Selected Title s i n This Serie s
168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8 167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s o f fractals, 199 7 166 Kenj i Ueno , A n introductio n t o algebrai c geometry , 199 7 165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n Galois theory, 199 7 164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis , 199 7 163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko , Probability theory : Collectio n o f problems, 199 7 162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d method s i n linea r statistical models , 199 7 161 Michae l Blank , Discretenes s an d continuit y i n problems o f chaotic dynamics , 199 7 160 V . G . Osmolovskit , Linea r an d nonlinea r perturbation s o f the operator div , 199 7 159 S . Ya . Khavinson , Bes t approximatio n b y linea r superposition s (approximat e nomography), 199 7 158 Hidek i Omori , Infinite-dimensiona l Li e groups , 199 7 157 V . B . Kolmanovski t an d L . E . ShaYkhet , Contro l o f systems wit h aftereffect , 199 6 156 V . N . Shevchenko , Qualitativ e topic s i n intege r linea r programming , 199 7 155 Yu . Safaro v an d D . Vassiliev , Th e asymptoti c distributio n o f eigenvalue s o f partia l differential operators , 199 7 154 V . V . Prasolo v an d A . B . Sossinsky , Knots , links , braid s an d 3-manifolds . A n introduction t o the ne w invariant s i n low-dimensiona l topology , 199 7 153 S . Kh . Aranson , G . R . Belitsky , an d E . V . Zhuzhoma , Introductio n t o th e qualitative theor y o f dynamical system s o n surfaces , 199 6 152 R . S . Ismagilov , Representation s o f infinite-dimensiona l groups , 199 6 151 S . Yu . Slavyanov , Asymptoti c solution s o f the one-dimensiona l Schrodinge r equation , 1996 150 B . Ya . Levin , Lecture s o n entire functions , 199 6 149 Takash i Sakai , Riemannia n geometry , 199 6 148 Vladimi r I . Piterbarg , Asymptoti c method s i n the theory o f Gaussian processe s an d fields, 199 6 147 S . G . Gindiki n an d L . R . Volevich , Mixe d proble m fo r partia l differentia l equation s with quasihomogeneou s principa l part , 199 6 146 L . Ya . Adrianova , Introductio n t o linea r system s o f differentia l equations , 199 5 145 A . N . Andriano v an d V . G . Zhuravlev , Modula r form s an d Heck e operators, 199 5 144 O . V . Troshkin , Nontraditiona l method s i n mathematical hydrodynamics , 199 5 143 V . A . Malyshe v an d R . A . Minlos , Linea r infinite-particl e operators , 199 5 142 N . V . Krylov , Introductio n t o th e theor y o f diffusio n processes , 199 5 141 A . A . Davydov , Qualitativ e theor y o f control systems , 199 4 140 Aizi k I . Volpert , Vital y A . Volpert , an d Vladimi r A . Volpert , Travelin g wav e solutions o f parabolic systems , 199 4 139 I . V . Skrypnik , Method s fo r analysi s o f nonlinea r ellipti c boundary valu e problems , 199 4 138 Yu . P . Razmyslov , Identitie s o f algebras an d thei r representations , 199 4 137 F . I . Karpelevic h an d A . Ya . Kreinin , Heav y traffi c limit s fo r multiphas e queues , 199 4 136 Masayosh i Miyanishi , Algebrai c geometry , 199 4 135 Masar u Takeuchi , Moder n spherica l functions , 199 4 134 V . V . Prasolov , Problem s an d theorem s i n linear algebra , 199 4 133 P . I . Naumki n an d I . A . Shishmarev , Nonlinea r nonloca l equation s i n the theor y o f waves, 199 4 132 Hajim e Urakawa , Calculu s o f variations an d harmoni c maps , 199 3 131 V . V . Sharko , Function s o n manifolds : Algebrai c an d topologica l aspects , 199 3 130 V . V . Vershinin , Cobordism s an d spectra l sequences , 199 3 (Continued in the back of this publication) This page intentionally left blank Introduction t o Complex Analysis This page intentionally left blank 10.1090/mmono/168
Translations o f MATHEMATICAL MONOGRAPHS
Volume 16 8
Introduction t o Complex Analysis
Junjiro Noguch i
Translated by Junjiro Noguchi
•^y^TPHTO Z MIT W 3//i^^zy& America n Mathematical Societ y Providence, Rhode Islan d
%DED Editorial Boar d Shoshichi Kobayash i (Chair ) Masamichi Takesak i
FUKUSO KAISEK I GAIRO N (Introduction t o comple x analysis ) by Junjir o Noguchi
Copyright © 199 3 by Shokab o Publishin g Company , Ltd . Originally publishe d i n Japanes e b y Shokab o Publishin g Company , Ltd. , Tokyo , 199 3
