Synopis: The first course in a two semester sequence. The plan is to cover a large part of the classical theory of functions of a single complex variable. “Calculus meets complex numbers” might serve as a starting description of complex analysis, but it doesn't do justice to the subject. 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. Complex analysis feeds into all the central areas of mathematics:
Algebra: polynomials furnish basic instances of complex differentiable functions, and the field of complex numbers is algebraically closed. Algebraic geometry is therefore a closely allied subject.
Geometry: where its derivative is non-zero, a single variable complex differentiable function is locally angle-preserving. Moreover, every non-empty proper open subset of the complex plane admits a canonical hyperbolic metric which tends to be contracted by complex differentiable function.
Topology: all orientable surfaces can be given `complex structures’ turning them into Riemann surfaces on which one can mix methods from complex analysis and algebraic topology.
Analysis: besides the obvious connection with calculus, complex analysis turns out to have a rich interaction with potential theory. I’ll probably pursue this more in the second semester than the first.
Instructor: Jeffrey Diller (click for contact info, list of my papers, etc.)
Official Time and place: Tuesdays and Thursdays 12:30-1:45 PM in Hayes-Healy 229.
Office hours: Wednesdays 5-6 and Thursdays 4-5 PM.
Textbook: A Course in Complex Analysis by Saeed Zakeri. Originally, I was going to teach this course sans textbook, but I decided (once again) that it’d be good to have a default reference. No single textbook can be expected to present all points of view on this subject, so below are several other sources that I like. You’ll find I tend to chart my own way in lectures anyhow.
Complex Analysis by Bak and Newman. You can download an electronic copy of this one for free from the ND Libraries website. It’ll be my backup for topics Zakeri doesn’t sufficient cover.
Function Theory of One Complex Variable (3rd edition) by Robert E. Greene and Steven G. Krantz. The book I used last time I taught this class.
Visual Complex Analysis by Tristan Needham. A long, leisurely and philosophical take on complex analysis aimed at undergraduates. It’s become quite popular in recent years.
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.
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.
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.”
Topics list: I hope to cover (most of) chapters 1-7 of the textbook and then additional material as time allows. The ever-evolving schedule page linked at the top will give a detailed list of topics, readings, etc.
Homework: Homework problems will account for 50% of your grade in this course. I'll assign new problems by noon (most) every Friday and expect you to upload a copy of your solutions by midnight the following Thursday. Note that we might or might not grade all solutions. Regardless, I plan to at least sketch out 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. Texed homeworks will be very welcome!
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 person. The midterm will be given outside class from 6-8 PM on Tuesday, October 13 in HH 117. The final is scheduled for Friday, December 16 from 10:30-12:30 in HH 229.
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: I don’t intend to have any particular masking policy for class this semester, but please bring any health concerns to my attention. If you believe you have contracted covid (or any other contagious illness that your instructor and classmates would do well to avoid), please let me know and absent yourself until it’s prudent to return. I’ll make sure you find out what happened in the classes you miss.