Complex Analysis
MA 505 - Spring 2026
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: SH 419
e-mail: tilley@wpi.edu
Phone: (508) 831-6664
Office Hours: T: 3:00-3:50pm ;
F: 3:00-3:50pm
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
Complex Analysis, S. Lang, Springer, Fourth Edition.
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.
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 red ink, scanned in grey-scale (not color) into a PDF file, and uploaded to the Canvas site. You are strongly encouraged to work on the homework problems together, but the written solutions should be done individually. Copying homework, either from a human, a text, or a bot, 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 3/04/2026, and the second will be on 4/15/2026. 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 include a project proposal, a short presentation, and a final report.
Grade Breakdown:
Homework: 25%
Exams: 50%
Project: 25%
Academic Policies
Grading Policy: Students have two business days after grades are posted to contact the instructor about potential errors in grading any assignment (homework or exams) after receiving their graded work via Canvas. Beyond this time, the grade on that assignment is final.
Generative AI Policy: Students should feel free to find online resources that describe the topics covered here that may present the material in a different style or structure that may be helpful. These resources are designed with the intention to aid students in learning the material. Generative AI tools like Gemini, Chat-GPT, or Copilot, are more general use tools to help the user find potentially relevant information quickly and in a format that appears conversational. Do: experiment with it and use if for brainstorming, and do cite the source/engine when communicating the output, Do not: give an assignment prompt as input and paste the output as your submission. Students who submit AI-generated content as their own work will receive no credit for that work.
Accommodations: Students with approved academic accommodations should plan to submit their accommodation letters through the Office of Accessibility Services Student Portal. Should you have any questions about how accommodations can be implemented in this particular course, please contact me as soon as possible. Students who are not currently registered with the Office of Accessibility Services (OAS) but who would like to find out more information about requesting accommodations, documentation guidelines, and what the accommodated interactive process entails should plan to contact OAS either by email: AccessibilityServices@wpi.edu, by phone (508) 831-4908, or by stopping by the office on the 5th floor of Unity Hall.
Approximate Lecture Schedule (Updated 1/14/2026)
Week 1: Ullrich: Appendices A-B, Chapter 0,
Week 2: Ullrich: Appendix C, Power Series
Week 3: Cauchy-Riemann, Exponentials, Logs, Harmonic Functions I
Week 4: Cauchy-Goursat, Cauchy Integral Formula
Week 5: Cauchy Integral Formula for Derivatives and Consequences
Week 6: Taylor and Laurent Series, Zeros of Analytic Functions, Analytic Continuation
Week 7: Exam 1 (3/04/2026)
Week 8: Singularities, Winding Number, Cauchy Residue Theorem, Applications
Week 9: Laplace Transforms, Argument Principle, Rouche's Theorem
Week 10: Biliear Transformations, Schwartz's Theorem, Poisson Integral Formula
Week 11: Runge's Theorem and Mittag-Leffler Theorem.
Week 12: Exam 2 (4/15/2026)
Week 13: Project Week
Week 14: Presentation Week