Lang Department of Mathematics Yale University New Haven, CT 06520 U.S.A Editorial Board P R Halmos F W Gehring c Managing Editor Department of Mathematics University of Santa Clara Santa Clara, CA 95053 U.S.A Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A Department of Mathematics University of California at Berkeley Berkeley, CA 94720 U.S.A C Moore AMS Subject Classification: 30-01 Library of Congress Cataloging in Publication Data Lang, Serge Complex analysis (Graduate texts in mathematics; 103) Includes index Functions of complex variables Mathematical analysis I Title II Series QA331.L255 1985 515.9 84-21274 The first edition of this book was published by Addison-Wesley Publishing Co., Menlo Park, CA, in 1977 © 1977, 1985 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc in 1985 Softcover reprint of the hardcover 2nd edition 1985 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC Typeset by Composition House Ltd., Salisbury, England 543 I ISBN 978-1-4757-1873-7 ISBN 978-1-4757-1871-3 (eBook) DOI 10.1007/978-1-4757-1871-3 Foreword The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level Somewhat more material has been included than can be covered at leisure in one term, to give opportunities for the instructor to exercise his taste, and lead the course in whatever direction strikes his fancy at the time A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc.) and I would recommend to anyone to look through them More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues The systematic elementary development of formal and convergent power series was standard fare in the German texts, but only Cartan, in the more recent books, includes this material, which I think is quite essential, e.g., for differential equations I have written a short text, exhibiting these features, making it applicable to a wide variety of tastes The book essentially decomposes into two parts The first part, Chapters I through VIII, includes the basic properties of analytic functions, essentially what cannot be left out of, say, a onesemester course I have no fixed idea about the manner in which Cauchy's theorem is to be treated In less advanced classes, or if time is lacking, the usual vi FOREWORD hand waving about simple closed curves and interiors is not entirely inappropriate Perhaps better would be to state precisely the homological version and omit the formal proof For those who want a more thorough understanding, I include the relevant material Artin originally had the idea of basing the homology needed for complex variables on the winding number I have included his proof for Cauchy's theorem, extracting, however, a purely topological lemma of independent interest, not made explicit in Artin's original Notre Dame notes (cf collected works) or in Ahlfor's book closely following Artin I have also included the more recent proof by Dixon, which uses the winding number, but replaces the topological lemma by greater use of elementary properties of analytic functions which can be derived directly from the local theorem The two aspects, homotopy and homology, both enter in an essential fashion for different applications of analytic functions, and neither is slighted at the expense of the other Most expositions usually include some of the global geometric properties of analytic maps at an early stage I chose to make the preliminaries on complex functions as short as possible to get quickly into the analytic part of complex function theory: power series expansions and Cauchy's theorem The advantages of doing this, reaching the heart of the subject rapidly, are obvious The cost is that certain elementary global geometric considerations are thus omitted from Chapter I, for instance, to reappear later in connection with analytic isomorphisms (Conformal Mappings, Chapter VII) and potential theory (Harmonic Functions, Chapter VIII) I think it is best for the coherence of the book to have