Translated fro m th e Japanes e b y Junjir o Noguch i
2000 Mathematics Subject Classification. Primar y 30-01 .
Library o f Congres s Cataloging-in-Publicatio n Dat a Noguchi, Junjiro , 1948 - [Fukuso kaisek i gairon. English ] Introduction t o comple x analysi s / Junjir o Noguch i ; translated b y Junjir o Noguchi . p. cm . — (Translation s o f mathematical monograph s ; v. 168 ) Includes bibliographica l reference s (p . - ) and index . ISBN 0-8218-0377- 8 (alk . paper ) 1. Functions o f complex variables . 2 . Mathematical analysis . I . Title. II . Series . QA331.7.N6413 199 7 515'.9-—dc21 97-1439 2 CIP
AMS softcove r ISB N 978-0-8218-4447- 2
Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 199 8 by the America n Mathematica l Society . Al l rights reserved . Reprinted b y the America n Mathematica l Society , 2008 . Translation authorize d b y Shokab o Publishin g Co. , Ltd . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 3 1 2 1 1 1 0 09 0 8 TO AKIKO This page intentionally left blank Contents
Preface x i
Chapter 1 . Comple x Number s 1 1.1. Comple x Number s 1 1.2. Plan e Topolog y 3 1.3. Sequence s an d Limit s 9 Problems 1 5
Chapter 2 . Comple x Functions 1 7 2.1. Comple x Function s 1 7 2.2. Sequence s o f Complex Function s 1 9 2.3. Serie s o f Functions 2 3 2.4. Powe r Serie s 2 5 2.5. Exponentia l Function s an d Trigonometri c Function s 3 0 2.6. Infinit e Product s 3 4 2.7. Rieman n Spher e 3 7 2.8. Linea r Transformation s 4 0 Problems 4 6
Chapter 3 . Holomorphi c Function s 4 9 3.1. Comple x Derivative s 4 9 3.2. Curvilinea r Integral s 5 4 3.3. Homotop y o f Curves 6 2 3.4. Cauchy' s Integra l Theore m 6 7 3.5. Cauchy' s Integra l Formul a 7 2 3.6. Mea n Valu e Theorem an d Harmoni c Function s 8 1 3.7. Holomorphi c Function s o n the Rieman n Spher e 8 7 Problems 8 9
Chapter 4 . Residu e Theorem 9 1 4.1. Lauren t Serie s 9 1 4.2. Meromorphi c Function s an d Residu e Theore m 9 4 4.3. Argumen t Principl e 9 9 x CONTENT S
4.4. Residu e Calculu s 10 6 Problems 11 4
Chapter 5 . Analyti c Continuatio n 11 7 5.1. Analyti c Continuatio n 11 7 5.2. Monodrom y Theore m 12 4 5.3. Universa l Coverin g and Rieman n Surfac e 13 0 Problems 13 8
Chapter 6 . Holomorphi c Mapping s 14 1 6.1. Linea r Transformation s 14 1 6.2. Poincar e Metri c 14 4 6.3. Contractio n Principl e 14 9 6.4. Th e Rieman n Mappin g Theore m 15 3 6.5. Boundar y Correspondenc e 15 7 6.6. Universa l Coverin g o f C \ {0,1 } 16 3 6.7. Th e Little Picard Theore m 16 7 6.8. Th e Bi g Picard Theore m 17 0 Problems 17 4
Chapter 7 . Meromorphi c Function s 17 9 7.1. Approximatio n Theore m 17 9 7.2. Existenc e Theorem s 18 5 7.3. Riemann-Stieltjes ' Integra l 19 2 7.4. Meromorphi c Function s o n C 19 4 7.5. Weierstrass ' Product 20 2 7.6. Ellipti c Function s 21 0 Problems 22 6
Hints and Answer s 22 9
References 24 1
Index 24 5
Symbols 24 9
Correction Lis t 251 Preface
Complex analysi s i s a n activ e researc h subjec t b y itself , but , eve n more , i t provides the foundations fo r broad areas of mathematics, and plays an importan t role in the application s o f mathematics to engineering . This boo k i s intende d t o describ e a classica l introductor y par t o f comple x analysis fo r universit y student s i n the science s o r engineering , an d i n particula r to serv e a s a text o r a reference boo k fo r juniors, seniors , o r first-year graduat e students in the sciences. Th e prerequisites are elementary calculus, from the real numbers throug h differentiatio n an d integration , an d linea r algebra ; se t theor y and som e genera l topolog y ar e not required , bu t woul d b e helpful . Historically , the contents of this volume had been discovered by