Description
Book SynopsisPresents 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. This book is suitable for a first-year course in complex analysis.
Table of ContentsPart 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.