covered in one sweep the basic analytic material before dealing with these more geometric global topics Since the proof of the general Riemann mapping theorem is somewhat more difficult than the study of the specific cases considered in Chapter VII, it has been postponed to the second part The second part of the book, Chapters IX through XIV, deals with further assorted analytic aspects of functions in many directions, which may lead to many other branches of analysis I have emphasized the possibility of defining analytic functions by an integral involving a parameter and differentiating under the integral sign Some classical functions are given to work out as exercises, but the gamma function is worked out in detail in the text, as a prototype The chapters in this part are essentially logically independent and can be covered in any order, or omitted at will In particular, the chapter on analytic continuation, including the Schwarz reflection principle, and/or the proof of the Riemann mapping theorem could be done right after Chapter VII, and still achieve great coherence As most of this part is somewhat harder than the first part, it can easily be omitted from a course addressed to undergraduates In the FOREWORD VII same spirit, some of the harder exercises in the first part have been starred, to make their omission easy In this second edition, I have rewritten many sections, and I have added some material I have also made a number of corrections whose need was pointed out to me by several people I thank them all I am much indebted to Barnet M Weinstock for his help in correcting the proofs, and for useful suggestions SERGE LANG Prerequisi tes We assume that the reader has had two years of calculus, and has some acquaintance with epsilon-delta techniques For convenience, we have recalled all the necessary lemmas we need for continuous functions on compact sets in the plane We use what is now standard terminology A function f:S - T is called injective if x t= y in S implies f(x) t= f(y) It is called surjective if for every z in T there exists XES such that f(x) = z If f is surjective, then we also say that f maps S onto T If f is both injective and surjective then we say that f is bijective Given two functions f, defined on a set of real numbers containing arbitrarily large numbers, and such that g(x) ~ 0, we write f« or f(x) «g(x) for x - 00 to mean that there exists a number C > such that for all x sufficiently large, we have If(x) I ~ Cg(x) Similarly, if the functions are defined for x near 0, we use the same symbol « for x - to mean that there If(x) I ~ Cg(x) x PREREQUISITES for all x sufficiently small (there exists {) > such that if Ix I < {) then If(x) I ~ Cg(x» Often this relation is also expressed by writing f(x) = O(g(x»), which is read: f(x) is big oh of g(x), for x + 00 or x -+ as the case may be We use ]a, b[ to denote the open interval of numbers a < x < b Similarly, [a, b[ denotes the half-open interval, etc [XIV, §4] 355 BEHAVIOR AT THE BOUNDARY L' Figure passing through t/!n(v n), as shown on Fig Then the beginning and end points of an lie at the same distance from the origin (This is what we wanted to achieve to make the next step valid.) Let T be rotation by the angle 2n/M If we rotate an by T iterated M times, i.e take then we obtain a closed curve, lying inside the annulus Finally, define the function h*(w) = h(w)h(w), and the function G(w) = h*(w)h*(Tw)· h*(T M - lW) Figure £n < IwI < 356 THE RIEMANN MAPPING THEOREM [XIV, §4] Let B be a bound for 1h( w) I, wED Each factor in the definition of G is bounded by B2 For any w on the above closed curve, some rotation Tkw lies on (Tn' and then we have Therefore G is bounded on the closed curve by For each ray L from the origin, let WL be the point of L closest to the origin, and lying on the closed curve Let W be the union of all segments [0, wL ] open on the right, for all rays L Then W is open, and the boundary of