Bena Tshishiku

Bena Tshishiku

Mathematician

Complex analysis (Math 1460)

Undergraduate course, Brown University, 2026

Announcements

  • (9/25). HW3 is posted below due next friday Oct 2 by 5pm.
  • (9/20). Some solutions to HW1 are posted below. (I cannot promise that I will post solutions to every assignment.)
  • (9/18). Office hours for the semester have been set for Tu 5:30-6:30 in Salomon 004 (me) and Th 5:30-6:30 in Salomon 202 (Sarah).

Course summary

This is an undergraduate course about the complex numbers ℂ and the study of functions ℂ → ℂ that are complex differentiable (these functions are called “holomorphic”). This is a rich and beautiful subject with connections to many other areas of math and beyond. We will focus on the theoretical aspects of complex analysis, although we will also discuss computations and applications.

Some highlights/goals of this course:

  • learn foundational results about holomorphic functions (e.g. Cauchy-Riemann equations, Cauchy integral formula, Riemann mapping theorem)
  • learn applications of complex analysis to other areas (fundamental theorem of algebra, contour integration)
  • strengthen your understanding of calculus/analysis by seeing it “in action”
  • improve your ability to read/write proofs and critique arguments
  • expose you to things you haven’t seen and challenge you through problem solving.

Some experience with writing proofs will help.

Textbook

Gamelin, Complex Analysis

We will mainly cover Parts 1 and 2, but may do some of Part 3.

Grading

participation 5%, homework 10%, quizzes 20%, 1st midterm 20%, 2nd midterm 20%, final exam 25%,

  • a grade of 92% or higher is guaranteed an A
  • a grade of 84% or higher is guaranteed an B
  • a grade of 76% or higher is guaranteed an C

For students taking this course S/NC, a minimum grade of 70% and a minimum participation grade of 70% is necessary to receive a grade of S.

Contact information

  • (Instructor) Bena Tshishiku (bena_tshishiku at brown.edu)
  • TA: Sarah Yan (sarah_yan at brown.edu)

Course events

  • Lectures: MW 8:30-9:50 in Bio-Med B13.
  • Bena’s office hour: Tu 5:30-6:30 in Salomon 004

  • Sarah’s office hour: Th 5:30-6:30 in Salomon 202

  • Midterm 1: Oct 14, in class
  • Midterm 2: Nov 23, in class
  • Final exam: Dec 15, 9am

Homework

There will be weekly assignments posted below. The homework is designed to increase your engagement with the material, with your peers, and with me. Doing the homework should be a human activity.

Collaboration is encouraged, but you should write your solutions alone and you must acknowledge the students you worked with. If you use AI on the homework, you must acknowledge this as well.

For any solution you submit, you should understand it well enough that you can explain it to someone else and answer questions about it. If you find yourself writing down things that you can’t explain, you should go back and think more about the problem.

Homework will be submitted on Gradescope. You can type your solutions or write them by hand (please be neat).

Late homework policy: As a general rule, late homework is not accepted. I know things happen, so I will drop your lowest assignment. For exceptional circumstances, please email me as early as possible.

Homework assignments

HW1, solutions

HW2

HW3

AI use

There are “good” (or perhaps reasonable) and “bad” (=counterproductive to learning) ways to use AI in this course.

Reasonable: ask it

  • questions about notation or definitions
  • about concepts that you’re confused about
  • to give you practice problems to test your knowledge

Counterproductive:

  • Feed it homework problems.
  • Feed it your solutions and to check for correctness. Learning to critique your own arguments is an objective of this course. Before submitting work, you should check it by returning to the definitions, testing the argument on examples, verifying that every hypothesis has been used appropriately, and discussing with me, Sarah, or your classmates.
  • Use it when you are stuck on a problem. (Better: read the book, do an example, talk to someone, think!)
  • Trusting it blindly as an authority (you should not trust any one source blindly).
  • Use it to eliminate productive struggle. Sometimes it’s helpful to be stuck on a problem and think your way out of it. You can learn a lot by making mistakes, testing ideas, trying different methods, thinking of examples, looking for connections. You will learn more if you do not outsource these steps to an LLM.