the beginning of the twentieth century, s o they ar e not ver y new. Nevertheless , the author trie d to arrange th e presentation an d choos e terminology t o agre e with moder n wor k i n the field. Many book s hav e bee n written o n this subject , i n many languages . Roughl y speaking, they can be divided into two types. On e tries to give an understandin g of th e subjec t usin g intuitiv e arguments . Th e othe r put s mor e emphasi s o n rigorous proofs, presenting the subject a s a fundamental theor y o f mathematics. This book fall s i n the latter group . I t i s well-known that ther e i s some difficult y in dealin g wit h curves , relate d t o Cauchy' s integra l theorem , a poin t o n whic h several published book s are somewhat unsatisfactory . T o deal with it rigorously , we give detailed description s o f the homotop y o f plane curves. Thi s material o n curves i s something th e reade r ma y wel l hav e encountere d i n previou s courses , so i t i s scattered throug h th e tex t wher e needed . Reader s whos e mathematica l background alread y includes this material may simply confirm the theorems an d go on. Since residu e theore m i s important bot h i n pur e mathematic s an d i n appli - cations, w e giv e a fairl y detaile d explanatio n o f ho w t o appl y i t t o numerica l calculations; thi s shoul d b e sufficien t fo r thos e wh o ar e studying comple x anal - ysis fo r applications . After thi s book , th e studen t shoul d b e read y t o tak e u p valu e distributio n theory, th e theor y o f Rieman n surfaces , comple x analysi s i n severa l variables , the theory o f complex manifolds, an d other subjects. Comple x analysi s will also
xi Xll PREFACE provide fundamenta l method s fo r th e theor y o f differential equations , algebrai c geometry, numbe r theory , an d othe r fields. Th e autho r wrot e Chapter s 6 and 7 with these transitions i n mind . The autho r wil l b e ver y please d i f thi s boo k help s student s t o understan d the classica l theor y o f comple x analysis , to relis h it s beauty, an d t o maste r th e rigorous treatment o f mathematical demonstrations . This volum e i s based o n lecture s fo r third-yea r student s give n b y the autho r at th e Tokyo Institute o f Technology. Th e clas s contained no t just mathematic s majors, but also physics and applied physics students. Th e author is very gratefu l to al l o f them . Finally, bu t no t least , th e autho r woul d lik e t o expres s hi s dee p thank s t o Professor Mitsur u Ozawa , wh o le d hi m t o comple x analysis , an d t o Professo r Shingo Murakami, who recommended hi m to write this book. H e is also obliged to Mr. Shuj i Hosoki , o f the Shokab o publishing company , fo r the proofreading .
April, 199 3 a t Ohokayam a Junjiro Noguch i
Added i n English translation : The publicatio n o f thi s Englis h translatio n wa s mad e possibl e b y th e sug - gestion an d th e recommendatio n o f Professo r Katsum i Nomizu . Th e autho r expresses hi s sincere gratitude t o him . February, 199 7 a t Ohokayam a Junjiro Noguch i This page intentionally left blank References
In writin g thi s boo k th e autho r ha s referre d t o a numbe r o f book s alread y published. Som e o f them ar e listed below : [1] A . Hurwitz and R. Courant, Funktionentheorie, Springe r Verlag, Berlin , 1929. [2] L . Ahlfors , Comple x Analysis , McGraw-Hill , Aucklan d e t al. , 1979 ; Japanese Translatio n b y K. Kasahara , Gendaisugakusha , Tokyo , 1982 . [3] H . Cartan , Theori e Elementair e d e Fonction s Analytique s d'Un e o u Plusieures Variables Complexes, Hermann, Paris, 1961 ; Japanese Trans- lation b y R . Takahashi , Iwanami Shoten , Tokyo , 1965. [4] K . Kasahara, Comple x Analysis-Analytic Function s o f One Variable (i n Japanese), Jikkyoshuppansha, Tokyo , 1978 . [5] Y . Komatsu , Introductio n t o Analysi s [I ] (i n Japanese), Hirokaw a Sho - ten, Tokyo , 1962 . [6] Y . Komatsu , Functio n Theor y (i n Japanese) , Asakur a Shoten , Tokyo , 1960. [7] Y . Komatsu, Exercise s in Function Theory (i n Japanese), Asakura Sho - ten, Tokyo , 1960 . [8] R . Takahashi , Comple x Analysi s (i n Japanese) , Universit y o f Toky o Press, Tokyo , 1990 . [9] T . Ochia i an d J . Noguchi , Geometri c Functio n Theor y i n Severa l Com - plex Variable s (i n Japanese) , Iwanam i Shoten , Tokyo , 1984 ; Englis h Translation b y Noguch i an d Ochiai , Amer . Math . Soc , Providence , Rhode Island, 1990 . For the readers who want to advanc e to a further stud y o n complex analysis , the author would like to give a list of books, which is not intended to be complete. For Riemann surfaces , I recommend th e following . [10] H . Weyl , Di e Ide e de r Riemannsch e Flache , B.G . Teubner , Stuttgart , 1913; Japanese Translation b y J. Tamura , Iwanam i Shoten , 1974 . (Thi s is a famou s classica l book, settling the concep t o f Riemann surfaces. )
241 242 REFERENCES
[11] K . Iwasawa, Algebraic Functions (i n Japanese), Iwanami Shoten, Tokyo, 1952; 2nd ed., 1972 ; English Translation b y G. Kato, Amer. Math. Soc , Providence, Rhod e Island , 1993 . (Thi s i s a famou s introductor y boo k from algebra. ) [12] R.C . Gunning , Lecture s o n Rieman n Surfaces , Princeto n Universit y Press, Princeton , 1966 . (Comprehensiv e lectur e notes , usin g shea f the - ory.) [13] Y . Kusunoki , Functio n Theor y (i n Japanese), Asakur a Shoten , Tokyo , 1973. (Thi s i s a nic e introductor y boo k o f close d an d ope n Rieman n surfaces t o the researc h level. ) [14] Y . Imayosh i an d M . Taniguchi , A n Introduction t o Teichmulle r Space s (in Japanese), Nipponhyoronsha, Tokyo, 1989; English Translation by Y. Imayoshi an d M . Taniguchi , Springer-Verlag , Toky o et al. , 1992 . (Thi s deals with the deformation theor y o f the comple x structure o f Rieman n surfaces, whic h i s called Teichmulle r theory. ) For value distribution theory , I recommen d [15] R . Nevanlinna, L e Theoreme de Picard-Borel et la Theorie de Fonctions Meromorphes, Gauthier-Villars , Paris , 1939 . (Thi s i s a monograp h b y the creator o f Nevanlinna theory, and i t clearly unrolls the developmen t of the mathematical theor y i n fron t o f your eyes. ) [16] W.K . Hayman , Meromorphi c Functions , Oxfor d Universit y Press , Ox - ford, 1964 . (Thi s is an introduction to Nevanlinna theory in one variable, up to the researc h level. ) [17] M . Ozawa , Moder n Functio n Theor y I - Theor y o f Valu e Distributio n (in Japanese), Morikitashuppansha, Tokyo , 1976 . (Thi s i s a monograp h at th e researc h level. ) [18] S . Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marce l Dekker, Ne w York , 1970 . (Thi s i s a n introductio n t o th e theor y o f Kobayashi hyperboli c manifold s b y Kobayash i himself , an d ha s ha d a deep influenc e o n comple x analysis sinc e then. ) [19] W . StoU , Value Distribution Theor y fo r Meromorphi c Maps , Aspects o f Math. E7, Vieweg , Braunschweig, 1985 . [20] P.A . Griffiths , Entir e Holomorphi c Mapping s i n One an d Severa l Com - plex Variables , Ann . Math . Studie s 85 , Princeto n Universit y Press , Princeton, 1976 . For complex analysis in several variables and the theory o f complex manifolds , I recommen d [21] L . Hormander , Introductio n t o Comple x Analysi s i n Severa l Variables , van Nostrand , Ne w York , 1966 . (Thi s i s a n introductio n t o th e the - ory fro m th e viewpoin t o f ellipti c differentia l equation s an d functiona l analysis. Thi s boo k establishe d on e research directio n o f the theory. ) REFERENCES 243
