CONTENTS 15

55.4 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . .174755.4.1 Regular And Singular Points . . . . . . . . . . . . . . . . . . .174755.4.2 Continuation Along A Curve . . . . . . . . . . . . . . . . . . .1749

55.5 The Picard Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . .175055.5.1 Two Competing Lemmas . . . . . . . . . . . . . . . . . . . . .175155.5.2 The Little Picard Theorem . . . . . . . . . . . . . . . . . . . . .175555.5.3 Schottky’s Theorem . . . . . . . . . . . . . . . . . . . . . . . .175555.5.4 A Brief Review . . . . . . . . . . . . . . . . . . . . . . . . . .176055.5.5 Montel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .176155.5.6 The Great Big Picard Theorem . . . . . . . . . . . . . . . . . .1763

55.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1764

56 Approximation By Rational Functions 176756.1 Runge’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1767

56.1.1 Approximation With Rational Functions . . . . . . . . . . . . .176756.1.2 Moving The Poles And Keeping The Approximation . . . . . . .176856.1.3 Merten’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . .176956.1.4 Runge’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .1773

56.2 The Mittag-Leffler Theorem . . . . . . . . . . . . . . . . . . . . . . . . .177656.2.1 A Proof From Runge’s Theorem . . . . . . . . . . . . . . . . . .177656.2.2 A Direct Proof Without Runge’s Theorem . . . . . . . . . . . .177756.2.3 Functions Meromorphic On Ĉ . . . . . . . . . . . . . . . . . . .177856.2.4 Great And Glorious Theorem, Simply Connected Regions . . . .1779

56.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1782

57 Infinite Products 178557.1 Analytic Function With Prescribed Zeros . . . . . . . . . . . . . . . . . .178857.2 Factoring A Given Analytic Function . . . . . . . . . . . . . . . . . . . .1794

57.2.1 Factoring Some Special Analytic Functions . . . . . . . . . . . .179557.3 The Existence Of An Analytic Function With Given Values . . . . . . . .179757.4 Jensen’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .180157.5 Blaschke Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1804

57.5.1 The Müntz-Szasz Theorem Again . . . . . . . . . . . . . . . . .180657.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1808

58 Elliptic Functions 181758.1 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1818

58.1.1 The Unimodular Transformations . . . . . . . . . . . . . . . . .182258.1.2 The Search For An Elliptic Function . . . . . . . . . . . . . . .182558.1.3 The Differential Equation Satisfied By ℘ . . . . . . . . . . . . .182758.1.4 A Modular Function . . . . . . . . . . . . . . . . . . . . . . . .182958.1.5 A Formula For λ . . . . . . . . . . . . . . . . . . . . . . . . . .183658.1.6 Mapping Properties Of λ . . . . . . . . . . . . . . . . . . . . .183858.1.7 A Short Review And Summary . . . . . . . . . . . . . . . . . .1846

58.2 The Picard Theorem Again . . . . . . . . . . . . . . . . . . . . . . . . .1850