COURSE: MATH 5410-001, Call # 83993

TIME AND PLACE: 3:45-5:05 TR, Gilbreath Hall room 304

INSTRUCTOR: Dr. Robert Gardner OFFICE HOURS: TBA

OFFICE: Room 308F of Gilbreath Hall

PHONE: 439-6979 (308F Gilbreath), Math Department Office 439-4349

E-MAIL:gardnerr@etsu.edu
WEBPAGE: http://faculty.etsu.edu/gardnerr/gardner.htm (see my webpage for a copy of this course syllabus, copies of the classnotes in PDF, and updates for the course).

TEXT: Functions of One Complex Variable, Second Edition, by John Conway.

PREREQUISITE: Formally, the prerequisite is an undergraduate real analysis class or advanced calculus class. Practically, what is necessary is some exposure to (and a reasonable recollection of) the topology of ℝ (open and closed sets, limit points, connectedness, compactness, completeness, lub and sup, glb and inf, sequences and series of real numbers, convergence, uniform convergence, comparison tests, Cauchy sequences), and properties of functions of a real variable (continuity, differentiability, power series representation). It is also assumed that the student has been exposed to some elementary properties of the complex numbers (algebra, geometry, roots of unity, modulus). Even some of the topics will be addressed in this class. You can find background material on Complex Variables (MATH 4337/5337) in my online notes for Complex Variables.

CLASS NOTES: We will use projected digital notes for the component of the lecture consisting of definitions, statements of theorems, and some examples. Proofs of the vast majority of theorems, propositions, lemmas, and corollaries are available in Beamer presentations and will be presented in class as time permits. The white board will be used for marginal notes and additional examples and explanation. Copies of the notes are online for Complex Analysis Class Notes. It is strongly recommended that you get printed copies of the overheads before the material is covered in class. This will save you from writing down most notes in class and you can concentrate on listening and supplementing the notes with comments which you find relevant. You should read the online notes to be covered in class before each class (we may not have class time to cover every little detail in the online notes). Try to understand the definitions, the examples, and the meanings of the theorems. After each class, you should read the section of the book covered in that class, paying particular attention to examples and proofs.

ABOUT THE COURSE: Complex analysis is basically the study of analytic functions. As we will see, a function of a complex variable is often much better behaved than a function of a real variable! We will introduce the complex numbers as an extension of the real numbers. We explore the complex plane and give a geometric interpretation of results whenever possible. We study metric spaces in general, but with an eye towards the complex field. Analytic functions are defined, their series representations are explored, and Mobius transformation are analyzed. Finally, we introduce integration of complex functions, prove the Fundamental Theorem of Algebra, and the Maximum Modulus Theorem. If time permits, I will discuss some of my research results which are related to topics in the class. We may have the opportunity to briefly explore applications of this material, but this is definitely a pure math class and our concentration will be on theory.

OUTLINE: Our tentative outline is:
Introduction. Introduction to Math Philosophy and Meaning, formalism, David Hilbert, Gottlob Frege, Bertrand Russell, Principia Mathematica, Russell's Paradox, Kurt Godel, well-formed formulas, complete axiomatic systems, undecidability, misinterpretations of undecidability, Adam Sokal, Peano's axioms of arithmetic.
Chapter I. The Complex Number System: Introduction to the complex plane, real and imaginary parts, modulus, polar representation, extended complex plane, Riemann sphere.
Chapter II. Metric Spaces and Topology of ℂ: Extensions of several ideas from ℝ to ℂ and other metric spaces, open and closed sets, connectedness, sequences, completeness, compact sets, continuity, convergence, uniform convergence.
Chapter III. Elementary Properties and Examples of Analytic Functions: Series, convergence of series, differentiability, analytic functions, mappings, Mobius transformations.
Chapter IV. Complex Integration: Riemann-Stieltjes integrals, power series, zeros of analytic functions, Fundamental Theorem of Algebra, Maximum Modulus Theorem, winding number, Cauchy's Integral Formula, properties of path integrals, Open Mapping Theorem.
(My plan for Complex Analysis 2 [MATH 5520] is to cover chapters V, VI, VII, IX, and XI.)

GRADING: Homework (H) to be turned in will be assigned regularly. We will have two tests (T₁ and T₂) and your average will be computed as follows: AVERAGE = (3H + T₁ + T₂)/5. Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate. Based on the assignment of grad points by ETSU, the plus and minus grades should be given on a 3 point subscale. For example, a B+ corresponds to an average of 87, 88, or 89; an A- corresponds to an average of 90, 91, or 92; an A corresponds to an average of 93 to 100 (ETSU does not grant A+ grades, nor D- grades), etc.

HOMEWORK: YOU MUST SHOW ALL DETAILS ON THE HOMEWORK PROBLEMS!!! Justify every step and claim you make - this is how you convince me that you know what you are doing. You may find some answers online, but these rarely sufficiently justify all steps and are unacceptable as homework solutions.