[22] A . Weil , Introductio n a I'Etud e de s Variete s kahleriennes , Hermann , Paris, 1958 . (Thi s i s a comprehensiv e cours e i n th e theor y o f Kahle r manifolds b y one o f the greatest mathematician s o f this century. Math - ematically i t i s very nourishing . [23] S . Nakano, Complex Function Theory i n Several Variables - Differentia l Geometric Approach (i n Japanese), Asakura Shoten, Tokyo, 1982. (Thi s is a goo d boo k o n functio n theor y o n Kahle r manifold s b y th e author , who is famous fo r th e Kodaira-Nakano vanishin g theorem. ) [24] T . Nishino, Function Theory of Several Complex Variables (in Japanese), Tokyo Universit y Press , Tokyo , 1997 . (Th e author , wh o wa s a studen t of K. Oka, the founder o f the theory, gives a treatise based on Oka's idea and work . A n Englis h translation i s in preparation. ) This page intentionally left blank Index
Abel's continuit y theorem , 4 6 complex number , 1 absolute convergence , 13, 23, 35 complex plane , 2 absolute value , 2 complex Poisso n integral , 8 2 accumulation point , 4 complex torus , 21 2 act (action) , 13 2 conformal mapping , 5 4 addition theore m fo r th e pe-function , 22 6 conformal metric , 13 7 adjoint harmoni c function , 8 5 conformal pseudo-metric , 13 7 analytic, 2 8 conformality, 5 4 analytic continuation , 117 , 12 4 conjugate, 2 analytic curve , 12 1 connected, 5 annul us, 9 1 connected component , 9 arc, 6 constant curve , 7 arc wise connected, 8 continuous, 1 8 area, 13 8 contraction principle , 149 , 151 argument, 3 converge, 9 , 1 9 argument principle , 10 0 converge absolutely , 13 , 23 Ascoli-Arzela theorem , 2 0 converge uniformly , 1 9 at mos t a pole , 9 3 convergent powe r series , 2 5 automorphic form , 22 2 correspondence o f circle to circle , 4 2 big Picard theorem , 17 1 counting function , 19 7 biholomorphic mapping , 137 covering, 5 , 13 0 Borel exceptiona l value , 20 2 covering mapping , 13 0 boundary, 5 covering transformation group , 13 3 boundary correspondence , 15 7 cross ratio, 4 7 boundary distance , 2 2 curve, 6 boundary point , 5 curvilinear integral , 58, 70 , 8 8 bounded, 4 , 1 7 deck transformatio n group , 13 3 branch, 76 , 126 defect, 20 2 branch point , 12 8 degree, 21 4 canonical product , 20 5 derivative, 4 9 Casorati-Weierstrass' theorem , 11 8 derived function , 4 9 Cauchy condition , 1 3 differential, 87 , 8 8 Cauchy product , 1 3 Dirichlet problem , 8 5 Cauchy sequence , 1 0 discrete, 5 Cauchy's integra l formula , 73, 7 7 disk, 3 Cauchy's integra l theorem , 7 0 disk neighborhood , 4, 39 , 135 Cauchy-Riemann equations , 5 0 disk o f definition , 12 4 characteristic function , 197 distance, 3 circle, 5 , 3 9 domain, 5 , 8 7 circle o f convergence, 2 5 domain o f existence, 19 1 closed, 4 doubly periodi c function , 21 2 closed curve , 6 elliptic, 4 8 closed set , 5 elliptic curve , 21 2 closure, 4 elliptic function , 210 , 212, 22 4 compact, 1 0 elliptic integral , 22 4 complete, 147, 14 8 end, 7 complex coordinate , 2 entire function , 11 8 complex derivative , 4 9 equicontinuous, 2 0 complex differentiable , J±9, 8 7 exceptional value , 16 7 complex function , 1 7 exponent o f convergence, 20 0
245 246 INDEX
