MTH 562 Complex Analysis

 Department of Mathematics, University of Rhode Island
 Fall Semester, 2000
 Prof. Orlando Merino - Tyler Hall 220 - 874 4442 -

Class Notes


 In this course we shall develop the theory of complex functions in a mathematically rigorous way. The prerequisites are a course such as undergraduate analysis MTH 435/436 or equivalent. 
 Evaluation is based on homework (75%) and a final -take home- exam (25 %).
 The text is Functions of One Complex Variable, by John B. Conway (Second Ed.)
 We shall cover chapters I--V of the text, and sections 1,2 of VI if time permits:

The complex number system (the complex plane, roots of complex numbers, the extended plane and stereographic projection), Metric Spaces and the topology of C (compactness, continuity and uniform convergence) Elementary properties of analytic functions (power series, analytic functions, Cauchy Riemann Equations, branches of the logarithm, linear fractional transformations, conformal functions), complex integration (Riemann Stieltjes integrals, zeros of analytic functions, Liouville's Theorem, The Fundamental Theorem of Algebra, the index of a closed curve, Cauchy's theorem, the open mapping theorem, Goursat's theorem), Singularities (Laurent expansions, classification of singularities, Casorati-Weierstrass Theorem, residues, the argument principle, Rouche's Theorem). The Maximum Modulus Theorem and Schwarz's lemma.