Harold P. Boas

Math 617
Theory of Functions of a Complex Variable I
Fall 2021

Course Information

Course Data

Instructor Details

Course Description

Intended primarily for graduate students in mathematics, this course addresses the theory of functions of one complex variable. The basic objects of study are holomorphic functions (complex-analytic functions). The course covers the representation of holomorphic functions by power series and by integrals; complex line integrals, Cauchy’s integral formula, and some applications; singularities of holomorphic functions, Laurent series, and computation of definite integrals by residues; the maximum principle and Schwarz’s lemma; and conformal mapping.

The qualifying examination in complex analysis is associated with this course and the sequel (Math 618).

Course Prerequisites

The official prerequisite for this course is Math 410 (real calculus in Euclidean space). The essential background that you need is some facility with proofs in the \(\varepsilon\)–\(\delta\) style.

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 box headed “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 I–VI.

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 should read each section of the textbook before the class meeting in which that section is on the agenda. During class, be prepared to solve problems and to discuss the topics.

The following schedule is subject to revision if circumstances change.

Week 1
Monday 30 August: Sections I.1 and I.2, the complex numbers.
Wednesday 1 September: Sections I.3 and I.4: polar representation.
Friday 3 September: Sections I.5 and I.6, stereographic projection.
Week 2
Monday 6 September: Section II.1, metric spaces.
Wednesday 8 September: Section II.2, connectedness.
Friday 10 September: Section II.3, completeness.
Week 3
Monday 13 September: Section II.4, compactness.
Wednesday 15 September: Section II.5, continuity.
Friday 17 September: Section II.6, uniform convergence.
Week 4
Monday 20 September: Section III.1, power series.
Wednesday 22 September: Section III.2, complex differentiability.
Friday 24 September: Section III.2, representation of analytic functions by power series.
Week 5
Monday 27 September: Section III.2, Cauchy–Riemann equations.
Wednesday 29 September: Section III.3, conformality.
Friday 1 October: Section III.3, Möbius transformations. First take-home exam is due. The due date for the first take-home exam is changed to Monday, October 4.
Week 6
Monday 4 October: Section III.3, properties of Möbius transformations.
Wednesday 6 October: Section IV.1, the Stieltjes integral.
Friday 8 October: Section IV.1, line integrals.
Week 7
Monday 11 October: Section IV.1, primitives.
Wednesday 13 October: Section IV.2, Cauchy’s formula for disks.
Friday 15 October: Section IV.2, consequences of Cauchy’s formula for disks.
Week 8
Monday 18 October: Section IV.3, zeros of analytic functions.
Wednesday 20 October: Section IV.3, theorems implying that an analytic function is constant.
Friday 22 October: Section IV.4, the index of a closed curve.
Week 9
Monday 25 October: Section IV.5, Cauchy’s integral theorem and formula.
Wednesday 27 October: Section IV.5, Morera’s theorem.
Friday 29 October: Section IV.6, homotopy version of Cauchy’s theorem.
Week 10
Monday 1 November: Section IV.6, simple connectivity.
Wednesday 3 November: Section IV.7, counting zeros.
Friday 5 November: Section IV.8, Goursat’s proof of Cauchy’s theorem.
Week 11
Monday 8 November: Section V.1, isolated singularites. Second take-home exam is due.
Wednesday 10 November: Section V.1, Laurent series.
Friday 12 November: Section V.2, the residue theorem.
Week 12
Monday 15 November: Section V.2, computation of integrals via residues.
Wednesday 17 November: Section V.2, more computations via residues.
Friday 19 November: Section V.3, Rouché’s theorem and the argument principle.
Week 13
Monday 22 November: Section VI.1, the maximum principle.
Wednesday 24 November: Reading Day, classes do not meet.
Friday 26 November: Thanksgiving Holiday, classes do not meet.
Week 14
Monday 29 November: Section VI.2, the Schwarz lemma.
Wednesday 1 December: Section VI.3, the three circles theorem.
Friday 3 December: Section VI.4, Phragmén–Lindelöf theory.
Week 15
Monday 6 December: review of Math 617.
Wednesday 8 December: preview of Math 618.
Final Exam
Scheduled by the registrar for 8:00–10:00 a.m. on Monday, December 13.

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.