Instructor: Luke Rogers
Instructor's Office Hours: Tuesday 3:15-4:15pm, Friday 1-2pm.
Lectures: Tuesdays and Thursdays, 2:00-3:15 in MSB411.
Textbook: Complex Analysis, by Lars Ahlfors.
Grading Scheme: 50% homework + 50% final exam, or final exam grade, whichever is better.
The textbook I have chosen for this course is a classic, and a standard text in the field, but it is not the easiest text to read. I chose it deliberately, in part for exactly the reason that when you are reading it you will sometimes need to reformulate what Ahlfors is saying into more formal language so that you can verify all steps in the proofs. I could have chosen a book in which the proofs are more formal and the reasoning laid down more concretely, but my experience has been that this often leads students into the trap of reading only for a superficial line-by-line understanding of arguments rather than the ability to develop the arguments from the conceptual basis. The former level of understanding is, frankly, insufficient for a research career in mathematics. The one advantage of this text is that if you can turn Ahlfors' exposistion into formal statements of theorems and proofs that satisfy you, then you probably actually understand what is going on. If you cannot, then you ought to be aware that you do not know what is going on.
One question I have already been asked is whether you need the text. The strict answer is no - you can learn everything from other books (there are many on this topic, and quite a number in the library). I will assign some questions from the text, but you can copy them out from someones text (mine, for example) if you need to. However, I do expect that you will find the text useful if you use it in the manner described above. If you intend a career in mathematics it is also an investment. With that said, it is expensive, and if you want to share a copy (even a library copy) with a friend (or two friends) that should suffice.
Our goal is to cover most of the material in the first 6 chapters of the text. In any complex analysis course one must begin by dealing with basic algebraic properties of the complex numbers and the geometry of these operations in the complex plane (a week or so). The bulk of the course is then concerned with the many geometric, topological and analytic properties of holomorphic (analytic) functions. Following the general flow of the text we will look first at some geometric mapping properties of certain classes and examples of holomorphic functions (probably about 3 weeks), then examine complex integration and the associated topological notions (about 4 weeks), use these results to obtain analytic properties and representations as series and products (3-4 weeks), and then return to more general mapping properties (1-2 weeks). Some topics may be ommitted (likely candidates are sections 2.4, 3.4, 3.5 of Chapter 3, sections 4.1-4.4 of Chapter 5) and the amount of Chapter 6 that is covered will depend on the time available, although sections 1.1-1.4 of Chapter 6 will definitely appear.
It has been pointed out to me that this is a lot of homework. Probably true, and we should talk about it in class - remind me! None of these problems is intended to be insanely hard, so as a first approximation, if you are not getting anywhere on a problem and have worked on it for 30-40 minutes, it might be worth putting it aside until you can talk to me or a fellow student and get some suggestions.
| Week | Topics | Reading | Exercises |
| 1 | Complex Numbers: Algebra and Geometric representation. | Chapter 1 | 1.1.1: 2,3 1.1.2: 4 1.1.4: 3,4 1.1.5: 1,3,4 1.2.1: 1,2 1.2.2: 2,4 1.2.3: 1,5 1.2.4: 5 Due Friday 2/20/09 HW1 Solutions |
| 2 | Complex functions: analyticity. | Chapter 2.1 and 2.2 | 2.1.2: 4,5 2.1.4: 1,4,6 Due Friday 2/27/09 |
| 3 | Complex functions: power series, exponential, logarithm. | Chapter 2.2 and 2.3 | 2.2.3: 3,4,6 2.2.4: 2,4,8 2.3.2: 2,3 2.3.4: 4,5,6 Due Friday 2/27/09 HW2 Solutions |
| 4 | No Classes this week - I am away. Classes moved to 2/6 and 2/20 Topological background, arcs and curves Conformal mappings |
Chapter 3.1 and 3.2 | 3.1.2: 6 3.1.3: 4,5,7 3.2.2: 1 Due Thursday 3/5/09 HW3 Solutions |
| 5 | Linear fractional transformations, Standard conformal maps. | Chapter 3.3 and 3.4 | 3.3.1: 1 3.3.3: 1,5,7,8 3.4.2: 2,3,7 Due Thursday 3/19/09 HW4 Solutions |
| 6 | Line Integrals, Greens theorem, Cauchy theorem for rectangle | Chapter 4.1 | 4.1.3: 2,3,6,8 Due Thursday 3/26/09 | 7 | Cauchy theorem on disc, Cauchy integral formula | Chapter 4.2 | 4.2.2: 1,2 Due Thursday 3/26/09 HW5 Solutions 4.2.3: 1,2,5 Due Thursday 4/2/09 |
8 | Local properties of analytic functions | Chapter 4.3 | 4.3.2: 1,3,5,6 Due Thursday 4/2/09 HW6 Solutions 4.3.3: 1,4 Due Thursday 4/9/09 HW7 Solutions |
9 | General Cauchy theorem | Chapter 4.4 | 4.3.4: 1,2,3 4.4.7: 3,5 4.5.2: 1,2 Due Thursday 4/16/09 HW8 Solutions |
10 | Residue calculus, Harmonic Functions | Chapter 4.5, 4.6 | 4.5.3: 1,3,4 Due Thursday 4/23/09 HW9 Solutions |
11 | Power series and partial fraction expansions | Chapter 5.1, 5.2 | 4.6.2.2 5.1.1.1 5.1.2.3 Due Thursday 4/30/09 HW10 Solutions |
12 | Normal families, Riemann mapping theorem | Chapter 5.5, 6.1 |