Complex Analysis [Hardcover]

An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic. Throughout, exercises range from the very simple to the challenging. The book is based on lectures given by the author at several universities, including UCLA, Brown University, La Plata, Buenos Aires, and the Universidad Autonomo de Valencia, Spain.The book provides an introduction to complex analysis for students withsome familiarity with complex numbers from high school. The bookconsists of three parts. The first part comprises the basic core of acourse in complex analysis for junior and senior undergraduates. Thesecond part includes various more specialized topics as the argumentprinciple, the Schwarz lemma and hyperbolic geometry, the Poissonintegral, and the Riemann mapping theorem. The third part consists ofa selection of topics designed to complete the coverage of allbackground necessary for passing PhD qualifying exams in complexanalysis. Topics selected include Julia sets and the Mandelbrot set,Dirichlet series and the prime number theorem, and the uniformizationtheorem for Riemann surfaces. The three geometries, spherical,euclidean, and hyperbolic, are stressed. Exercises range from the verysimple to the quite challenging, in all chapters. The book is based onlectures given over the years by the author at several places,particularly the Interuniversity Summer School at Perugia (Italy), andalso UCLA, Brown University, Valencia (Spain), and La Plata(Argentina).lS$