MA 4291: Applied Complex Variables

C-Term - Spring 2022
Department of Mathematical Sciences
Worcester Polytechnic Institute

Instructor: Prof. B.S. Tilley
Department of Mathematical Sciences
Worcester Polytechnic Institute
Office: SH 105A
Phone: (508) 831-6664
e-mail: tilley@wpi.edu
Tilley Home Page
Office Hours: TF: 3:00-4:50pm or by appt.

TA: Dane Johnson
e-mail: djohnson3@wpi.edu
Office: SH 204
Office Hours: TBA

Textbook: Complex Variables and Applications, Eighth Edition, J.W. Brown and R.V. Churchill, McGraw-Hill, ISBN: 978-0-07-305194-9.

Room: AK 233

Course Description:

This course provides an introduction to the ideas and techniques of complex analysis that are frequently used by scientists and engineers. The presentation will follow a middle ground between rigor and intuition. Topics covered include: complex numbers, analytic functions, Taylor and Laurent expansions, Cauchy integral theorem, residue theory, and conformal mappings.

Learning Objectives:

  1. Perform complex arithmetic and determine values of functions with multiple branches

  2. Determine continuity, differentiability and analyticity of a function calculate the derivative where it exists.

  3. Apply conformal mapping techniques to applications involving Laplace's equation in two dimensions.

  4. Evaluate contour integrals using parameterization, and using the Cauchy-Goursat theorem over multiply-connected domains.

  5. Articulate the impact of the Cauchy integral formula on Liouville's Theorem, the Fundamental Theorem of Algebra, and the Maximum Modulus Theorem.

  6. Find the Taylor and Laurent series of a function and determin its circle or annulus of convergence.

  7. Compute the residue of a function and use residue theory to evaluate a contour integral or an integral over the real line.

  8. Find the Laplace transform of a function, and the inverse Laplace transform of a function.

Course Organization

The class meets five times per week (lecture and discussion sections). Instead of using the discussion section for solving problems, the first 10 minutes of every meeting will be devoted to questions on the assigned problems. Lectures and Discussion will be in person during the first two weeks of term, but videos of all lectures/discussion can be found through Echo 360 on the course Canvas site. In case of a WPI-defined significant weather event, the lecture will be given remotely, either through Echo 360 or via Zoom. Check the Canvas course site for further details.

Exam Policy

Both exams will be given in person during lecture/discussion time on 2/3/2022 and 3/4/2022. Prior to the start of each exam, you must place all of your belongings (e.g., cell phone, study materials, etc.) in your backpack and under your desk, so that no items are visible during the exam. All exams this term are closed book, closed notes, and no electronic devices are permitted. You will be required to show your WPI ID when turning in your exam to verify your identity.

Grade Weights

  • Written Homework: 30%

  • Exam 1 (2/3/2022): 35%

  • Exam 2 (3/4/2022): 35%

Electronics Policy

All lectures (audio and video) are captured through course capture and can be found on the course Canvas page. NO recording of audio or video by students is allowed during lecture or during conference. Laptops, phones, and tables should be turned off during the lecture and conference sessions. If you take notes (typing/stylus only) using these devices during lecture, then you should sit in the back of the room, since screen activity is distracting to your neighbors.

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 with disabilities 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 Daniels 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.