Harold P. Boas

Math 618
Theory of Functions of a Complex Variable II
Spring 2022

Course Information

Course Data

Instructor Details

Course Description

This course is a sequel to Math 617. Topics include infinite products, the Weierstrass factorization theorem, Mittag-Leffler’s theorem, normal families, proof of the Riemann mapping theorem, analytic continuation, Runge’s approximation theorem, harmonic functions, and Picard’s theorems.

The qualifying examination in complex analysis is associated with Math 617 and Math 618.

Course Prerequisite

The prerequisite for this course is Math 617 or the equivalent.

Course Objectives

By the end of the course, you should be able to

Textbook

The required textbook is Functions of One Complex Variable I, second edition, by John B. Conway, published by Springer-Verlag in 1978. Since the campus library subscribes to a collection of Springer books, registered TAMU students can download a pdf copy of the textbook for free. You may need your NetID password to download the book. If the preceding link does not work for you, then locate the entry for the book in the TAMU library catalogue and click on the link labeled “Connect to the full text of this electronic book.” At the indicated link, there is also an option to purchase a paper copy for $24.99 (plus tax), much cheaper than the list price. (Look for the link “MyCopy Softcover.”) You may also be able to find an inexpensive used copy from your favorite book vendor.

The course material is contained in Chapters VII–XII.

The textbook author has posted a list of corrections, most of which have been incorporated into the latest (seventh) printing of the second edition, which is the version you should have if you downloaded an electronic pdf copy. I have posted some additional comments and corrections.

Grading Policy

Course letter grades are assigned using the standard scale (60% is passing, 70% or higher earns a C, 80% or higher earns a B, 90% or higher earns an A).

The categories contributing to the course grade have the following weights.

Late Work Policy

The expectation is that you will meet announced deadlines for submission of assignments. I recognize that extraordinary circumstances may arise, so I will accept late work for partial credit. I have configured Canvas to apply a penalty of 10% per day for late submissions. (To handle partial days, Canvas uses the ceiling function \(\lceil\cdot\rceil\), the smallest integer greater than or equal to the number, so the grade penalty for a submission \(x\) days late is \(\lceil x\rceil \times{}\)10%, capped at 100%.)

Work submitted to make up for an excused absence is not considered late and is exempted from the late work policy. See Student Rule 7.

Course Schedule

You are encouraged to read each section of the textbook before the class meeting in which that section is on the agenda. The following schedule is subject to revision if circumstances change.

Week 1
Monday 17 January: Martin Luther King, Jr. Holiday. Classes do not meet.
Wednesday 19 January: Section VII.1, the metric space of continuous functions.
Friday 21 January: Section VII.1, the Arzelà–Ascoli theorem.
Week 2
Monday 24 January: Section VII.2, convergence and compactness in the space of holomorphic functions.
Wednesday 26 January: Section VII.3, convergence and compactness in the space of meromorphic functions.
Friday 28 January: Section VII.4, the Riemann mapping theorem.
Week 3
Monday 31 January: Section VII.5, infinite products.
Wednesday 2 February: Section VII.5, the Weierstrass factorization theorem.
Friday 4 February: Section VII.6, the sine function.
Week 4
Monday 7 February: Section VII.7, the Gamma function.
Wednesday 9 February: Section VII.7, uniqueness theorems for \(\Gamma\).
Friday 11 February: Section VII.8, the zeta function.
Week 5
Monday 14 February: Section VII.8, more on \(\zeta\).
Wednesday 16 February: Section VIII.1, Runge’s approximation theorem.
Friday 18 February: Section VIII.1, more on Runge’s theorem.
Week 6
Monday 21 February: Section VIII.2, simple connectivity.
Wednesday 23 February: Section VIII.3, Mittag-Leffler’s theorem.
Friday 25 February: Section IX.1, the Schwarz reflection principle.
Week 7
Monday 28 February: Section IX.2, analytic continuation along paths.
Wednesday 2 March: Section IX.3, the monodromy theorem.
Friday 4 March: Section IX.4, topological spaces.
Week 8
Monday 7 March: Section IX.5, the sheaf of germs of holomorphic functions.
Wednesday 9 March: Section IX.6, the concept of a Riemann surface.
Friday 11 March: Section IX.6, more on Riemann surfaces.
Spring Break
Classes do not meet the week of March 14–18.
Week 9
Monday 21 March: Section IX.7, covering spaces.
Wednesday 23 March: Section X.1, harmonic functions.
Friday 25 March: Section X.2, the Poisson integral.
Week 10
Monday 28 March: Section X.3, subharmonic functions.
Wednesday 30 March: Section X.4, the Dirichlet problem.
Friday 1 April: Section X.5, Green’s function.
Week 11
Monday 4 April: Section XI.1, Jensen’s formula.
Wednesday 6 April: Section XI.2, genus and order.
Friday 8 April: Section XI.3, Hadamard’s factorization theorem.
Week 12
Monday 11 April: Section XII.1, Bloch’s theorem.
Wednesday 13 April: Section XII.2, Picard’s little theorem.
Friday 15 April: Reading Day. Classes do not meet.
Week 13
Monday 18 April: Section XII.3, Schottky’s theorem.
Wednesday 20 April: Section XII.4, Picard’s great theorem.
Friday 22 April: Reserved for student presentations.
Week 14
Monday 25 April: Reserved for student presentations.
Wednesday 27 April: Reserved for student presentations.
Friday 29 April: Reserved for student presentations.
Week 15
Monday 2 May: Reserved for student presentations.
Tuesday 3 May (redefined as Friday): Reserved for student presentations.
Wednesday 4 May: Reading Day. Classes do not meet.

