This text on complex variables is geared toward graduate students and undergraduates who have taken an introductory course in real analysis. It is a substantially revised and updated edition of the popular text by Robert B. Ash, offering a concise treatment that provides careful and complete explanations as well as numerous problems and solutions.
An introduction presents basic definitions, covering topology of the plane, analytic functions, real-differentiability and the Cauchy-Riemann equations, and exponential and harmonic functions. Succeeding chapters examine the elementary theory and the general Cauchy theorem and its applications, including singularities, residue theory, the open mapping theorem for analytic functions, linear fractional transformations, conformal mapping, and analytic mappings of one disk to another. The Riemann mapping theorem receives a thorough treatment, along with factorization of analytic functions. As an application of many of the ideas and results appearing in earlier chapters, the text ends with a proof of the prime number theorem.
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The General Cauchy Theorem
Applications of the Cauchy Theory
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A(fl analytic function analytic logarithm analytic on ft apply Assume bounded Cauchy-Riemann equations Cauchy's integral formula Cauchy's theorem circle closed curve closed path compact subsets complex numbers constant on ft continuous argument continuous function converges absolutely converges uniformly D(zq defined entire function equivalent exp(z f(zo finite function g g is analytic half plane harmonic functions hence hypothesis identity theorem implies integral formula Lemma Let ft limit point linear fractional transformation maximum principle obtain one-to-one analytic map open set open subset point in ft pole of order polygonal polynomial power series prime number theorem Problem Proof Prove radius of convergence rational function real numbers removable singularity residue result follows Riemann mapping theorem Rouche's theorem Runge's theorem sequence zn series converges simply connected subsets of ft Theorem Let Theorem Suppose uniformly on compact zero of order zo)n