Classical Topics in Complex Function Theory

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1 Reinhold Remmert Classical Topics in Complex Function Theory Translated by Leslie Kay With 19 Illustrations Springer

2 Preface to the Second German Edition Preface to the First German Edition Acknowledgments Advice to the reader vii vii x x A Infinite Products and Partial Fraction Series 1 1 Infinite Products of Holomorphic Functions 3 1. Infinite Products 4 1. Infinite products of numbers 4 2. Infinite products of functions 6 2. Normal Convergence 7 1. Normal convergence 7 2. Normally convergent products of holomorphic functions 9 3. Logarithmic differentiation The Sine Product sint^z = -KZ\^=X (\-z 1 jv 2 ) Standard proof Characterization of the sine by the duplication formula Proof of Euler's formula using Lemma *. Proof of the duplication formula for Euler's product, following Eisenstein On the history of the sine product 17

3 xii Contents 4*. Euler Partition Products Partitions of natural numbers and Euler products Pentagonal number theorem. Recursion formulas for p(n) and a(n) Series expansion of n^li(l + Q v ~ z ) m powers of z On the history of partitions and the pentagonal number theorem 24 5*. Jacobi's Product Representation of the Series /(*,?):= ~-oo</ 1 ' 2 * 1 ' 1. Jacobi's theorem Discussion of Jacobi's theorem On the history of Jacobi's identity. 28 Bibliography 30 2 The Gamma Function The Weierstrass Function A(z) = ze< z rl>i(l + z/v)e- z / v The auxiliary function #(*):= *n7=i(l + */")e-* / " The entire function A(z) := e lz H{z) The Gamma Function Properties of the F-function Historical notes The logarithmic derivative The uniqueness problem Multiplication formulas 45 6*. Holder's theorem 46 7*. The logarithm of the F-function Euler's and Hankel's Integral Representations of T(z) Convergence of Euler's integral Euler's theorem 51 3*. The equation 52 4*. Hankel's loop integral Stirling's Formula and Gudermann's Series Stieltjes's definition of the function/i(z) Stirling's formula Growth of \T(x + iy)\ for \y\ -> oo 59 4*. Gudermann's series 60 5*. Stirling's series 61 6*. Delicate estimates for the remainder term 63 7*. Binet's integral 64 8*. Lindelof's estimate The Beta Function Proof of Euler's identity Classical proofs of Euler's identity 69 Bibliography 70 25

4 xiii 3 Entire Functions with Prescribed Zeros The Weierstrass Product Theorem for C Divisors and principal divisors Weierstrass products Weierstrass factors The Weierstrass product theorem Consequences On the history of the product theorem Discussion of the Product Theorem Canonical products Three classical canonical products The CT-function The p-function 85 5*. An observation of Hurwitz 85 Bibliography '. 86 4* Holomorphic Functions with Prescribed Zeros The Product Theorem for Arbitrary Regions Convergence lemma The product theorem for special divisors The general product theorem Second proof of the general product theorem Consequences Applications and Examples Divisibility in the ring O(G). Greatest common divisors Examples of Weierstrass products On the history of the general product theorem Glimpses of several variables Bounded Functions on E and Their Divisors Generalization of Schwarz's lemma Necessity of the Blaschke condition Blaschke products Bounded functions on the right half-plane 102 Appendix to Section 3: Jensen's Formula 102 Bibliography _ Iss'sa's Theorem. Domains of Holomorphy Iss'sa's Theorem Bers's theorem Iss'sa's theorem Proof of the lemma Historical remarks on the theorems of Bers and Iss'sa 110 5*. Determination of all the valuations on M(G)... Ill

5 2. Domains of Holomorphy A construction of Goursat Well-distributed boundary sets. First proof of the existence theorem Discussion of the concept of domains of holomorphy Peripheral sets. Second proof of the existence theorem On the history of the concept of domains of holomorphy Glimpse of several variables Simple Examples of Domains of Holomorphy Examples for E Lifting theorem Cassini regions and domains of holomorphy 122 Bibliography 123 Functions with Prescribed Principal Parts Mittag-Leffler's Theorem for C Principal part distributions Mittag-Leffier series Mittag-Leffler's theorem Consequences Canonical Mittag-Leffier series. Examples On the history of Mittag-Leffler's theorem for C Mittag-Leffler's Theorem for Arbitrary Regions Special principal part distributions Mittag-Leffler's general theorem 132! 3. Consequences... \ On the history of Mittag-Leffler's general theorem Glimpses of several variables 135 3*. Ideal Theory in Rings of Holomorphic Functions Ideals in O{G) that are not finitely generated Wedderburn's lemma (representation of 1) Linear representation of the gcd. Principal ideal theorem Nonvanishing ideals Main theorem of the ideal theory of O(G) On the history of the ideal theory of holomorphic functions Glimpses of several variables 141 Bibliography 142

