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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. The Geometry of addition and multiplication538. Logarithms54 Chapter 3.

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 includes a brief discussion of non-Euclidean geometry. The section on quaternions

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