Complex Analysis
MA 505 - Spring 2023
Department of Mathematical Sciences
Worcester Polytechnic Institute
Department of Mathematical Sciences
Worcester Polytechnic Institute
Instructor: Prof. B.S. Tilley
Department of Mathematical Sciences
Worcester Polytechnic Institute
Tilley Home Page
Office: SL 405D
e-mail: tilley@wpi.edu
Phone: (508) 831-6664
Office Hours: T: 3:00-3:50pm ;
R: 3:00-4:50pm
or by appointment
Textbook:
Complex Made Simple, D.C. Ullrich, American Mathematical Society, (2008).
Additional References:
Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, L.V. Ahlfors, McGraw Hill
Introduction to Complex Variables and Applications, R.V. Churchill, McGraw-Hill.
Complex Variables: Introduction and Applications, M.J. Ablowitz and A.S. Fokas, Cambridge
Theory of Functions, E.C. Titschmarsh, Oxford
Functions of a Complex Variable: Theory and Technique, G.F. Carrier, M. Krook, and C.E. Pearson, SIAM
Course Description: This course will provide a rigorous and thorough treatment of the theory of functions of one complex variable. The topics to be covered include complex numbers, complex differentiation, the Cauchy-Riemann equations, analytic functions, Cauchy’s theorem, complex integration, the Cauchy integral formula, Liouville’s theorem, the Gauss mean value theorem, the maximum modulus theorem, Rouche's theorem, the Poisson integral formula, Taylor-Laurent expansions, singularity theory, conformal mapping with applications, analytic continuation, Schwarz’s reflection principle and elliptic functions. (Prerequisite: knowledge of undergraduate analysis.)
Class Expectations: As a graduate-level mathematics course, collaborative learning and active engagement are expected. Collaborative learning meas that students collaborate together to learn the material in the course. Active engagement by students means that students accept the responsibility for their own learning of the material and do not perceive the instructor (professor) as a source of all knowledge.
In order to meet these expectations, the classroom environment must be professional and supportive. Students are expected to treat each other with mutual respect, provide constructive feedback to other students, and to realize that as humans we all need guidance at times.
Students should also be able to communicate the mathematics they are learning, in both written and oral form, to others. Every student in class will have to give a (very short) oral presentation before the class detailing the answer to a homework question at least twice over the semester.
Special Arrangements: If you need course adaptations or accommodations because of a disability, or if you have medical information to share with me, please make an appointment with me as soon as possible. My office location and office hours are listed above. If you have not already done so, students who believe that they may need accommodations in this class are encouraged to contact the Office of Accessibility Services (OAS) as soon as possible to ensure that these accommodations are implemented in a timely fashion. The OAS is in Unity Hall, (508) 831-4908. Students who need accommodations for exams are required to make the arrangements to take these exams at the Exam Proctoring Center (EPC) on the day of the exam.
Grades: Grades will be determined by the following deliverables
Homework: There will be weekly homework assignments that are due in class the following week. The assignments should be done in pencil. You are strongly encouraged to work on the homework problems together, but the written solutions should be done individually. Copying homework is cheating, and will be dealt with according to the academic integrity policies of WPI. I anticipate ten (10) homework assignments this term.
Exams: There are going to be two exams. The first will be on 2/20/2023, and the second will be on 4/10/2023. These exams are open book, open notes, no internet access.
Project: There will be a final individual project which is due the final week of the semester. The goal of this project is for each student to learn an advanced topic in complex analysis individually, be able to present the material in a lecture format, and to write report on this topic. Details of this project will be available before the first exam, but the components will include a project proposal,
Grade Breakdown:
Homework: 25%
Exams: 50%
Project: 25%
Approximate Lecture Schedule (Updated 3/27/2023)
Lecture 1: Ullrich: Appendices A-B, Chapter 0,
Lecture 2: Ullrich: Appendix C, Power Series
Lecture 3: Cauchy-Riemann, Exponentials, Logs, Harmonic Functions I
Lecture 4: Cauchy-Goursat, Cauchy Integral Formula
Lecture 5: Cauchy Integral Formula for Derivatives and Consequences
Lecture 6: Taylor and Laurent Series, Zeros of Analytic Functions, Analytic Continuation
Lecture 7: Exam 1 (2/20/2023)
Lecture 8: Singularities, Winding Number, Cauchy Residue Theorem, Applications
Lecture 9: Laplace Transforms, Argument Principle, Rouche's Theorem
Lecture 10: Biliear Transformations, Schwartz's Theorem, Poisson Integral Formula
Lecture 11: Runge's Theorem and Mittag-Leffler Theorem.
Lecture 12: Exam 2 (4/10/2023)
Lecture 13: Project Day
Lecture 14: Presentation Day