6 xv B Mapping Theory The Theorems of Montel and Vitali Montel's Theorem Montel's theorem for sequences Proof of Montel's theorem Montel's convergence criterion Vitali's theorem 150 5*. Pointwise convergent sequences of holomorphic functions Normal Families Montel's theorem for normal families Discussion of Montel's theorem On the history of Montel's theorem 154 4*. Square-integrable functions and normal families *. Vitali's Theorem Convergence lemma Vitali's theorem (final version) On the history of Vitali's theorem 158 4*. Applications of Vitali's theorem Interchanging integration and differentiation Compact convergence of the F-integral Miintz's theorem Consequences of a Theorem of Hurwitz 162 Bibliography The Riemann Mapping Theorem Integral Theorems for Homotopic Paths Fixed-endpoint homotopic paths Freely homotopic closed paths Null homotopy and null homology Simply connected domains 171 5*. Reduction of the integral theorem 1 to a lemma *. Proof of Lemma 5* The Riemann Mapping Theorem Reduction to Q-domains Existence of holomorphic injections Existence of expansions Existence proof by means of an extremal principle On the uniqueness of the mapping function Equivalence theorem On the History of the Riemann Mapping Theorem Riemann's dissertation Early history From Caratheodory-Koebe to Fejer-Riesz

7 xvi Contents 4. Caratheodory's final proof Historical remarks on uniqueness and boundary behavior Glimpses of several variables Isotropy Groups of Simply Connected Domains Examples The group Aut a G for simply connected domains G/.C 189 3*. Mapping radius. Monotonicity theorem 189 Appendix to Chapter 8: Caratheodory-Koebe Theory Simple Properties of Expansions Expansion lemma Admissible expansions. The square root method *. The crescent expansion The Caratheodory-Koebe Algorithm Properties of expansion sequences Convergence theorem Koebe families and Koebe sequences Summary. Quality of convergence Historical remarks. The competition between Caratheodory and Koebe The Koebe Families K m and K.^ A lemma The families /C m and /C^ 199 Bibliography for Chapter 8 and the Appendix Automorphisms and Finite Inner Maps Inner Maps and Automorphisms Convergent sequences in Hoi G and Aut G Convergence theorem for sequences of automorphisms Bounded homogeneous domains 205 4*. Inner maps of H and homotheties Iteration of Inner Maps Elementary properties H. Cartan's theorem The group Aut a G for bounded domains The closed subgroups of the circle group 209 5*. Automorphisms of domains with holes. Annulus theorem Finite Holomorphic Maps Three general properties Finite inner maps of E Boundary lemma for annuli 213

8 xvii 4. Finite inner maps of annuli Determination of all the finite maps between annuli 216 4*. Rado's Theorem. Mapping Degree Closed maps. Equivalence theorem Winding maps Rado's theorem Mapping degree Glimpses 221 Bibliography 221 C Selecta The Theorems of Bloch, Picard, and Schottky Bloch's Theorem Preparation for the proof Proof of Bloch's theorem 227 3*. Improvement of the bound by the solution of an extremal problem 228 4*. Ahlfors's theorem 230 5*. Landau's universal constant Picard's Little Theorem Representation of functions that omit two values Proof of Picard's little theorem Two amusing applications Schottky's Theorem and Consequences Proof of Schottky's theorem Landau's sharpened form of Picard's little theorem Sharpened forms of Montel's" and Vitali's theorems Picard's Great Theorem Proof of Picard's great theorem On the history of the theorems of this chapter Bibliography Boundary Behavior of Power Series Convergence on the Boundary Theorems of Fatou, M. Riesz, and Ostrowski A lemma of M. Riesz Proof of the theorems in A criterion for nonextendibility 248 Bibliography for Section Theory of Overconvergence. Gap Theorem 249

9 xviii Contents 1. Overconvergent power series Ostrowski's overconvergence theorem Hadamard's gap theorem Porter's construction of overconvergent series On the history of the gap theorem On the history of overconvergence Glimpses 255 Bibliography for Section A Theorem of Fatou-Hurwitz-Polya Hurwitz's proof Glimpses 259 Bibliography for Section An Extension Theorem of Szego Preliminaries for the proof of (Sz) A lemma Proof of (Sz) An application Glimpses 265 Bibliography for Section Runge Theory for Compact Sets Techniques Cauchy integral formula for compact sets Approximation by rational functions Pole-shifting theorem Runge Theory for Compact Sets Runge's approximation theorem Consequences of Runge's little theorem Main theorem of Runge theory for compact sets Applications of Runge's Little Theorem Pointwise convergent sequences of polynomials that do not converge compactly everywhere Holomorphic imbedding of the unit disc in C Discussion of the Cauchy Integral Formula for Compact Sets Final form of Theorem Circuit theorem 285 Bibliography Runge Theory for Regions Runge's Theorem for Regions Filling in compact sets. Runge's proof of Mittag-Leffler's theorem Runge's approximation theorem Main theorem of Cauchy function theory 292

10 xix 4. On the theory of holes On the history of Runge theory Runge Pairs Topological characterization of Runge pairs Runge hulls Homological characterization of Runge hulls. The Behnke-Stein theorem Runge regions 298 5*. Approximation and holomorphic extendibility Holomorphically Convex Hulls and Runge Pairs Properties of the hull operator Characterization of Runge pairs by means of holomorphically convex hulls 302 Appendix: On the Components of Locally Compact Spaces. Sura- Bura's Theorem Components Existence of open compact sets Filling in holes Proof of Sura-Bura's theorem 305 Bibliography Invariance of the Number of Holes Homology Theory. Separation Lemma Homology groups. The Betti number Induced homomorphisms. Natural properties Separation of holes by closed paths Invariance of the Number of Holes. Product Theorem for Units On the structure of the homology groups The number of holes and the Betti number Normal forms of multiply connected domains (report) On the structure of the multiplicative group O(G) X Glimpses 318 Bibliography 318 Short Biographies 321 Symbol Index 329 Name Index 331 Subject Index 337

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