The Spring 2018 Complex Analysis 2 class.
COURSE: MATH 5520
TIME: 2:15-3:35 TR; PLACE: 341 Sam Wilson Hall
INSTRUCTOR: Dr. Robert Gardner; OFFICE: Room 308F of Gilbreath Hall
OFFICE HOURS: 3:35-4:00 TR; PHONE: 439-6979 (Math Office 439-4349)
E-MAIL: gardnerr@etsu.edu
WEBPAGE: www.etsu.edu/math/gardner/gardner.htm (see my webpage for a copy of this course syllabus and updates for the course).
TEXT: Functions of One Complex Variable, Second Edition, by John Conway.
PREREQUISITE: Complex Analysis 1 (MATH 5510).
ABOUT THE COURSE: We continue our exploration of functions of a complex variable. We'll study complex integration, singularities and Laurent series, and look at the various versions of the Maximum Modulus Theorem and some of its implications (not all of which are in the Conway book). If time permits, we will look at additional topics, such as spaces of analytic functions, the Riemann Zeta Function, analytic continuation, and Riemann surfaces.
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 at: http://faculty.etsu.edu/gardnerr/5510/notes.htm and http://faculty.etsu.edu/gardnerr/5510/notes2.htm 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.
OUTLINE:
Our tentative outline is:
Chapter 4. Complex Integration (continued): Zeros of analytic functions, Fundamental Theorem of Algebra, Maximum Modulus Theorem, winding number, Cauchy's Integral Formula, properties of path integrals, Open Mapping Theorem.
Chapter 5. Singularities: Classification of singularities, Laurent series, residues, integrals, meromorphic functions, argument principle, Rouche's Theorem.
Chapter 6. Maximum Modulus Theorem: Versions of Max Mod Theorem, Schwarz's Lemma,
Hadamard's Three Circles Theorem (maybe), Pragmen-Lindelof Theorem (maybe).
Chapter 7. Compactness and Convergence in the Spaces of Analytic Functions: Spaces of continuous and analytic functions, Arzela-Ascoli Theorem, spaces of meromorphic functions, Riemann Mapping Theorem, Weierstrass Factorization Theorem, the Gamma Function, the Riemann Zeta Function, the Riemann Hypothesis.
Other Possible Topics: Analytic continuation and Riemann surfaces (Chapter IX), entire functions (Chapter XI). We may also consider some of my research areas on the location of zeros of a polynomial in terms of coefficients (so called "Enestrom-Kakeya Theorem" type results) and Bernstein inequalities for polynomials.
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: http://www.etsu.edu/academicintegrity/ (last accessed 8/29/2017). 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. McIntosh). We will have two in-class tests. To address potential academic misconduct during the test, I will wander the room and may request to see the progress of your work on the test while you are taking it. You will not be allowed to access your phone during the tests. You will not be allowed to stop during a test to go to the bathroom, unless you have presented a documented medical need beforehand.
GRADING: Homework will be assigned and collected regularly. 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 passing graduate grades lower than C), etc. Remember that the lowest passing grade in a graduate course is a C, so you need an average of 73% on all assignments in order to pass this class.
IMPORTANT DATES: (see http://www.etsu.edu/etsu/academicdates.aspx for the official ETSU calendar; accessed 8/28/2017): UPDATE
OTHER RESOURCES. The following may be useful in Complex Analysis 2.
HOMEWORK.The following homework is assigned:
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Bob Gardner's webpage
Last Updated: April 9, 2018.