Elementary Theory of Analytic Functions of One or Several Complex Variables Cartan Henri

Specifikacia Elementary Theory of Analytic Functions of One or Several Complex Variables Cartan Henri

CHAPTER I. POWER SERIES IN ONE VARIABLEI. Formal power series 2. Convergent power series 3. Logarithmic and exponential functions 4. Analytic functions of one variableExercisesCHAPTER II. HOLOMORPHIC FUNCTIONS; CAUCHY'S INTEGRALI. Curvilinear integrals; primitive of a closed form 2. Holomorphic functions; fundamental theoremsExercisesCHAPTER III. TAYLOR. AND LAURENT EXPANSIONSI. 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 residuesExercisesCHAPTER IV. ANALYTIC FUNCTIONS OF SEVERAL VARIABLES; HARMONICI. 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 variablesExercises "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)ExercisesCHAPTER VI. HOLOMORPHIC TRANSFORMATIONSI. 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 surfacesExercisesCHAPTER VII. HOLOMORPHIC SYSTEMS OF DIFFERENTIAL EQUATIONSI. Existence and uniqueness theorem 2. Dependence on parameters and on initial conditions 3. Higher order differential equationsExercisesSOME NUMERICAL OR QUANTITATIVE ANSWERSTERMINOLOGICAL INDEXNOTATIONAL INDEX