exponential function , 3 0 Laplace equation , 8 2 exterior point , 5 Laplacian, 8 2 finite length , 5 5 lattice, 21 2 fixed point , 4 7 lattice point , 21 2 Fubini-Study metric , 13 8 Laurent series , 9 3 function element , 12 4 length, 54 , 13 8 fundamental domain , 16 4 length parameter , 5 6 fundamental group , 13 0 length parametrization , 5 6 fundamental periods , 21 2 lifting, 13 3 fundamental theore m o f algebra, 8 0 limit, 9 , 1 7 Gaussian curvature , 13 9 limit function , 1 9 Gaussian plane , 2 line segment, 6 5 general curve , 6 linear transformation , 40, 14 1 generators, 21 2 Liouville's theorem, 7 9 gro*up, 40 Lipschitz' condition , 5 6 Hadamard's thre e circle s theorem, 9 0 little Picar d theorem , 16 7 harmonic function , 8 2 logarithmic branc h point , 12 8 Harnack inequality , 8 6 logarithmic differential , 9 9 hermitian metric , 13 7 loxodromic, 4 8 hermitian pseudo-metric , 13 7 majorant, 2 4 holomorphic, 49 , 87, 13 6 majorant test , 2 4 holomorphic differential , 8 8 maximum function , 19 8 holomorphic function , 13 6 maximum principle , 7 9 holomorphic loca l coordinate, 13 5 mean valu e theorem, 81 , 84 holomorphic transformation , 13 7 meromorphic differential , 9 5 holomorphic transformatio n group , 13 7 meromorphic function , 9 4 homeomorphism, 39, 13 1 Mittag-Leffler's theorem , 18 5 homotopic, 6 3 Mobius transformation , 4 0 homotopic t o a point, 6 3 modular group , 4 6 homotopy, 6 3 monodromy theorem , 12 5 homotopy class , 6 3 Montel, 16 8 Hurwitz's theorem , 10 1 Morera's theorem , 7 7 hyperbolic, 48 , 147 , 14 9 multi-valued, 7 5 hyperbolic distance , 14 5 mutual reflections , 42 , 4 3 hyperbolic geodesic , 14 6 natural boundary , 19 1 hyperbolic length , 145 , 14 9 neighborhood, 4 hyperbolic metric , 14 9 negative variation, 19 3 identity theorem , 2 9 Nevanlinna's defec t relation , 20 2 imaginary unit , 1 Nevanlinna's firs t mai n theorem , 19 7 indefinite integral , 7 2 Nevanlinna's inequality , 19 8 infinite product , 13 , 34 non-Euclidean, 147 infinite produc t o f functions , 3 6 non-singular, 7 , 12 1 infinity, 3 8 normal, 16 8 initial point , 7 normal family , 15 3 injective, 1 3 north pole , 3 7 interior point , 5 number o f zeros of / surrounde d b y C, 10 1 inverse, 1 ^-invariant, 21 4 inverse functio n theorem , 10 3 1-valued, 7 5 isolated essentia l singularity , 93 , 17 1 open, 4 isolated point , 4 open covering , 5 Jensen's formula , 19 4 open mappin g theorem , 10 3 Jordan curve , 7 open se t in , 5 Jordan's theorem , 6 6 orbit, 13 2 Koebe's function , 17 5 order, 93 , 20 0 lambda function , 16 7 order function , 197 INDEX 247 order of tu-point, 10 2 Riemann surfac e o f th e invers e function , order o f the branch , 12 8 127 orientation, 8 Riemann's extensio n theorem , 11 7 parabolic, 4 8 Riemann's zet a function , 20 9 parameter, 6 Riemann-Stieltjes' integral , 19 3 parameter change , 6 ring domain, 9 1 partial sum , 12 , 2 3 rotation number , 76 , 9 6 partition, 5 4 Rouche's theorem , 10 1 partition point , 5 4 Runge increasin g covering, 18 4 period, 21 0 Runge's theorem , 18 3 period group , 21 0 Schwarz' reflection principle , 11 9 period parallelogram , 21 2 Schwarz-Christoffel's formula , 17 6 Picard exceptional value , 20 1 Schwarz-Pick's lemma , 15 1 piecewise continuousl y differentiable , 7 Schwarzian derivative , 17 5 piecewise linea r curve (Streckenzug) , 9, 6 5 sequence, 9 Poincare metric , 139 , 144* 148 sequence o f complex functions , 1 9 point a t infinity , 3 8 series, 1 2 Poisson integral , 8 2 series of functions , 2 3 Poisson kernel , 8 2 series o f order change , 1 3 polar coordinate , 3 simple curve , 7 pole, 9 3 simply connected , 6 3 positive orientation , 157 , 16 2 south pole , 3 7 positive variation , 19 3 special linea r group , 4 0 power series, 2 5 star-shaped, 6 3 power serie s expansion, 2 8 stereographic projection , 3 8 preserve the orientation , 15 7 Stolz' domain, 4 6 primitive function , 62 , 8 8 subgroup, 4 3 principal congruenc e subgroup , 4 6 subsequence, 1 0 principal part , 18 5 sum, 8 principal value , 10 8 surjective, 1 3 principle o f th e permanenc e o f th e func - Taylor series , 2 8 tional relation , 13 0 terminal point , 7 Prinzip vo n de r Gebietstreu e (ope n map - topology, 4 ping theorem), 10 3 total variation , 19 2 projective specia l linea r group , 4 1 transcendental, 119 , 17 4 proximity function , 19 7 transitively, 4 4 Puiseux series , 10 5 trigonometric functions , 3 1 Puiseux serie s expansion, 10 6 uniformization o f Riemann surfaces , 13 6 pull-back, 13 7 uniformization theorem , 15 6 purely imaginary , 2 uniformly bounded , 2 0 radius o f convergence, 2 5 uniformly continuous , 1 8 ratio o f the circumferenc e o f a circle to it s uniqueness o f analyti c continuation , 11 7 diameter, 3 3 unit circle , 5 rational function , 9 4 unit disk , 3 refinement, 5 5 univalent, 10 2 reflection, 4 2 universal covering , 13 2 reflection points , 12 2 universal coverin g mapping , 13 2 reflection principle , 4 3 upper hal f plane , 4 5 relative topology , 5 variation, 192 , 19 3 relatively compact , 1 2 w-point, 10 2 removable singularity , 11 8 Weierstrass' canonica l form , 22 0 residue, 9 5 Weierstrass' irreducible factor , 20 3 Riemann mappin g theorem , 15 4 Weierstrass' M-test , 2 4 Riemann sphere , 37 , 39 Weierstrass' pe-function, 21 7 Riemann surface , 127 , 135 Weierstrass' product , 202 , 205 248 INDEX
Weierstrass' theorem , 18 9 zero, 9 3 Symbols
G,9 elemen t d(z; &D) boundary distance , 2 2 U union 5L(2,C), 4 0 fl intersectio n P5L(2,C),41 N natura l number s (positiv e inte - Aut(A(l)), 4 3 gers) Aut(C), 4 3 Z integer s H = {z G C;Im z > 0 } upper hal f Z+ non-negativ e integer s plane, 4 5 Q rationa l number s Aut(H), 4 5 R rea l number s SX(2;R), 4 5 C comple x numbers, 1 5L(2,Z),46 c* = c \ {0} r(n), 4 6 Aut( • ) automorphism grou p <9* = <9/<9z , 50 max maximu m dx = a/a*, so min minimu m (d), 5 4 mod congruenc e L(C; (<*)), 54 [x] = max{ n G Z,n ^ x} Gauss ' |(d)|, 5 4 symbol £(C) lengt h o f curve, 5 5 n ( ) = n !
^W, 19 3 I log+ x = log max{x, 1}, 19 6 m(r,/), 19 7 n(r,f), 19 7 N(r,f), 19 7 r(r,/),197 M(r, /) maximu m function, 19 8 Pf order , 20 0 E(z\p), 20 3 IT* 207 T(z) Gamm a function, 20 7 C(z) Riemann's zeta function, 20 9 Q[vi,W2] perio d parallelogram, 211, 212 ^[^1,^2] lattice , 21 2 OL(2,Z), 213 deg/, 21 4 £', 21 6 p(z) Weierstrass ' p (pe ) function , 217 Correction Lis t Introduction t o Comple x Analysi s (Version 1998 ) p. 6 , T 8 : continuou s functio n => increasin g continuou s functio n p. 17 , T 5 : < 0 => < e p. 28 , i n FIGUR E 12 : |b | == » | 6 - a\ p. 29 , 1 8 - 9 : |6 | =* \b ~ a\ ( 2 places ) p. 29 , | 11:2 } => a domain D p. 30 , t 11 : e . => bas e e.
P- 32 , | 10 : ( 2n_2) => (4n-2 ) p. 32 , | 6 : mea n => intermediat e p. 36 , t 14 : serie s == > sequence p. 49 , 1 12: /(a + z ) - f(z) = * /( a + h ) - /(a ) p. 55 , I 20 : (0^) - 0fe-i)) 2 = > (Mtj) ~ ^litj-i)) 2 p. 57 , T 5 ~ 9 : tj^ - 1 ==> ^>-i ( 3 places ) p. 66 , t 1 - homotopi c to ==> a change o f parameter o f p. 69 , |8: \a\ => |a-z 0| p. 72 , | 13,15 : zi => z (2places ) p. 73 , 1 3 \a — w\. Th e =^ > \a — w\. Le t / b e a holomorphic functiono n o n D. Th e p. 76 , T 2 (*2), => fe)<0, p. 77 , | 1 ^Ife-i^j] = > V#j- i +e,tj -e ] p. 83 , 1 5 dz = » 2d 2 p. 85 , T 7 Delete " ft(e w) " p. 95 , 1 7 P- 05, T 13 : y^ = > ^/a~^ P- 10, In FIGUR E 44 : y = > 1 P- 14, | 3 : 2T T => 2TT Z P- 24, | 4 : polynomia l =4 > rational functio n P- 29, | 10 : functio n = > function s P- 30, T 3 : f : D ^ D => f : D -^ D' P- °y' ^ U ' a(z ) ^ 4a(z ) P- 59, T 10 ~ 9: an injective => a bijective P- 73, 1 9: a =* (3 P- 75, I 18 : (f) = > (f ) P- 76, T 1 : l\/27 r =» 2\/27 r P- 80, 1 9,13: g = » && ( 2 places ) P- 83, 1 19: D => C P- 86, | 4 : / = » A p. 207 , t 10 : n\ =» ( n - 1) ! p. 217 , T 13: p{z) 2 = » p'(z) 2 p. 218 , t 1 : z 2 => di p. 219 , | 6 : polynomial s =$> rationa l function s p. 229 , T 12 : v^/ = > y/y/2
