Elementary Theory of Analytic Functions of One or Several Complex Variables (eBook)

Henri Cartan (Autor)

eBook Download: EPUB

2013
240 Seiten
Dover Publications (Verlag)
9780486318677 (ISBN)

Infos

Basic treatment includes existence theorem for solutions of differential systems where data is analytic, holomorphic functions, Cauchy's integral, Taylor and Laurent expansions, more. Exercises. 1973 edition.

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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.

Inhalt

CHAPTER I. POWER SERIES IN ONE VARIABLE I. Formal power series2. Convergent power series3. Logarithmic and exponential functions 4. Analytic functions of one variable ExercisesCHAPTER II. HOLOMORPHIC FUNCTIONS; CAUCHY'S INTEGRAL I. Curvilinear integrals; primitive of a closed form2. 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' lemma4. Laurent's expansion5. 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 variables2. Analytic functions 3. Harmonic functions of two real variables 4. Poisson's formula; Dirichlet's problem5. 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 functions3. Infinite products of holomorphic functions4. Compact subsets of H(D) ExercisesCHAPTER VI. HOLOMORPHIC TRANSFORMATIONSI. General theory ; examples2. "Conformal representation ; automorphisms of the plane, the Riemann sphere, the open disc"3. Fundamental theorem of conformal representation4. Concept of complex manifold ; integration of differential forms5. Riemann surfaces ExercisesCHAPTER VII. HOLOMORPHIC SYSTEMS OF DIFFERENTIAL EQUATIONSI. Existence and uniqueness theorem2. Dependence on parameters and on initial conditions3. Higher order differential equations ExercisesSOME NUMERICAL OR QUANTITATIVE ANSWERSTERMINOLOGICAL INDEXNOTATIONAL INDEX

Produkt-Details

• Erscheint lt. Verlag: 22.4.2013
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• ISBN-13: 9780486318677 / 9780486318677