Complex AnalysisThe book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. It conists of sixteen chapters. The first eleven chapters 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 in the book include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The three geometries, spherical, euclidean, and hyperbolic, are stressed. Exercises range from the very simple to the quite challenging, in all chapters. The book is based on lectures given over the years by the author at several places, including UCLA, Brown University, the universities at La Plata and Buenos Aires, Argentina; and the Universidad Autonomo de Valencia, Spain. |
Contents
The Square and Square Root Functions | |
Trigonometric and Hyperbolic Functions | |
The CauchyRiemann Equations | |
Fractional Linear Transformations | |
Harmonic Conjugates | |
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analytic function annulus boundary bounded domain branch Cauchy integral Cauchy-Riemann equations closed path coefficients compact subset complex numbers component conformal map conformal self-map constant continuous function converges normally converges uniformly coordinate disk critical point decomposition defined derivative differentiable disk centered entire function Example Exercise extended complex plane finite number fixed point fractional linear transformation function f(z Green's function harmonic function Hence Hint hyperbolic identity interval inverse isolated singularity Julia set lemma Let f(z line segment maximum principle meromorphic function metric obtain open unit disk parameter piecewise smooth polynomial power series expansion proof punctured radius of convergence rational function real axis Riemann mapping theorem Riemann surface satisfies Section sector sequence series converges simple pole Suppose f(z Suppose that f(z tends unique unit circle upper half-plane vertical z-plane zeros