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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.

lytic function introduced in Chapter 5 all lead to the same class of functions. This material forms the basis for both the theory and application of complex analysis. In short, Chapter 5 considers derivatives and Chapter 6 considers integrals. Chapter 7 o ers many applications of the Cauchy theory to ordinary integrals.

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