Complex Analysis

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). A native of Minnesota, the author did his undergraduate work at YaleUniversity and his graduate work at UC Berkeley.

After spending sometime at MIT and at the Universidad Nacional de La Plata (Argentina), hejoined the faculty at UCLA in 1968. The author has published a numberof research articles and several books on functional analysis andanalytic function theory. he is currently involved in the CaliforniaK-12 education scene.