Math 416, Spring 2021

Complex Analysis


Instructor                   John E. McCarthy
Class                           TuTh 11.30-12.45

Office Hours              M: 4-5 Tu: 2.30-3.15 Th: 1.00-1.45  Zoom Office Hours
Phone                          935-6753

 Modality: This will be decided in consultation with the class. My initial plan is to give in person classes to the

majority of enrolees, and to live stream the classes for those who cannot attend in person. However, once I know who has signed up, I will discuss with the enrolees what works best for them.

Exams    There will be two exams in the course:

                        1) Exam 1       Midterm - Friday March 19.
                        2) Exam 2       Final exam. Tuesday May 11, 1.00-3.00.


There will be weekly homework sets during the semester, assigned on Tuesday and due the following Tuesday.

Study/Wellness Days

The following dates are either Study days in the Mathematics and Statistics Department, or Wellness Days in Arts and Sciences:

Tuesday Feb 9
Tuesday Mar 2
Wednesday Mar 3
Monday Mar 22
Monday April 12

On these days, there are no office hours or classes. Any assignment due that day is extended 24 hours
 (so the one assigned Feb 2 is due Feb 10, and the one assigned Feb 23 is due Mar 4).


Math 318, or permission of instructor.


Complex Analysis is an essential tool in (almost) all areas of modern mathematics. It started with
Tartaglia's solution of the cubic - in order to find the real roots of a real cubic polynomial, the formula
requires complex numbers. The fundamental theorem of algebra says that every complex polynomial
can be factored into linear factors. This means that every matrix has complex eigenvalues, though not necessarily real ones.
Analytic functions of complex variables - functions that can locally be written as power series - are the heart of the subject.
They are both flexible and rigid, in ways we will discuss, and make the subject very attractive.



  1. Complex Numbers. De Moivre's Formula
  2. Complex Differentiation. Cauchy-Riemann equations.
  3. Harmonic Functions
  4. Linear fractional transformations
  5. Exponential and logarithmic functions
  6. Power series
  7. Complex Integration
  8. Cauchy's theorem
  9. Some of the many consequences of Cauchy's theorem - Liouville's theorem, Maximum modulus theorem, Schwarz's lemma
  10. Harmonic functions redux
  11. Laurent series. Singularities and Poles.
  12. Residue theorem. Definite integrals.
  13. Rouche's theorem. Riemann mapping theorem.
  14. Homotopy version of Cauchy's theorem.

Basis for Grading

Attendance and class participation will be 30% of your grade, homework will be 30%, the midterm will be 15%, and the final will be 25%.


Homework is an extremely important part of the course. Whilst talking to other people about it is not dis-allowed, too often this degenerates into one person solving the problem, and other people copying them (often justified to themselves by saying "I provide the ideas, X does the details" - but the details are the key. If you can't translate the idea into a real proof, you don't understand the material well enough). So I shall introduce the following rules:
(a) You can only talk to some-one else about a problem if you have made a genuine effort to solve it yourself.

(b) You must write up the solutions on your own. Suspiciously similar write-ups will receive 0 points.

(c) You may not get solutions from the internet.

Academic Integrity

Academic Integrity is especially important when we are struggling to allow remote learning.

During exams, you may not get help from anyone else, or seek solutions online.

I will ask you to write and sign the following Honor Statement on each exam:

I affirm that I have not given or received any unauthorized help on this assignment, and that this work is my own.


I expect you to attend class every day, and to participate in class discussions.

It is important to stay abreast of the material. At the beginning of class I will call on people to give definitions or state theorems; this will be part of your grade. I may call on you at any time to answer a question.

Class etiquette (in person): don't be disruptive or discourteous. No beeping, ringing, crunching, rustling, leaving early or arriving late. No texting, sleeping, checking your phone.

Class etiquette (online): please stay fully engaged - no multitasking. Except for compelling reasons, please keep your video on.

Texts             Complex Function Theory by Donald Sarason (AMS, 2007)


Additional Reading        

Any book on complex analysis in Olin library will contain all the material we cover.
Find one whose style you like and check it out (some of them are ebooks).


Accommodations based upon sexual assault:

The University is committed to offering reasonable academic accommodations to students who are victims of sexual assault.  Students are eligible for accommodation regardless of whether they seek criminal or disciplinary action.  Depending on the specific nature of the allegation, such measures may include but are not limited to: implementation of a no-contact order, course/classroom assignment changes, and other academic support services and accommodations.  If you need to request such accommodations, please direct your request to Kim Webb, Director of the Relationship and Sexual Violence Prevention Center, or Jen Durham Austin, Support Services Counselor. Both Kim Webb and Jen Durham Austin are confidential resources; however, requests for accommodations will be shared with the appropriate University administration and faculty.  The University will maintain as confidential any accommodations or protective measures provided to an individual student so long as it does not impair the ability to provide such measures.

If a student comes to me to discuss or disclose an instance of sexual assault, sex discrimination, sexual harassment, dating violence, domestic violence or stalking, or if I otherwise observe or become aware of such an allegation, I will keep the information as private as I can, but as a faculty member of Washington University, I am required to immediately report it to my Department Chair or Dean or directly to Ms. Jessica Kennedy, the University's Title IX Director.  If you would like to speak with directly Ms. Kennedy directly, she can be reached at (314), or by visiting the Title IX office in Umrath Hall.  Additionally, you can report incidents or complaints to the Office of Student Conduct and Community Standards or by contacting WUPD at (314) 935-5555 or your local law enforcement agency. See: Title IX

You can also speak confidentially and learn more about available resources at the Relationship and Sexual Violence Prevention Center by calling (314) 935-3445 for an appointment or visiting the 4th floor of Seigle Hall.  See: RSVP Center

Bias Reporting:
The University has a process through which students, faculty, staff and community members who have experienced or witnessed incidents of bias, prejudice or discrimination against a student can report their experiences to the University's Bias Report and Support System (BRSS) team. 

Mental Health:
Mental Health Services' professional staff members work with students to resolve personal and interpersonal difficulties, many of which can affect the academic experience. These include conflicts with or worry about friends or family, concerns about eating or drinking patterns, and feelings of anxiety and depression. 

Center for Diversity and Inclusion (CDI):
The Center of Diversity and Inclusion (CDI) supports and advocates for undergraduate, graduate, and professional school students from underrepresented and/or marginalized populations, creates collaborative partnerships with campus and community partners, and promotes dialogue and social change.  One of the CDI's strategic priorities is to cultivate and foster a supportive campus climate for students of all backgrounds, cultures and identities.