Problems and Solutions for Complex Analysis

I Complex Numbers and Functions.- I.1 Definition.- I.2 Polar Form.- I.3 Complex Valued Functions.- I.4 Limits and Compact Sets.- I.6 The Cauchy-Riemann Equations.- II Power Series.- II.1 Formal Power Series.- II.2 Convergent Power Series.- II.3 Relations Between Formal and Convergent Series.- II.4 Analytic Functions.- II.5 Differentiation of Power Series.- II.6 The Inverse and Open Mapping Theorems.- III Cauchy’s Theorem, First Part.- III.1 Holomorphic Functions on Connected Sets.- III.2 Integrals over Paths.- III.5 The Homotopy Form of Cauchy’s Theorem.- III.6 Existence of Global Primitives Definition of the Logarithm.- III.7 The Local Cauchy Formula.- IV Winding Numbers and Cauchy’s Theorem.- IV.2 The Global Cauchy Theorem.- V Applications of Cauchy’s Integral Formula.- V.1 Uniform Limits of Analytic Functions.- V.2 Laurent Series.- V.3 Isolated Singularities.- VI Calculus of Residues.- VI.1 The Residue Formula.- VI.2 Evaluation of Definite Integrals.- VII Conformal Mappings.- VII.2 Analytic Automorphisms of the Disc.- VII.3 The Upper Half Plane.- VII.4 Other Examples.- VII.5 Fractional Linear Transformations.- VIII Harmonic Functions.- VIII.1 Definition.- VIII.2 Examples.- VIII.3 Basic Properties of Harmonic Functions.- VIII.4 The Poisson Formula.- VIII.5 Construction of Harmonic Functions.- IX Schwarz Reflection.- IX.2 Reflection Across Analytic Arcs.- X The Riemann Mapping Theorema.- X.1 Statement of the Theorem.- X.2 Compact Sets in Function Spaces.- XI Analytic Continuation along Curves.- XI.1 Continuation Along a Curve.- XI.2 The Dilogarithm.- XII Applications of the Maximum Modulus Principle and Jensen’s Formula.- XII.1 Jensen’s Formula.- XII.2 The Picard-Borel Theorem.- XII.6 The Phragmen-Lindelof and Hadamard Theorems.- XIII Entire and Meromorphic Functions.- XIII.1 Infinite Products.- XIII.2 Weierstrass Products.- XIII.3 Functions of Finite Order.- XIII.4 Meromorphic Functions, Mittag-Leffler Theorem.- XV The Gamma and Zeta Functions.- XV.1 The Differentiation Lemma.- XV.2 The Gamma Function.- XV.3 The Lerch Formula.- XV.4 Zeta Functions.- XVI The Prime Number Theorem.- XVI.1 Basic Analytic Properties of the Zeta Function.- XVI.2 The Main Lemma and its Application.