Instructor: Jeffrey Diller (click for contact info, list of my papers, etc.)
Official Time and place: Tuesdays and Thursdays 12:45-2 PM in Hayes-Healy 117.
Office hours: Starting Wed August 19 I’ll hold regular office hours Wed from 4-6. My plan is to send you all an invite to a recurring zoom mtg that starts 4 on Wed. If no one shows by 4:30, I’ll shut it down, so please email if you plan to come later than that, and I can restart the mtg if needs be.
Textbook: Function Theory of One Complex Variable (3rd edition) by Robert E. Greene and Steven G. Krantz. No single textbook can be expected to present all points of view on this subject, so if you don't find Greene and Krantz helpful, I'd suggest looking at these two reserve books, or browsing the library stacks for math books whose numbers begin with QA331. I’ll point you to three other sources in particular:
Complex Analysis by Lars Ahlfors. Ahlfors was one of the greatest complex analysts of the twentieth century, and his book is a classic, albeit maybe slightly dated by now.
Complex Analysis by Stein and Shakarchi. This was written for an undergraduate course at Princeton, part of a four course sequence in analysis, but it’s really at the graduate level and quite good.
Course notes by Terry Tao, who is arguably the world’s best living analyst. His account of things is almost always interesting and worth reading. Beware, however, that since the notes are in blog form with last-written entries appearing at the top, you have to scroll down to the bottom to get to the first “chapter.”
Turns out we’re not doing course reserves this semester, so I’m working on getting the libraries to get electronic access to the first two books.
What is complex analysis? “Calculus meets complex numbers” might serve as a starting description of complex analysis, but this doesn't do justice to the potency of the combination. The notion of “imaginary” number has been around since at least the Renaissance. But systematic attempts to take it seriously and to integrate it into algebra, analysis, and geometry only really got going in the nineteenth century with the work of Cauchy, Riemann and others. Many facts (e.g. the prime number theorem) that ostensibly belong to other areas of mathematics are difficult, if not impossible, to state or prove without complex analysis. And many physical theories (e.g. signal processing, quantum mechanics) are most naturally expressed in terms of complex analysis. In the first term of this two semester sequence, I hope to present a large part of the ``classical (i.e. 19th century) theory'' of complex analysis.
What this course will cover: Topics for the first semester are fairly standard. I hope to cover chapters 1-7 of the textbook. A more precise list of topics, in roughly the order we'll meet them, is as follows.
Geometry and arithmetic of complex numbers.
Definition and basic properties of complex analytic functions.
Contour integrals and Cauchy's Theorems.
Consequences and applications of Cauchy's Integral formula, including but not limited to
Liouville's theorem;
the maximum principle;
isolated singularities;
Calculus of residues;
The general form of Cauchy's theorems.
Conformal Mappings
Normal families and the Riemann mapping theorem.
(The Poincare metric)
(Schwarz-Christoffel transformations)
Harmonic Functions
(Subharmonic functions and the Dirichlet problem)
(Monodromy, Elliptic modular functions, and Picard's Theorems)
Parenthetic topics are things I'd like to cover if time permits. Time is, however, a rather unforgiving taskmaster.
Homework: Homework problems will account for 50% of your grade in this course. I'll assign new problems by noon every Friday and expect you to upload a copy of your solutions by noon the following Friday. Note that we might or might not grade all solutions. Regardless, I plan to at least write up solutions to all the problems and to make them available to you. I strongly encourage you to collaborate with your fellow students when solving homework problems, but you must write up solutions yourself. That is, you may not copy from someone else’s solutions.
Exams: There will be a midterm and final exam in this course. They'll be worth 20% and 30% of your grade, respectively. Both will be in take home format. The midterm will be given to you after class on Thursday Sept 17 to be completed in the next 48 hours. The final will be posted by 5 PM on 11/14/20. You will have until 5 PM 11/17 to complete it. You are welcome to consult with me or the textbook or your class notes for exams. Anything else (e.g. working with other students, looking on the web or at other textbooks, etc) will be regarded as cheating.
Necessary Background: Prior exposure to complex analysis is helpful, but not necessary. Mostly what I expect is familiarity with understanding and writing mathematical proofs, particularly the epsilon/delta sort that arise in undergraduate analysis (or advanced Calculus) courses. Familiarity with topology of R^n (e.g. open, closed, compact, and connected sets and the theorems concerned with them) is somewhere between helpful and necessary.
Any instance of cheating will be dealt with according to Notre Dame’s Academic Code of Honor.
Covid-19 stuff: if you’re like me, you’ve gotten way more email than a person can absorb regarding steps we’re to take to mitigate the spread of covid-19. If I see any problems in this direction concerning our class, I’ll try to point them out. Please don’t hesitate to return the favor if you see me doing something I shouldn’t or not doing something I should or some such. Concerning specifics, let me only note here that you’re supposed to pick a seat and stick with it in the first week of class, reporting where you sat according to these directions.