W consists of points of the closed curve By the maximum modulus principle, it follows that IG(O) I ~ maxi G(w) I, where the max is taken for w on the closed curve Letting n tend to infinity, we see that G(O) = 0, whence h(O) = 0, a contradiction which proves the lemma, and therefore also the theorem Theorem 4.4 Let U be bounded Let J: U -+ D be an isomorphism with the disc, and let oc 1, OC be two distinct boundary points oj U which are accessible Suppose J extended to OC and OC2 by continuity Then Proof We suppose J(oc ) = J(oc ) After multiplying J by a suitable constant, we may assume J(OCi) = -1 Let g: D -+ U again be the inverse function of f Let Y1, Y2 be the curves defined on an interval [a, b] such that their end points are oc , oc , respectively, and Yi(t) E U for i = 1, and t E [a, b], t =1= b There exists a number c with a < c < b such that if c < t < b, and there exists () such that and not intersect the disc D( -1, ()) as shown on Fig 10 Let A({)) =D II D( -1, ()) [XIV, §4] 357 BEHAVIOR AT THE BOUNDARY o Figure 10 Then A(c5) is described in polar coordinates by and - cp(r) ;£ e ;£ cp(r) Note that cp(r) < n12 with an appropriate function cp(r) Area g(A(c5)) = If We have: Ig'(z) 12 dy dx A(6) (6f'l'(r) = JIo - 'I'(r) Ig'( -1 + re i6 )12r de dr For each r < c5 let Wi' w2 be on !(Yl), !(Y2)' respectively, such that IWi + 11 = r, i = 1,2, and Then where the integral is taken over the circular arc from Wi to W2 in D, with center - For < r < c5 we get whence by the Schwarz inequality, we find I !Xl - !X 4nr 12 :;;; r f'l'(r) -'I'(r) Ig'( -1 + re i6 ) 12 de 358 THE RIEMANN MAPPING THEOREM [XIV, §4] We integrate both sides with respect to r from to (j The right-hand side is bounded, and the left-hand side is infinite unless (Xl = (X2 This proves the theorem The technique for the above proof is classical It can also be used to prove the continuity of the mapping function at the boundary Cf for instance Hurwitz-Courant, Part III, Chapter 6, §4 APPENDIX Cauchy's Formula for COO Functions Let D be an open disc in the complex numbers, and let DC be the closed disc, so the boundary of DC is a circle Cauchy's formula gives us the value as an integral over the circle C: if f is holomorphic on DC, that is on some open set containing the closed disc But what happens if f is not holomorphic but merely smooth, say its real and imaginary parts are infinitely differentiable in the real sense? There is also a formula, which unfortunately is not usually taught in basic courses, although it gives a beautiful application of several notions which arise in both real and complex analysis, and advanced calculus We shall give this theorem here, together with an application, which occurs in the theory of partial differential equations Let us write z = x + iy We introduce two new derivatives Let where f1 = Re f and f2 = 1m f are the real and imaginary parts of f respectively We say that f is Coo if f1,J2 are infinitely differentiable in the naive sense of functions of two real variables x and y In other words, all partial derivatives of all orders exist and are continuous We write f E Coo(DC) to mean that f is Coo on some open set containing DC For such f we define and of = ~ (Of oz ox + i Of) oy 360 CAUCHY'S FORMULA FOR COO FUNCTIONS [Appendix] Symbolically, we put a 1(0 0) - and 1- oz - ax oy The Cauchy-Riemann equations can be formulated neatly by saying that f is holomorphic if and only if of = az See Chapter VIII, §1 We shall need the Stokes-Green formula for a simple type of region In advanced calculus, one integrates expressions where P, Q are COO functions, and C is some curve The Stokes-Green theorem relates such integrals over a boundary to a double integral f'rJ (OQ _ OP) B oy aX dx dy taken over a region B which is bounded by the curve C statement is this The precise Stokes-Green Formula Let B be a region of the plane, bounded by a finite number of curves, oriented so that the region lies to the left of each curve Let y be the boundary, so oriented Let P, Q have continuous first partial derivatives on B and its boundary Then It is useful to express the Stokes-Green formula in terms of the derivatives Ojoz and a/oz Writing dz = dx + idy and dz = dx - idy, we can solve for dx and dy in terms of dz and dz, to give dx = !