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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. Complex numbers and geometry591. Lines, circles, and balls592. Analytic geometry623. Quadratic polynomials634. Linear fractional transformations695. The Riemann sphere73 Chapter 4. Power series expansions751. Geometric series752. The radius of convergence783.

An Introduction to Complex Analysis and Geometry John P. D’Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 61801

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