Elementary Theory of Analytic Functions of One or Several Complex Variables

CHAPTER I. POWER SERIES IN ONE VARIABLE I. Formal power series 2. Convergent power series 3. Logarithmic and exponential functions 4. Analytic functions of one variable Exercises CHAPTER II. HOLOMORPHIC FUNCTIONS; CAUCHY'S INTEGRAL I. Curvilinear integrals; primitive of a closed form 2. Holomorphic functions; fundamental theorems Exercises CHAPTER III. TAYLOR. AND LAURENT EXPANSIONS I. Cauchy's inequalities; Liouville's theorem 2. Mean value property and the maximum modulus principle 3. Schwarz' lemma 4. Laurent's expansion 5. Introduction of the point at infinity. Residue theorem 6. Evaluation of integrals by the method of residues Exercises CHAPTER IV. ANALYTIC FUNCTIONS OF SEVERAL VARIABLES; HARMONIC I. Power series in several variables 2. Analytic functions 3. Harmonic functions of two real variables 4. Poisson's formula; Dirichlet's problem 5. Holomorphic functions of several complex variables Exercises "CHAPTER V. CONVERGENCE OF SEQUENCES OF HOLOMORPHIC OR MEROMORPHIC FUNCTIONS ; SERIES, INFINITE PRODUCTS ; NORMAL FAMILIES" I. Topology of the space C(D) 2. Series of meromorphic functions 3. Infinite products of holomorphic functions 4. Compact subsets of H(D) Exercises CHAPTER VI. HOLOMORPHIC TRANSFORMATIONS I. General theory ; examples 2. "Conformal representation ; automorphisms of the plane, the Riemann sphere, the open disc" 3. Fundamental theorem of conformal representation 4. Concept of complex manifold ; integration of differential forms 5. Riemann surfaces Exercises CHAPTER VII. HOLOMORPHIC SYSTEMS OF DIFFERENTIAL EQUATIONS I. Existence and uniqueness theorem 2. Dependence on parameters and on initial conditions 3. Higher order differential equations Exercises SOME NUMERICAL OR QUANTITATIVE ANSWERS TERMINOLOGICAL INDEX NOTATIONAL INDEX