(dz + dz) and dy = 2i (dz - dz) [Appendix] CAUCHY'S FORMULA FOR COO FUNCTIONS 361 Then P dx + Q dy = dz + h dz, where g, h are suitable functions Let us write symbolically dz " dz = -2i dx dy Then by substitution, we find the following version of the Stokes-Green Formula: {g dz + h dz = 51 (~~ -~~) dz " dz Remark Directly from the definition of oloz and oloz one verifies that the usual expression for df is given by of of_ of of oz dz + oz dz = ox dx + oy dy Consider the special case where B = B(a) is the region obtained from the disc DC by deleting a small disc of radius a centered at a point zo, as shown on the figure c; Q B(a) Figure Then the boundary consists of two circles C and C;;, oriented as· shown so that the region lies to the left of each curve We have written C;; to indicate the circle with clockwise orientation, so that the region B(a) lies to the left of C;; As before, C is the circle around D, oriented counterclockwise Then the boundary of B(a) can be written 362 CAUCHY'S FORMULA FOR COO FUNCTIONS [Appendix] We shall deal with integrals where q>(z) is a smooth function, and where Zo is some point in the interior of the disc Such an integral is an improper integral, and is supposed to be interpreted as a limit · I1m a-+O fi () q> dz 1\ dz , Z - Zo Z B(a) where B(a) is the complement of a disc of radius a centered at Zo0 The limit exists, as one sees by using polar coordinates Letting Z = Zo + re i6 with polar coordinates around the fixed point Zo, we have dx dy = rdrdO Z - Zo = re i6 , so r cancels and we see that the limit exists, since the integral becomes simply ~nd fL q>(z) dr dO The region B(a) is precisely of the type where we apply the StokesGreen formula Cauchy's Theorem for Coo functions Let f E COO (DC) and let Zo be a point in the interior D Let C be the circle around D Then i J(Zo) = ~ J(z) dz 2m c z - Zo + ~ f'r o~ dz 2m JD az 1\ dz z - Zo Proof Let a be a small positive number, and let B(a) be the region obtained by deleting from D a disc of radius a centered at zo Then aJ/oz is Coo on B(a) and we can apply the Stokes-Green formula to J(z) dz = g(z) dz z - Zo over this region Furthermore Note that this expression has no term with dz -o (-1-) =0 iJZ z - Zo and og oj oz = oz z - Zo [Appendix] CAUCHY'S FORMULA FOR C«> 363 FUNCTIONS because 1/(z - zo) is holomorphic in this region Then by Stokes-Green we find J(z) dz e; z - + Zo J(z) dz = _ c z - f'r o~ dz " dz JB( ) oZ ZO Z - Zo The limit of the integral on the right-hand side as a approaches is the double integral (with a minus sign) which occurs in Cauchy's formula We now determine the limit of the curve integral over c; on the left-hand side We parametrize C (counterclockwise orientation) by Os S 2n Then dz = aiei8 dz, so J(z) dz c; z - =- Zo Since J is continuous at Zo, J(z) dz e" z - =- Zo "J(ZO + aei8 )i dO we can write where h(a, 0) is a function such that lim h(a, 0) = ,,"'0 uniformly in O Therefore lim J(z) dz = -2nij(zo) - lim e; z - Zo ,,"'0 "h(a, O)i dO = - 2nij(zo)· Cauchy's formula now follows at once Remark Suppose that J is holomorphic on D Then oj oz = 0, and so the double integral disappears from the general formula to give the Cauchy formula as we encountered it previously 364 CAUCHY'S FORMULA FOR COO FUNCTIONS [Appendix] ° Remark Consider the special case when the function f is on the boundary of the disc Then the integral over the circle C is equal to 0, and we obtain the formula f(zo) = f'rJ ~ 2m D o~ dz Adz OZ Z - Zo This allows us to recover the values of the function from its derivative of/oz Conversely, one has the following result Theorem Let g E