ACADEMIC MISCONDUCT: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Some of the homework problems are fairly standard for this class, and you may find proofs online or in an online version of the solutions manual. The online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class. If I get homework from two (or more) of you that is virtually identical, then neither of you will get any credit. If you copy homework solutions from an online source, then you will get no credit. These are examples of plagiarism and I will have to act on this as spelled out on ETSU's "Academic Integrity @ ETSU" webpage (last accessed 2/19/2021). To avoid this, do not copy homework and turn it in as your own!!! Even if you collaborate with someone, if you write the homework problems out in such a way that you understand all of the little steps and details, then it will be unique and your own work. If your homework is identical to one of your classmates, with the exception of using different symbols/variables and changing "hence" to "therefore," then we have a problem! If you copy a solution from a solution manual or from a website, then we have a problem! I will not hesitate to charge you with academic misconduct under these conditions. When such a charge is lodged, the dean of the School of Graduate Studies is contacted. Repeated or flagrant academic misconduct violations can lead to suspension and/or expulsion from the university (the final decision is made by the School of Graduate studies and the graduate dean, Dr. McGee).

DESIRE2LEARN: I will not much rely on the Desire2Learn ("elearn") website. Instead, I will simply post all material directly on the internet. However, I will post your grades and homework solutions on D2L.

SYLLABUS ATTACHMENT: You can find an on-line version of the university's syllabus attachment (which contains general information concerning advisement, honor codes, dropping, etc.; last accessed 1/5/2021).

COVID-19 SAFETY POLICIES: You can find material on ETSU's COVID-19 polcies on the Modified Stage 4 Frequently Asked Questions (FAQs) webpage (last accessed 7/26/2021; document last updated 6/21/2021)). This document includes the following questions/comments:

• 6(c). Faculty may not ask students if they are vaccinated.
• 6(d). Faculty may not require students to wear a face-covering.
• 11(b). Any member of the university community who has been fully vaccinated but wishes to continue wearing a facial is encouraged to do so. We ask that you provide a supportive culture for those who continue this practice.
• 11(c). Individuals should not ask a person’s COVID-19 vaccination status...
• 18. Will students be required to get a COVID-19 Vaccine?
  (a) Not at this time. However, students are strongly encouraged to get a vaccine.
• 48(b). If you have any symptoms of COVID-19, do not report to campus for work or class. You should immediately notify your supervisor and/or professor and follow guidelines for seeking medical care and self-quarantining.

IMPORTANT DATES: (see the official ETSU calendar for more details; accessed 2/19/2021):

• Monday, August 23 = First day of classes.
• Sunday, August 29 = Last day to register or add without departmental permit.
• Wednesday, September 1 = Last day for graduate students to apply for spring 2022 graduation and to submit committee forms.
• Sunday, September 5 = Last day to drop without a grade of "W."
• Monday, September 6 = Labor Day (univeristy closed, no classes).
• Tuesday, September 7 = Begin late add with dean's permission.
• Monday and Tuesday, October 11 and 12 = Break Day (no classes).
• Monday, October 18 = Last day to drop without dean's permission.
• Monday, October 18 = Last day to schedule oral defense of thesis for December 2021 graduation.
• Monday, November 1 = Last day to complete oral defense of thesis for December 2021 graduation.
• Monday, November 8 = Deadline to upload approved thesis for December 2021 graduation.
• Thursday, November 11 = Veterans Day holiday (university closed, no classes).
• Wednesday, Thursday, and Friday, November 24, 25, and 26 = Thanksgiving holiday (univerity closed, no classes).
• Wednesday, December 1 = Last day to withdraw from the university.
• Friday, December 3 = Last day of classes.

OTHER RESOURCES. The following may be of interest:

1. To access the Mathematical Reviews: Go to the Sherrod Library online catalog. Click the "Title" tab and enter "Mathematical Reviews." Select "MathSciNet [Electronic Resource]" and follow the links. You will be asked to enter your user ID and password (the same you use for your ETSU e-mail). You are then redirected to MathSciNet and can freely use it and even download PDF versions of some of the papers you find! Of course, this protocol will work for any electronic journal available through the Sherrod Library.
2. Geometry of Polynomials, Mathematical Monographs and Surveys #3, on GoogleBooks by Morris Marden, AMS, 1986. See Chapter II "The Critical Points of a Polynomial." For a study of the location of the zeros of a polynomial in terms of the coefficients, see Chapter VII.
3. Information on the Nova episode The Mathematical Mystery Tour from 1985 can be found on my webpage A Mathematical Mystery Tour 25th Anniversary Webpage . The webpage includes a link to the YouTube site which contains the show.
4. A more contemporary documentary is The Story of Maths, a BBC production from 2008. This is a four part series and all of this series is available online: The Story of Maths. (In fact, this is posted on an excellent website which includes a number of documentaries and is a "must visit": Top Documentary Films!) Also of interest is the Wikipedia page for The Story of Maths. The series is available from a number of online streaming video sources, as revealed by a Google search. This websites were all active as of 6/29/2019.
5. A lighter documentary on the history of math is The Story of One, narrated by Monty Python-er Terry Jones. It is also a BBC production and dates from 2005. If you are involved in teaching, this is a nice documentary which should be accessible by highs schoolers and middle schoolers. It is on TopDocumentaryFilms at: http://topdocumentaryfilms.com/story-of-one/ (accessed 6/29/2019).
6. Syllabus for Complex Analysis 2 (MATH 5520), spring 2022.
7. A Beamer presentation of some of the work I have done with past graduate students in complex analysis is: Dr. Bob's Favorite Results on Complex Polynomials.

HOMEWORK.The following homework is assigned:

Return to Dr. Bob's webpage

Last updated: July 26, 2021.