University Policies

This section contains university-level policies. The TAMU Faculty Senate established the wording of these policies.

Attendance Policy

The university views class attendance and participation as an individual student responsibility. Students are expected to attend class and to complete all assignments.

Please refer to Student Rule 7 in its entirety for information about excused absences, including definitions, and related documentation and timelines.

Makeup Work Policy

Students will be excused from attending class on the day of a graded activity or when attendance contributes to a student’s grade, for the reasons stated in Student Rule 7, or other reason deemed appropriate by the instructor.

Please refer to Student Rule 7 in its entirety for information about makeup work, including definitions, and related documentation and timelines.

“Absences related to Title IX of the Education Amendments of 1972 may necessitate a period of more than 30 days for make-up work, and the timeframe for make-up work should be agreed upon by the student and instructor” (Student Rule 7, Section 7.4.1).

“The instructor is under no obligation to provide an opportunity for the student to make up work missed because of an unexcused absence” (Student Rule 7, Section 7.4.2).

Students who request an excused absence are expected to uphold the Aggie Honor Code and Student Conduct Code. (See Student Rule 24.)

Academic Integrity Statement and Policy

“An Aggie does not lie, cheat or steal, or tolerate those who do.”

“Texas A&M University students are responsible for authenticating all work submitted to an instructor. If asked, students must be able to produce proof that the item submitted is indeed the work of that student. Students must keep appropriate records at all times. The inability to authenticate one’s work, should the instructor request it, may be sufficient grounds to initiate an academic misconduct case” (Section 20.1.2.3, Student Rule 20).

You can learn more about the Aggie Honor System Office Rules and Procedures, academic integrity, and your rights and responsibilities at aggiehonor.tamu.edu.

Americans with Disabilities Act (ADA) Policy

Texas A&M University is committed to providing equitable access to learning opportunities for all students. If you experience barriers to your education due to a disability or think you may have a disability, please contact Disability Resources office. Disabilities may include, but are not limited to, attentional, learning, mental health, sensory, physical, or chronic health conditions. All students are encouraged to discuss their disability-related needs with Disability Resources and their instructors as soon as possible.

Disability Resources is located in the Student Services Building or at (979) 845-1637 or visit disability.tamu.edu.

Title IX and Statement on Limits to Confidentiality

Texas A&M University is committed to fostering a learning environment that is safe and productive for all. University policies and federal and state laws prohibit gender-based discrimination and sexual harassment, including sexual assault, sexual exploitation, domestic violence, dating violence, and stalking.

With the exception of some medical and mental health providers, all university employees (including full and part-time faculty, staff, paid graduate assistants, student workers, etc.) are Mandatory Reporters and must report to the Title IX Office if the employee experiences, observes, or becomes aware of an incident that meets the following conditions (see University Rule 08.01.01.M1):

Mandatory Reporters must file a report regardless of how the information comes to their attention — including but not limited to face-to-face conversations, a written class assignment or paper, class discussion, email, text, or social media post. Although Mandatory Reporters must file a report, in most instances, a person who is subjected to the alleged conduct will be able to control how the report is handled, including whether or not to pursue a formal investigation. The University’s goal is to make sure you are aware of the range of options available to you and to ensure access to the resources you need.

Students wishing to discuss concerns in a confidential setting are encouraged to make an appointment with Counseling and Psychological Services (CAPS).

Students can learn more about filing a report, accessing supportive resources, and navigating the Title IX investigation and resolution process on the University’s Title IX webpage.

Statement on Mental Health and Wellness

Texas A&M University recognizes that mental health and wellness are critical factors that influence a student’s academic success and overall wellbeing. Students are encouraged to engage in healthy self-care by utilizing available resources and services on your campus.

Students who need someone to talk to can contact Counseling & Psychological Services (CAPS) or call the TAMU Helpline (979-845-2700) from 4:00 p.m. to 8:00 a.m. weekdays and 24 hours on weekends. 24-hour emergency help is also available through the National Suicide Prevention Hotline (800-273-8255) or at suicidepreventionlifeline.org.