Coo(DC) be a Coo function on the closed disc the function f(w) = -2' m ff -g(z) -dz DZ-W 1\ Then dz is defined and Coo on D, and satisfies of ow = g(w) for WED The proof is essentially a corollary of Cauchy's theorem if one has the appropriate technique for differentiating under the integral sign However, we have now reached the boundary of this course, and we omit the proof Index A C Absolute convergence 52, 151,308 Absolute value Absolutely integrable 307 Accessible point 352 Accumulation point 21 Addition formula 299 Adherent 18 Algebraic function 333 Algebraic independence 262 Analytic 69, 145 Analytic continuation 92, 238, 324, 330 Analytic isomorphism 72,77, 159, 196, 200 Angle 9,35 Angle preserving 88 Annulus 151 Artin's proof 139 Automorphism 32, 159, 196,200 Canonical product 288 Casorati-Weierstrass 158 Cauchy-Riemann equations 33, 360 Cauchy sequence of functions 51 Cauchy sequence of numbers 19 Cauchy's formula 135, 144 Cauchy's theorem 129, 133, 362 Chain 131 Chain rule 30 Close together 113 Closed 18 Closed chain 131 Closed path 98, 116, 131 Compact 21,347 Compact support 309 Comparison test 52 Complex conjugate Complex differentiable 28 Complex number Composition of series 67 Conformal 36, 196,229 Conformal mapping 36, 196 Congruent mod lattice 293 Congruent power series 43 Conjugate Conjugation 197 Connected 89, 93 Connected along partition III Connected along path 330 Constant term 38 Continuous 19 B Beginning point 88 Bernoulli number 47 Binomial series 59,65 Blaschke product 279 Borel-Caratheodory theorem 273 Boundary point 18,351 Boundary value 226 Bounded set 18 366 Converge uniformly 51 Convergent power series 49, 55, 62 Convex 117 Convexity theorem 269 Convolution 245 Covering 25 Cross ratio 223 Curve 87 D Derivative 28, 147 Difference equation 48,61 Differentiable 28, 32 Differential equation 86 Differentiating under integral sign 309 Differentiation of power series 82 Dilogarithm 339 Dirac sequence 244 Direct analytic continuation 324 Dirichlet series 149 Disc 17,200 Discrete 91 Distance between sets 21 Dixon's proof 162 Domination 66, 264 E Elliptic functions 292 End point 88 Entire function 146,262,276,286 Equicontinuous 344 Equipotential 228 Essential singularity 158 Euler constant 315 Euler summation 316 INDEX H Hadamard entire function 275 Hadamard factorization 290 Hadamard's three circles 270 Hankel integral 321 Harmonic 224,241 Holomorphic 28, 145, 161 Holomorphic isomorphism 32 Homologous 128, 132 Homotopic 115,129 Hermite interpolation 261 I Imaginary part 5, 13 Infinity 161,217 Integral 94, 95, 109, 129 Integral evaluation 180 Interior 18, 173 Inverse function theorem 73 Inverse of number Inverse of power series 41, 73 Inversion 16,217 Isolated 91 Isolated singularity 155 Isomorphism 32, 72, 196,206 Isothermal 228 J Jensen's formula 257 L LI-norm L2-norm 150 150 Finite covering 25 Finite order 286 Fixed point 27,219 Flow lines 231 Formal power series 38 Fractional linear maps 28,215 Fundamental parallelogram 293 Laplace transform 310 Lattice 293 Laurent series 151 Legendre relation 305 Length 99 Level curve 228 Lim sup 55 Liouville's theorem 146 Local isomorphism 73 Locally constant 80 G M Gamma function 311 Goursat's theorem 104 Green's function 234 Green's theorem 243, 361 Maximum modulus principle 79,92, 255 Mean value theorem 243 Mellin transform 187,312 F 367 INDEX Meromorphic 157, 161,276,290,321, 322 Minimum modulus principle 288 Mittag-Leffler theorem 290 Monodromy theorem 333 Multiplication formula for gamma function 314 N Normal family 344 Normal vector 239 o Open disc 17 Open mapping 73, 77 Order of entire function 286 Order of pole 156 Order of power series 40, 46 P Paley-Wiener space 321 Partial sum 49 Path 89 Pathwise connected 90 Perpendicularity 229 Phragmen-Linde1of theorem 268, 272 Picard theorem 343 Point of accumulation 21,55 Poisson representation 249 Polar form Pole 156 P61ya 255 Potential function 227 Power series 38 Power