Complex Made Simple

Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from? In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work. Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains.Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem. This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis.

Table of Contents:
Part 1. Complex made simple: Differentiability and Cauchy-Riemann equations Power series Preliminary results on holomorphic functions Elementary results on holomorphic functions Logarithms, winding numbers and Cauchy's theorem Counting zeroes and the open mapping theorem Euler's formula for $\sin(z)$ Inverses of holomorphic maps Conformal mappings Normal families and the Riemann mapping theorem Harmonic functions Simply connected open sets Runge's theorem and the Mittag-Leffler theorem The Weierstrass factorization theorem Caratheodory's theorem More on$\mathrm{Aut}(\mathbb{D})$ Analytic continuation Orientation The modular function Preliminaries for the Picard theorems The Picard theorems Part 2. Further results: Abel's theorem More on Brownian motion More on the maximum modulus theorem The Gamma function Universal covering spaces Cauchy's theorem for non-holomorphic functions Harmonic conjugates Part 3. Appendices: Complex numbers Complex numbers, continued Sin, cos and exp Metric spaces Convexity Four counterexamples The Cauchy-Riemann equations revisited References Index of notations Index.