251 Selected Title s i n Thi s Serie s (Continued from the front of this publication)
129 Mitsu o Morimoto , A n introductio n t o Sato' s hyperfunctions , 199 3 128 V . P . Orevkov , Complexit y o f proof s an d thei r transformation s i n axiomati c theories , 1993 127 F . L . Zak , Tangent s an d secant s o f algebraic varieties , 199 3 126 M . L . Agranovskff , Invarian t functio n space s on homogeneou s manifold s o f Li e group s and applications , 199 3 125 Masayosh i Nagata , Theor y o f commutative fields, 199 3 124 Masahis a Adachi , Embedding s an d immersions , 199 3 123 M . A . Akivi s an d B . A . Rosenfeld , Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou , Theor y o f entire an d meromorphi c functions : deficien t an d asymptotic value s an d singula r directions , 199 3 121 LB . Fesenk o an d S . V . Vostokov , Loca l fields an d thei r extensions : A constructiv e approach, 199 3 120 Takeyuk i Hid a an d Masuyuk i Hitsuda , Gaussia n processes , 199 3 119 M . V . Karase v an d V . P . Maslov , Nonlinea r Poisso n brackets . Geometr y an d quantization, 199 3 118 Kenkich i Iwasawa , Algebrai c functions , 199 3 117 Bori s Zilber , Uncountabl y categorica l theories , 199 3 116 G . M . Fel'dman , Arithmeti c o f probability distributions , an d characterizatio n problem s on abelia n groups , 199 3 115 Nikola i V . Ivanov , Subgroup s o f Teichmiiller modula r groups , 199 2 114 Seiz o Ito , Diffusio n equations , 199 2 113 Michai l Zhitomirskif , Typica l singularitie s o f differential 1-form s an d Pfaffia n equations , 1992 112 S . A . Lomov , Introductio n t o the genera l theor y o f singular perturbations , 199 2 111 Simo n Gindikin , Tub e domain s an d th e Cauch y problem , 199 2 110 B . V . Shabat , Introductio n t o comple x analysi s Par t II . Function s o f several variables , 1992 109 Isa o Miyadera , Nonlinea r semigroups , 199 2 108 Take o Yokonuma , Tenso r space s an d exterio r algebra , 199 2 107 B . M . Makarov , M . G . Goluzina , A . A . Lodkin , an d A . N . Podkorytov , Selecte d problems i n rea l analysis , 199 2 106 G.-C . Wen , Conforma l mapping s an d boundar y valu e problems , 199 2 105 D . R . Yafaev , Mathematica l scatterin g theory : Genera l theory , 199 2 104 R . L . Dobrushin , R . Kotecky , an d S . Shlosman , Wulf f construction : A global shap e from loca l interaction, 199 2 103 A . K . Tsikh , Multidimensiona l residue s an d thei r applications , 199 2 102 A . M . Il'in , Matchin g o f asymptotic expansion s o f solution s o f boundary valu e problems , 1992 101 Zhan g Zhi-fen , Din g Tong-ren , Huan g Wen-zao , an d Don g Zhen-xi , Qualitativ e theory o f differential equations , 199 2 100 V . L . Popov , Groups , generators , syzygies , and orbit s i n invarian t theory , 199 2 99 Nori o Shimakura , Partia l differentia l operator s o f elliptic type , 199 2 98 V . A . Vassiliev , Complement s o f discriminants o f smooth maps : Topolog y an d applications, 199 2 (revise d edition , 1994 ) 97 Itir o Tamura , Topolog y o f foliations : A n introduction , 199 2 96 A . I . Markushevich , Introductio n t o th e classica l theor y o f Abelia n functions , 199 2 95 Guangchan g Dong , Nonlinea r partia l differentia l equation s o f secon d order , 199 1 94 Yu . S . Il'yashenko , Finitenes s theorem s fo r limi t cycles , 199 1 93 A . T . Fomenk o an d A . A . Tuzhilin , Element s o f the geometr y an d topolog y o f minimal surface s i n three-dimensional space , 199 1 (See the AM S catalo g fo r earlie r titles )