Syllabus – Math 331 – Complex Analysis – Spring 2004 Instructor: William T. Ross – 215 Jepson Hall – (804) 289 – 8090 Text: Fundamentals of Complex Analysis for Mathematics, Science, and Engineering – Saff and Snider - Prentice Hall – Third Edition Topics: This is an introduction to the theory of functions of one-complex variable. Topics include: complex numbers, analytic functions, the Cauchy-Riemann equations, the complex exponential, logarithmic, power, and trigonometric functions, contour integration, Cauchy's integral formula, estimates of analytic functions, maximum modulus theorem, power series and Laurent series, zeros and singularities, the residue theory, applications of the residue theory, the argument principle, Rouche's theorem, Riemann mapping theorem, Mobius transformations. Homework: Problem sets will be collected every day. Students are allowed to help each other as long as everyone does their share. Problem sets are due by 5:00 PM the day they are due and late assignments cannot be accepted. Exams: There will be two in-class exams (Feb 19 and April 1) and a comprehensive final exam (April 26 – 2 5 PM). Grades: To be determined as follows: Exam 1 – 25 % Exam 2 – 25 % Problem sets – 20% Final Exam – 30% Schedule: We will do our best to stick to the following schedule. Daily assignments will be posted on Blackboard. T 1/3 Complex numbers 1.1 – 1.3 R 1/5 Complex powers and roots 1.4 – 1.5 T 1/20 Topology of the complex plane, complex-valued functions 1.6 – 1.7, 2.1 – 2.2 R 1/22 Analytic and harmonic functions 2.3 – 2.5 T 1/27 Exp and Trig functions 3.1 – 3.2 R 1/29 Complex log 3.3 T 2/3 – Complex powers and inverse trig functions 3.5 R 2/5 Contour integrals and complex anti derivatives 4.1 – 4.3 T 2/10 Cauchy integral formula 4.4 R 2/12 Cauchy integral formula 4.5

T 2/17 Max-Mod and Liouville’s theorem 4.6 R 2/19 Exam 1 T 2/24 Sequences and series of complex numbers 5.1, 5.4 R 2/26 Sequences and series of complex functions 5.1, 5.4 T 3/2 Taylor series 5.2 R 3/4 Power series 5.3 T 3/9 Spring Break R 3/11 Spring Break T 3/16 Laurent series 5.5 R 3/18 Zeros and singularities 5.6 T 3/23 Residue theorem 6.1 R 3/25 Applications 6.2 T 3/30 Applications 6.3 R 4/1 Exam 2 T 4/6 Argument principle and Rouche’s theorem 6.7 R 4/8 Conformal mappings 7.1 – 7.2 T 4/13 Mobius transformations 7.3 – 7.4 R 4/15 Mobius transformations 7.3 – 7.4 T 4/20 Review for final exam R 4/22 Review for final exam M 4/26 Comprehensive final exam