# Math 617, Theory of Functions of a Complex Variable I, Fall 2006Harold P. Boas

## What's new

Wednesday, December 13
The final examination was given.
Tuesday, December 5
We discussed the plan for the final examination, and we completed the discussion of univalence of analytic functions.
Thursday, November 30
We continued the discussion of the argument principle and related topics, including Hurwitz's theorem and the characterization of local univalence of analytic functions. The assignment for next time (not to hand in) is Exercises 13.3 and 13.4.
Tuesday, November 28
We discussed the circle of ideas around the argument principle, Rouché's theorem, and the open mapping theorem. For next time, read section 12B and do exercises 12.2 and 12.3.
Tuesday, November 21
The third examination was given. Solutions are available.
Thursday, November 16
Students presented some additional calculations of real integrals via contour integration. The assignment is to continue reviewing for the examination to be given on Tuesday, November 21 on sections 8G through 11C.
Tuesday, November 14
Students presented solutions of some of the integration problems involving functions with branches. The assignment is to review for the examination to be given on Tuesday, November 21 on sections 8G through 11C.
Thursday, November 9
We solved problem 10 from the May 2005 qualifying examination, and we evaluated a real integral by using a keyhole contour. For next time, read sections 11B and 11C. Each student will prepare a problem from pages 109-110 to present in class.
Tuesday, November 7
We discussed different ways to define a complex logarithm function and the problem of branches. For next time, do exercises 10.3 (page 103), 10.8 (page 104), and 10.11 (page 105).
Thursday, November 2
We continued the discussion of contour integration applied to principal value integrals. For next time, do exercises 9.1a (page 90), 9.3 (page 93), 9.7b (page 95), and 9.11a (page 98).
Tuesday, October 31
The examinations were returned, and we looked at some examples of computing real integrals via residues. For next time, read section 8G and do exercises 8.14 and 8.18.
Thursday, October 26
The second examination was given. Solutions are available.
Tuesday, October 24
We discussed some applications of residues. The examination next class covers section 7 and parts A-F of section 8.
Thursday, October 19
We continued the discussion of isolated singularities, proved the Casorati-Weierstrass theorem, and discussed residues. For next time, do the following exercises: 8.4 parts d and e, 8.7, 8.10, and 8.12 parts h and i.
Tuesday, October 17
We discussed the classification of isolated singularities of analytic functions. For next time, do the following exercises: 8.1, 8.2, 8.3a, 8.3c, 8.4a.
Thursday, October 12
We discussed two of the homework problems and the convergence and analyticity of the integral defining the Gamma function in the right-hand half plane. For next time, read section 7H and do the following exercises: 7.17 and supplementary exercises 1 and 3 on page 76.
Tuesday, October 10
We discussed some consequences of Cauchy's integral formula: power series expansions, isolated zeroes of analytic functions, and the principle of persistence of functional relations. For next time, read section 7E and do the following exercises: supplementary exercise 1 on page 68, exercise 7.15 on page 69, and exercise 7.16 on page 71.
Thursday, October 5
I have posted solutions to the first examination. In class, we discussed consequences of Cauchy's integral formula. For next time, do the following exercises: 7.7, 7.9, 7.10, and 7.12 (all on page 64).
Tuesday, October 3
The graded examinations were returned, and we discussed Cauchy's integral formula. For next time, do the following exercises: 7.5 and 7.6 on page 63.
Monday, October 2
I have posted a draft of Chapter 2 of the textbook at the TAMU WebCT site.
Monday, September 25
The take-home examination is available. Our class will not meet this week because I am traveling to Amherst College to give a lecture. I will be back on Friday, and the exam is due by 4:00pm on Friday in my office (202 Milner). I expect to be in my office most of the day today, so you can drop by in case you have questions about the meaning of the problems on the exam.
Thursday, September 21
A student presented two solutions to exercise 4.9, and we discussed Goursat's proof of Cauchy's theorem and the statement of Morera's theorem.
Next week we will not have class because I will be giving a lecture at Amherst College on September 27. I will distribute the take-home examination on Monday, September 25, and it will be due in my office by 4:00pm on Friday, September 29. I will be available in my office for questions on Friday, September 22 until the seminar at 3:00pm and on Monday, September 25 most of the day.
Tuesday, September 19
We discussed complex line integrals, the statement of Cauchy's integral theorem, and the existence of logarithms. For next time, read section 6 and work the exercises (not to hand in).
Thursday, September 14
Students presented solutions of some of the exercises in section 4. Also, there was a handout about Mertens's theorem. For next time, do the following exercises: 5.2 on page 40 and 5.7 on page 45.
Tuesday, September 12
Students presented solutions of some of the exercises in section 4. For next time, do the following exercises: supplementary exercises 2 and 5 on page 35 and exercise 4.17 on page 37.
Thursday, September 7
We discussed uniform convergence of power series and its implications. For next time, each student will present one of the exercises from section 4 of chapter 1.
Tuesday, September 5
We discussed some of the previous homework problems as well as convergence of power series and Cauchy's formula for the radius of convergence. For next time, do the following exercises: 3.10 parts f and i (page 26), supplementary exercise 2 on page 26, and 3.13 on page 28.
Thursday, August 31
We discussed the notion of complex differentiability and the Cauchy-Riemann equations. For next time, do the following exercises: 2.2 (page 18; note that the end of the first line of the exercise should say f(0)=1, not f(0)=0), 2.5 (page 20), 2.7 and 2.9 (page 22).
Wednesday, August 30
I uploaded a new version of Chapter 1 with a correction of a typographical error in Exercise 2.2 on page 18.
Tuesday, August 29
We discussed the complex numbers from both the algebraic and the topological point of view, and we looked at some sample exercises from the book. The standing assignment is to read the book. To hand in at the beginning of next class, do the following exercises: supplementary exercise 2 on pages 7-8; exercise 1.17 on page 11; supplementary exercise 2c on page 12; and exercise 1.24 on page 14.
I have uploaded a new version of Chapter 1 with two typographical errors corrected (one on page 7 and one on page 13).
Monday, August 28
This site went live today. Watch for regular updates. The first-day handout is available online. Students who are registered in the class can download a pdf file of chapter 1 of the textbook at the TAMU WebCT site.