This course ended December 18, 2014. |
Department of Mathematics and Computer Science Hobart and William Smith Colleges Fall 2014. Instructor: David J. Eck (eck@hws.edu) Course Handout: http://math.hws.edu/eck/courses/math448_f14.html Monday, Wednesday, Friday, 12:20–1:10 PM Room Lansing 301 Scheduled Office Hours in Lansing 313 (but also check Lansing 310): Monday: 1:30–2:50 Tuesday: 11:00–1:00 Wednesday: 11:15–12:10 Friday: 11:15–12:10
Homework | ||
1. Due September 5 | 2. Due September 12 | 3. Due September 17 |
4. Due September 24 | 5. Due October 1 | 6. Due October 17 |
7. Due November 3 | 8. Due November 14 | 9. Due November 24 |
Welcome to the course!
The reading for the week is Chapter 1 (in the textbook Complex Analysis, 3rd edition, by Joseph Bak and Donald Newman). You should try to get a copy of the book and read Sections 1.1, 1.2, and 1.4 before class on Wednesday. Read 1.3 and 1.5 before class on Friday. I will probably ask for volunteers to do some small presentations on Friday. We will discuss this in class on Monday, along with a general discussion of how the course should work.
From Chapter 1, we still need to look (briefly) at Section 1.5, about stereographic projection, the point at infinity, and the Riemann sphere. We will move on to Chapter 2, and we should cover most or all of that chapter this week. Chapter 2 is about polynomials and power series over the complex numbers. It also introduces the complex derivative for functions in general. The first question is how to tell when a complex polynomial P(x,y) is actually a polynomial Q(x+iy) in a single complex variable. This is a somewhat unusual approach that will lead to some insight into the famous "Cauchy-Riemann equations," which determine whether a function is complex differentiable.
We will finish Chapter 2 and start Chapter 3 this week. Chapter thee introduces analytic functions. A function is analytic on an open set if it is differentiable at every point in that set. We will see that a function f(z) = u(x,y) + i*v(x,y) is analytic on an open set if and only if the partial derivatives u_{x}, u_{y}, v_{x}, v_{y} exist and are continuous and satisfy the Cauchy-Riemann equations on the open set. We will also define the standard functions e^{z}, cos(z), and sin(z).
We will be working on Chapter 4 this week. This chapter introduces line integrals and proves some basic results about them, including a complex analog of the Fundamental Theorem of Calculus and the existence of antiderivatives of entire functions. On Wednesday, we will have some student presentations of results from the textbook (see Homework 4).
We have finished Chapter 4. I will start the week by reviewing some of the results that we have seen so far. We will then move on to Chapter 5. In Chapter 5, we start seeing payoff for the work we have done. Chapter 5 deals with entire functions, but a lot of it will carry over to more general results about analytic functions on open sets. The major is result is the Cauchy Integral Formula for entire functions, which shows that the value of an analytic function at a point can be obtained by integrating along a circle that contains that point in its interior. From there, we will show that every analytic function is given by a power series with infinite radius of convergence. As a side benefit, we get a nice proof of the Fundamental Theorem of Algebra.
I've decided to skip Newton's method (from the end of Chapter 5), so we will move on to Chapter 6. The first two sections of that chapter essentially extend results from Chapter 5 about entire functions to the more general case of functions analytic on an open set. Since we already did the general case, there is nothing new for us in 6.1 and 6.2. The rest of the chapter, however, covers some interesting and important results, including the uniqueness theorem for analytic functions and the maximum modulus principle. I also want to spend some time on the topological background of this material, accumulation points and compact sets.
There is no class on Monday because of Fall break.
There is an in-class test on Monday of next week, and a take-home test will be handed out in class on the 17th. Aside from discussing the test, we will be working on Chapter 7.
Here is the study guide for the test that was handed out in class on the 15th.
There is an in-class test on Monday, and a take-home part that is due at the start of class on Friday.
For Wednesday and Friday, we will be moving on to Chapter 8, but I plan to deviate somewhat from the presentation in the book. In particular I will use a more common and intuitive definition of simply connected domain, and will give a more informal proof of the full version of the Cauchy-Goursat theorem.
We will work on Chapter 9 this week, covering isolated singularities and Laurent series.
The reading for the week is Chapter 10. We will cover residues and the Cauchy Residue Theorem, and we will look at several theorems that follow from it. Chapter 10 will probably carry over into the first part of next week.
You should choose a final presentation topic by the end of this week.
We will work on applications of the Residue Theorem to real integrals and series, Chapter 11 of the textbook. The homework for the week also has some information about the final presentation.
We will finish up Chapter 11 on Monday. (I don't plan to do the last section, on series involving binomial coefficients). After that, we will move on to Chapter 13, which introduces conformal mapping. We will spend the rest of the semester on conformal mapping and harmonic functions.
There were no classes on Wednesday or Friday, because of Thanksgiving. On Monday, students presented solutions to some problems, and we discussed linear fractional transformations.
There is a test on Wednesday. A study guide is available.
An here is the take-home part of the test, due next Monday.
Aside from the test, we will continue to investigate linear fractional transformations, conformal mappings, and automorphisms. For example, we will find all automorphisms of the unit disk and of the Riemann Sphere.
The takehome part of the second test is due on Monday. We will spend part of the week looking at at least part of the proof of the Riemann Mapping Theorem. In the remaining time, we will go back and review some of the major topics from the course.
Final presentations will take place during the scheduled final exam period, Thursday, December 18, from 1:30 to 3:30. You are required to be present for all presentations. You are strongly encouraged to come in before Thursday to give me a preview of your presentation and hear any suggestions that I might have.