Quizzes

There will be a quiz during class each Wednesday starting 9/16. The quizzes are designed to give you (and me) feedback about whether you are achieving the learning goals for the course. The material for the quiz will be the content from the previous week’s lectures.

Participation

At the beginning of every class there will be a small warm-up activity/exercise that will be turned in and graded based on completion only. These activities will have an educational component, but they are also there to encourage you to show up to class on time (I know it’s early).

Lecture notes

I am not planning to post any lecture notes. I believe it is part of your responsibility to come to class and write down what you think is important. This is a basic way to engage with the material.

Topic schedule (subject to change)

  • Week 1: Gamelin Chapter I, Section 1-2, 5
    • Wed (9/9). Overview, complex arithmetic, complex exponential
  • Week 2: Gamelin Chapter I, Sections 4,7,8; Chapeter II, Sections 1-3
    • Mon (9/14). Complex derivatives, square roots
    • Wed (9/16). holomorphic functions, open sets, CR equations
    • Fri (9/17). HW1 due.
  • Week 3: Gamelin Chapter 2, Sections 5; Chapter 3, Sections 1-3,
    • Mon (9/21). complex logarithm, Cauchy-Riemann equations
    • Wed (9/23). visualizing holomorphic functions, harmonic conjugates
    • Fri (9/24). HW2 due
  • Week 4: Gamelin Chapter 4, Sections 1-6
    • Mon (9/28). complex integration, Green’s theorem and Cauchy’s theorem, antiderivatives
    • Wed (9/30). Cauchy integral formula, holomorphic implies smooth, Liouville and Morera’s theorems
    • Fri (10/1). HW3 due
  • Week 5: Gamelin Chapter 5, Sections 4,7,8
    • Mon (10/5). Cauchy integral formula, computing with CIF, holomorphic implies analytic
    • Wed (10/7). residue calculus, fundamental theorem of algebra
    • Fri (10/8). HW4 due
  • Week 6: Gamelin Chapter 5, Section 8
    • Mon (10/12). no class (university holiday) Analytic continuation, residue theorem computations
    • Wed (10/14). Midterm 1
    • Fri (10/15).
  • Week 7:
    • Mon (10/19). analytic continuation, Gamma/zeta functions, more residue calculus
    • Wed (10/21). More on Gamma/zeta functions, summation with the residue theorem
    • Fri (10/22). HW5 due
  • Week 8: Ch 3, Section 5; Ch 9, Sections 1-2
    • Mon (10/26). More on Gamma/Zeta functions, maximum modulus principle
    • Wed (10/28). Maximum principle for harmonic functions, disk automorphisms, Singularities
    • Fri (10/29). HW6 due
  • Week 9: Chapter 6, Sections 1-2; Chapter 9, Sections 1-2
    • Mon (11/2). Singularities, Schwarz lemma, Mobius transformations, Laurent series decomposition
    • Wed (11/4). Laurent series decomposition, Singularities, Argument principle
    • Fri (11/5). HW7 due
  • Week 10: Ch 8, Sections 1-4
    • Mon (11/9). Argument principle, Rouche’s theorem, non-isolated singularities
    • Wed (11/11). Open mapping theorem, Maximum modulus (revisited), inverse mappings
    • Fri (11/12). HW8 due
  • Week 11: Chapter 11, Sections 2, 5, 6
    • Mon (11/16). SL(2,ℤ) and the upper half plane, Riemann mapping theorem
    • Wed (11/18). More SL(2,ℤ), Riemann mapping theorem
    • Fri (11/19). no HW due (prepare for midterm)
  • Week 12:
    • Mon (11/23). Midterm 2
    • Wed (11/25). No class (Thanksgiving holiday)
  • Week 13:
    • Mon (11/30).
    • Wed (12/2).
    • Fri (12/4). HW9 due
  • Week 14: Reading period
    • Mon (12/7).
    • Wed (12/9).
  • Week 15:
    • Tues (12/15). Final exam at 9am