Function Theory of One Complex Variable
Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples and exercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $H^p$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.
Comentarios de la gente - Escribir un comentario
No encontramos ningún comentario en los lugares habituales.
Complex Line Integrals
Applications of the Cauchy Integral
Meromorphic Functions and Residues
The Zeros of a Holomorphic Function
Holomorphic Functions as Geometric Mappings
Infinite Series and Products
Applications of Infinite Sums and Products
Rational Approximation Theory
Special Classes of Holomorphic Functions
The Prime Number Theorem
The Statement and Proof of Goursats Theorem
analytic continuation boundary bounded Calculate Cauchy integral formula Cauchy–Riemann equations Chapter circle closed curve compact sets compact subsets complex numbers conformal mapping conformally equivalent connected open set constant continuous function converges uniformly Corollary defined definition derivative differentiable domain entire function essential singularity example Exercise f is holomorphic fact Figure finite fixed follows function element function f harmonic function hence Hint holomorphic function holomorphically simply connected homotopic Laurent series Lemma Let f Let U C C linear fractional transformation meromorphic function neighborhood nonzero ÖD(P one-to-one open set piecewise pole polynomial proof of Theorem Proposition Prove that f removable singularity result Riemann mapping theorem sequence simply connected ſº subharmonic Suppose that f topological uniformly on compact unit disc vanishes zeros of f