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An Introduction to Complex Analysis and Geometry

An Introduction to Complex Analysis and GeometryJohn P. D AngeloDept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 2009 by John P. D AngeloContentsChapter 1. From the real numbers to the Complex numbers111. Introduction112. Number systems113. Inequalities and ordered fields164. The Complex numbers245. Alternative definitions ofC266. A glimpse at metric spaces30 Chapter 2. Complex numbers351. Complex conjugation352. Existence of square roots373. Limits394. Convergent infinite series415. Uniform convergence and consequences446. The unit circle and trigonometry507.

Chapter 8 introduces additional appealing topics such as the fundamental the-orem of algebra (for which we give three proofs), winding numbers, Rouche’s theo-rem, Pythagorean triples, conformal mappings, the quaternions, and (a brief men-tion of) complex analysis in higher dimensions. The section on conformal mappings

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