Math 4555: Complex Variables ~ Homepage

Math 4555 (Complex Variables), Fall 2025

Course Information
Instructor		Class Times		Office Hours
Evan Dummit edummit at northeastern dot edu		MW 2:50pm-4:30pm 241 Richards Hall		MW 1:30pm-2:30pm W 4:45pm-5:45pm 571 Lake Hall
For detailed information about the course, please consult the 4555 Course Syllabus. Note: any information given in class or on this webpage supersedes the written syllabus.
Problem Session Leader(s)			Recitation Times + Locations
Jordan Martino			F 1pm-3pm 143 Ryder Hall
We will use Piazza for any course-related discussion: here is the Piazza page.
All homework assignments will be posted on this webpage (see below). Homework assignments will be submitted via Gradescope, which is accessible through Canvas. Please submit scans of your homework pages by 11:59pm Eastern on the due date. Late assignments may be penalized at the grader's discretion.
Use of large-language models ("generative AI", such as ChatGPT or Claude) or equivalent technology in any manner is expressly prohibited in this course. This includes, but is not limited to, summarizing course information, asking for hints or solutions to course assignments, and general information retrival on course topics. Ask questions during class, in office hours, on Piazza, during problem sessions, or via email instead.
The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses. Reference texts are available upon request.

Homework Assignments and Exams
Some Tips on Problem Solving are available as suggestions for the homework assignments.
Homework #1, due Fri Sep 12th. (solutions) Homework #2, due Fri Sep 19th. (solutions) Homework #3, due Fri Sep 26th. [Note: Problem 5 is the challenge problem and problem 6 is a regular problem, so don't accidentally skip problem 6!] Homework #4, due Fri Oct 3rd.
Problems marked "Challenge" are optional, and worth much less than the other problems. They are encouraged for any student seeking an additional challenge -- and especially for students interested in further study of mathematics, and especially especially if you are seeking to go to graduate school in mathematics and might want me to write you a recommendation letter. But even if none of those apply, the problems are always worth reading at the very least.

Handouts / Lecture Notes
Handout	Topics
Chapter 1: Complex Numbers and Derivatives (19pp, v1.50, posted 9/1)	1.1 ~ Complex Arithmetic (Complex Numbers, Polar and Exponential Forms, Topology of C) 1.2 ~ Complex Derivatives (Limits, Complex Derivatives, Partial Derivatives and the Cauchy-Riemann Equations, Holomorphic Functions and Angles)
Chapter 2: Complex Power Series (24pp, v1.50, posted 9/11)	2.1 ~ Formal Power Series (Formal Power Series, Formal Laurent Series) 2.2 ~ Convergence of Power Series (Sequences and Series, Convergence of Power Series, Continuity and Differentiability, Analytic Functions) 2.3 ~ Elementary Functions as Power Series (Complex Exponentials, Trigonometric Functions, Logarithms and Powers)

Tips For Success In This Course
Attend Lecture	Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should review the notes and watch the lecture recording. You are responsible for all material covered in lecture.
Attend Problem Sessions	There are weekly problem sessions run by the course's TAs. The goal of the problem sessions is to provide you a location where you can work with other students on assignments, and also get assistance from a TA. It is highly recommended to attend at least one problem session per week to work on the homework problems.
Read the Lecture Notes	The lecture notes are a comprehensive source of material for the course. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics. Please note that the electronic notes are not identical to the material covered in class: this is by design, so as to provide you a slightly different perspective on the material.
Solve Homework Problems	Much of the learning in this course will take place as you solve the homework problems. Like many other activities, problem-solving and proof-writing are things that are learned by doing them, not by hearing someone else tell you about them or reading about them in a book. As such, the homework assignments are an integral part of the course, and are fundamental to learning the material. It is highly recommended that you look over the homework assignments as soon as they are available, and work on them well in advance of the deadline: many problems will take substantial time and effort to solve, and you should expect to spend as much time as you need to finish the assignments.
Attend Office Hours	Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor. Office hours will be the most effective when you have already put in effort to learn the material on your own (including trying to solve the homework problems), and when you come in with a list of specific questions or topics you are struggling with.

Course Schedule
The schedule is subject to change! All sections refer to the course lecture notes.
Week	Schedule
Week of Sep 1 (class starts 9/3)	§1.1.1: Complex Arithmetic §1.1.2: Polar and Exponential Forms No homework this week.
Week of Sep 8	§1.1.3: Topology of C §1.2.1: Limits §1.2.2: Complex Differentiation §1.2.3: The Cauchy-Riemann Equations Homework #1 due Friday 9/12 via Gradescope.
Week of Sep 15	§1.2.4: Holomorphic Functions and Angles §2.1.1: Formal Power Series §2.1.2: Formal Laurent Series §2.2.1: Convergent Sequences and Series Homework #2 due Friday 9/19 via Gradescope.
Week of Sep 22	§2.2.2: Convergent Power Series §2.2.3: Continuity and Differentiability of Power Series Homework #3 due Friday 9/26 via Gradescope.
Week of Sep 29	§2.3.1: Exponentials and Trigonometric Functions §2.3.2: Complex Logarithms §3.1.1: Line Integrals in C Homework #4 due Thursday 10/2 via Gradescope.
Week of Oct 6	§3.1.2: Evaluating Line Integrals §3.1.3: Path Independence §3.2.1: Cauchy's Integral Theorem via Green §3.2.2: Deformation of Contours Homework #5 due Friday 10/10 via Gradescope.
Week of Oct 13 (no class 10/13)	§3.2.3: Integration of Power Series §3.2.4: Winding Numbers Homework #6 due Friday 10/17 via Gradescope.
Week of Oct 20	§3.2.5: Cauchy's Integral Formula §3.2.6: Higher Derivatives and Series Expansions §4.1.1: The Cauchy Estimates Homework #7 due Friday 10/24 via Gradescope.
Week of Oct 27	§4.1.2: Entire Functions, Liouville's Theorem Review for midterm exam MIDTERM EXAM Wednesday 10/29 in class. No homework this week.
Week of Nov 3	§4.1.3: The Maximum Modulus Principle §4.2.1: Laurent Series Expansions §4.2.2: Zeroes of Holomorphic Functions §4.2.3: Removable Singularities, Poles, and Essential Singularities Homework #8 due Friday 11/7 via Gradescope.
Week of Nov 10	§4.3.1: Residues, The Residue Theorem §4.3.2: Residue Integrals: Circular Contours §4.3.3: Residue Integrals: Circular Contours with Detours §4.3.4: Residue Integrals: Other Contours (Rectangles, Keyhole) Homework #9 due Friday 11/14 via Gradescope.
Week of Nov 17	§5.1.1: Counting Zeroes and Poles §5.1.2: Rouché's Theorem §5.1.3: The Open Mapping Theorem, Local Invertibility Homework #10 due Friday 11/21 via Gradescope.
Week of Nov 24 (no class 11/26-11/30)	§5.2.1: The Point at Infinity and the Extended Complex Plane §5.2.2: Fractional Linear Transformations No homework this week.
Week of Dec 1 (class ends 12/3)	§5.2.3: Conformal Mapping §5.2.4: Analytic Continuation In-Class Component of Final Exam Wednesday 12/3 in class. Homework #11 due Friday 12/5 via Gradescope.
Take-Home Component of Final Exam due Thursday 12/11 via Gradescope.