series expansion 82 Primitive 84, 104, 118 Principal part 290 Principal value of log 121 Q Reflection 16, 324 Relatively compact 344 Removable singularity 155 Residue 165 Residue formula 166, 179 Riemann mapping theorem 197, 340 Riemann sphere 159 Rouch6's theorem 172 S Schwarz lemma 198, 257 Schwarz-Pick lemma 201 Schwarz reflection 324 Sigma function 303 Simple pole 157, 166 Simply connected 118 Singularity 155 Speed 99 Star shaped 118 Stirling formula 315 Stokes Green theorem 361 Straight line on sphere 218 Strict order 262, 286 Subcovering 25 Subdivision 139 Subfamily 25 Sup norm 51, 100 T Theta function 302 Translation 217 Triangle inequality Trigonometric integrals 185 U Uniform convergence 51, 147,307 Upper half plane 17,203 V Vector field 33, 225, 230 q-product 279 w R Weierstrass-Bolzano 21 Weierstrass product 280 Whittaker function 295, 303 Winding number 124 Radius of convergence 55 Rational values 265 Real analytic 327 Real part 5, 13 Rectangle 104 Refinement 112 z Zero of function 70 Zeta function 272, 303 Graduate Texts in Mathematics continuedfrom page ii 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVER/WATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELL/Fox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOV /MERZLJAKOV Fundamentals of the Theory of Groups 63 BoLLABAS Graph Theory 64 EDWARDS Fourier Series Vol I 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed 66 WATERHOUSE Introduction to Affine Group Schemes 67 SERRE Local Fields 68 WEIDMANN Linear Operators in Hilbert Spaces 69 LANG Cyclotomic Fields II 70 MASSEY Singular Homology Theory 71 FARKAS/KRA Riemann Surfaces 72 STILLWELL Classical Topology and Combinatorial Group Theory 73 HUNGERFORD Algebra 74 DAVENPORT Multiplicative Number Theory 2nd ed 75 HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras 76 IITAKA Algebraic Geometry 77 HECKE Lectures on the Theory of Algebraic Numbers 78 BURRIS/SANKAPPANAVAR A Course in Universal Algebra 79 WALTERS An Introduction to Ergodic Theory 80 ROBINSON A Course in the Theory of Groups 81 FORSTER Lectures on Riemann Surfaces 82 BoTT/Tu Differential Forms in Algebraic Topology 83 WASHINGTON Introduction to Cyclotomic Fields 84 IRELAND/RoSEN A Classical Introduction to Modern Number Theory 85 EDWARDS Fourier Series: Vol II 2nd ed 86 VAN LINT Introduction to Coding Theory 87 BROWN Cohomology of Groups 88 PIERCE Associative Algebras 89 LANG Introduction to Algebraic and Abelian Functions 2nd ed 90 BR0NDSTED An Introduction to Convex Polytopes 91 BEARDON On the Geometry of Discrete Groups 92 DIESTEL Sequences and Series in Banach Spaces 93 DUBROVIN/FoMENKO/NovlKov Modern Geometry - Methods and Applications Vol I 94 WARNER Foundations of Differentiable Manifolds and Lie Groups 95 SHIRYAYEV Probability, Statistics, and Random Processes 96 CONWAY A Course in Functional Analysis 97 KOBLITZ Introduction to Elliptic Curves and Modular Forms 98 BROCKER/tom DIECK Representations of Compact Lie Groups 99 GROVE/BENSON Finite Reflection Groups 2nd ed 100 BERG/CHRISTENSEN/RESSEL Harmonic Analysis on Semigroups: Theory of positive definite and related functions 101 EDWARDS Galois Theory 102 VARADARAJAN Lie Groups, Lie Algebras and Their Representations 103 LANG Complex Analysis 2nd ed 104 DUBROVIN/FoMENKO/NovlKov Modern Geometry - Methods and Applications Vol II ... Topology in Dimensions and continued after Index Serge Lang Complex Analysis Second Edition With 132 Illustrations Springer Science+Business Media, LLC Serge Lang Department of Mathematics Yale University... Library of Congress Cataloging in Publication Data Lang, Serge Complex analysis (Graduate texts in mathematics; 103) Includes index Functions of complex variables Mathematical analysis I Title II... will be called complex numbers I §1 Definition The complex numbers are a set of objects which can be added and multiplied, the sum and product of two complex numbers being also a complex number,