Applied Complex Analysis

Meetings

MWF, 11:45 - 12:40pm, broadcasted over Zoom, with recordings posted under password protection below (e-mail the instructor for access)

Instructor

Kevin Sackel, kevin.sackel(at)stonybrook.edu

Office Hours: Mondays 1-2pm (after class), Fridays 3-4pm

Grader

Jonathan Galván Bermúdez

Textbook

Complex Variables and Applications (9th Edition) by Brown and Churchill

Supplementary Material

Relevant lectures by the Youtube channel 3Blue1Brown

• Complex Number Fundamentals and Euler's Formula, two lectures which cover the basics of how complex numbers work

• Winding numbers and domain coloring, which beautifully illustrates how to think about the complex plane, and builds up to a "proof" of the Fundamental Theorem of Algebra. We will formalize this proof when we talk about Rouché's theorem later in the course. (This will be the second proof of the FToA in this course.)

• Visualizing the Riemann zeta function and analytic continuation, which captures many properties of complex analytic functions, focusing on the Riemann zeta function, but also including the function f(z) = z^2

Syllabus

Syllabus

Homework

All homework is due Monday at 11:00 PM on Blackboard unless otherwise noted.

Homework 0 (due Feb 1) - ungraded, but please complete - Please e-mail the instructor (no need to submit to Blackboard) with the following info:

• 1. Acknowledge that you have received the Blackboard announcement and have read the syllabus in its entirety (remember the syllabus is on the course webpage).

  2. Ask any preliminary questions you may have about what is in the syllabus. (Of course more questions may arise later in the course, at which point you should feel free to e-mail me.)

  3. Write a paragraph or two (or more!) about what you think this course will be about, and how that might fit in with what you already know. Of course, we will be using complex numbers, but try to think about what complex numbers might be useful for. Feel free especially to discuss how you think this course might fit in with the stories presented in other courses you've taken so far (e.g. algebra (linear or abstract), real analysis, geometry, physics). Incorrect guesses are more than welcome and encouraged - I am not expecting you to know the material of the course already, but I want to see what threads you may have picked up from previous courses. (Please try to not just regurgitate what is written in the course description!) The videos from 3Blue1Brown in the Supplementary Material section might be helpful in crafting your thoughts.

Homework 1 (due WEDNESDAY Feb 10): Solutions

Homework 2 (due Feb 15): Solutions

Homework 3 (due Feb 22): Solutions

Homework 4 (due Mar 1): Solutions

Homework 5 (due Mar 8): Solutions

Homework 6 (due WEDNESDAY Mar 17): Solutions

Homework 7 (due FRIDAY Mar 26): Solutions

Homework 8 (due Apr 12): Solutions

Homework 9 (due Apr 19): Solutions

Homework 10 (due Apr 26): Solutions

Homework 11 (due May 10):

Exams

Oral Exam 1 - 5-to-10 minute exam, week of March 29 (NB: Delayed one week from March 22)

• Logistics and general info

• Problems

Oral Exam 2 - 5-to-10 minute exam, week of May 3

• Logistics and general info

• Problems

• Your examiner

• Sign-ups: KEVIN and JONATHAN

Lectures

VIDEOS OF LECTURES AVAILABLE BY REQUEST

Lecture X (M, Feb 1): CANCELED (Snow Day)

Lecture 1 (W, Feb 3): Logistics, Overview, 1.1 (video, slides)

Lecture 2 (F, Feb 5): 1.2-1.6 (video, slides)

Lecture 3 (M, Feb 8): 1.5-1.10 (video, slides)

Lecture 4 (W, Feb 10): 1.9-1.11 (video, slides)

Lecture 5 (F, Feb 12): 1.11-1.12 (video, slides)

Lecture 6 (M, Feb 15): 1.12-2.13 (video, slides)

Lecture 7 (W, Feb 17): 2.14-2.15 (video, slides for the function z^2 from Sutherland, marked up version of Sutherland's slides, slides)

Lecture 8 (F, Feb 19): 2.15-2.17 (video, slides)

Lecture 9 (M, Feb 22): 2.17-2.20 (video, slides)

Lecture 10 (W, Feb 24): 2.20-2.22 (video, slides)

Lecture 11 (F, Feb 26): 2.22, 2.23, 2.25 (video, slides)

Lecture 12 (M, Mar 1): 2.25-2.27 (video, slides)

Lecture 13 (W, Mar 3): 2.27, 2.28, 3.30 (video, slides)

Lecture 14 (F, Mar 5): 3.30-3.33 (video, slides)

Lecture 15 (M, Mar 8): 3.33-3.35 (video, slides)

Lecture 16 (W, Mar 10): 3.35-3.37, 3.39 (video, slides)

Lecture 17 (F, Mar 12): 3.38-3.40 (video, slides)

Lecture 18 (M, Mar 15): 3.40-4.42 (video, slides)

Lecture 19 (W, Mar 17): 4.43-4.45, 4.47 (video, slides)

Lecture 20 (F, Mar 19): 4.47-4.49 (video, slides)

Lecture 21 (M, Mar 22): Recap, Finish 4.48-4.49 (video, slides)

Lecture 22 (W, Mar 24): 4.50 (video, slides)

Lecture 23 (F, Mar 26): 4.51-4.53 (video, slides)

Lecture 24 (M, Mar 29): 4.53-4.54 (video, slides)

Lecture 25 (W, Mar 31): 4.55, (mentioned but skipped 4.56), 4.57 (video, slides)

Lecture 26 (F, Apr 2): 4.57-4.58 (video, slides)

Lecture 27 (M, Apr 5): 4.59 (video, slides)

Lecture 28 (W, Apr 7): 5.60-5.61 (video, slides)

Lecture 29 (F, Apr 9): 5.62-5.63 (video, slides)

Lecture 30 (M, Apr 12): 5.64-5.67 (video, slides)

Lecture 31 (W, Apr 14): 5.68 (video, slides)

Lecture 32 (F, Apr 16): 5.69-5.71 (video, slides)

Lecture 33 (M, Apr 19): 5.72-5.73 (video, slides)

Lecture 34 (W, Apr 21): 6.74-6.75 (video, slides)

Lecture 35 (F, Apr 23): 6.76-6.77 (video, slides)

Lecture 36 (M, Apr 26): 6.78-6.80 (video, slides)

Lecture 37 (W, Apr 28): 6.81-6.83 (video, slides)

Lecture 38 (F, Apr 30): 6.83-6.84 (video, slides)

Lecture 39 (M, May 3): 6.84, 7.93 (video, slides)

Lecture 40 (W, May 5): 7.93-7.94, biholomorphic domains, Riemann mapping theorem (statement) (video, slides)

Lecture 41 (F, May 7): more on biholomorphic domains, harmonic measure, Riemann zeta function and hints of analytic number theory (video, slides)