Elementary Theory of Analytic Functions of One or Several Complex Variables

Noted mathematician offers basic treatment of theory of analytic functions of a complex variable, touching on analytic functions of several real or complex variables as well as the existence theorem for solutions of differential systems where data is analytic. Also included is a systematic, though elementary, exposition of theory of abstract complex manifolds of one complex dimension. Topics include power series in one variable, holomorphic functions, Cauchy's integral, more. Exercises. 1973 edition.

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
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