# Math 618, Theory of Functions of a Complex Variable II, Spring 2007Harold P. Boas

## What's new

Thursday, April 26
In our last class meeting for the semester, we discussed Runge's approximation theorem and its proof.
Tuesday, April 24
We proved the general case of the theorem of Weierstrass: on any planar region, there exists an analytic function with prescribed zeroes (subject to the necessary condition that the zeroes do not accumulate in the interior of the region).
Sunday, April 22
I have posted a note that patches the hole in the discussion of the example in Thursday's class.
Friday, April 20
I have posted the third take-home examination, which is due on Friday, April 27, by 4:00pm.
Thursday, April 19
We finished proving Hadamard's factorization theorem. In view of the upcoming examination, there is no homework assignment.
Tuesday, April 17
We continued the proof of Hadamard's factorization theorem, and in particular we derived the Poisson-Jensen formula. The assignment for next time is to do problem 1 from the May 2003 complex analysis qualifying examination.
Thursday, April 12
We derived a generalization of Jensen's formula to cover the case when the function has a zero at the origin, and we proved half of the Hadamard factorization theorem. The assignment for next time is to do problem 9 from the May 1999 qualifying examination and problem 8 from the January 2005 qualifying examination.
Tuesday, April 10
We proved Jensen's formula and started the proof of the Hadamard factorization theorem for entire functions of finite order. The assignment for next time is to do Exercise 32.1 and problem 7 from the May 1995 complex analysis qualifying examination.
Thursday, April 5
We proved the Weierstrass factorization theorem for entire functions and discussed the notion of the order of an entire function and the statement of Hadamard's factorization theorem. The assignment for next time is to do part (f) of Exercise 31.4 (which appeared on the June 1986 complex analysis qualifying examination) and problem 8 from the January 2004 qualifying examination.
Tuesday, April 3
We discussed the notion of convergence of infinite products in connection with the problem of reconstructing entire functions from their zeroes. The assignment for next time is to do Exercises 31.2 and 31.3.
Sunday, April 1
I have posted a pdf file of Chapter 5 of the textbook to WebCT.
Thursday, March 29
The second examination was returned, and we discussed some of the problems on the exam. In particular, we talked about the automorphisms of the annulus, Harnack's inequality, and Bohr's theorem. The assignment for next time is to read section 29 about Riemann surfaces.
Tuesday, March 27
We discussed normal families of analytic functions and one of Montel's theorems. The assignment for next class is to do the following two problems (not in the textbook): (1) Show that if a family of functions is pointwise equicontinuous, then the family is uniformly equicontinuous on every compact set. In other words, if the number δ in the definition of continuity can be taken to be independent of the function at an arbitrary point, then δ can be taken to be independent of both the function and the point as the point varies within a compact set. (2) Change the set-up in the proof of the Riemann mapping theorem to consider the family of analytic functions that (a) map a simply connected region D into the unit disk, (b) take a specified point to the origin, and (c) have the derivative normalized at the special point to be a positive real number. (The functions are not assumed to be one-to-one, which is the only difference from the set-up we used in class in the proof of the Riemann mapping theorem.) Show that there is an extremal function in this new family of functions and that the extremal function is again the Riemann mapping function (that is, the extremal function is both one-to-one and onto).
Thursday, March 22
We discussed two of the three steps in the proof of the Riemann mapping theorem.
Tuesday, March 20
I posted a new version of the take-home exam that corrects an error in the formula for the Poisson kernel: the coefficient of the cosine term should be 2rR instead of 2r. By popular demand, the due date for the exam is changed from Friday 23 March to Monday 26 March at 4pm.
In class, we discussed how the Riemann mapping theorem, the solvability of the Dirichlet problem, and the existence of the Green function are all essentially equivalent.
Monday, March 19
The second take-home exam is available.
Thursday, March 8
We continued the discussion of Möbius transformations. The assignment for spring break is to travel safely. Reminder: the second take-home examination immediately follows spring break.
Tuesday, March 6
Reminder: the second take-home examination will be given immediately following spring break. In class, we discussed the analytic automorphisms of the complex plane, of the extended complex plane, and of the unit disk. The assignment for next time is to do exercises 25.2, 25.3, and 25.5.
Thursday, March 1
We continued the discussion of the Dirichlet problem and its applications. In particular, we proved a removable singularities theorem for bounded harmonic functions. The assignment for next time is to read sections 20E,F and 22A,B,C and to do Exercise 22.1.
Tuesday, February 27
We discussed the solution of the Dirichlet problem on the unit disk via the Poisson integral. The assignment for next time is to do problem 1 from the May 2005 qualifying examination and problem 6 from the January 2004 qualifying examination.
Thursday, February 22
We discussed some of the problems from the first exam. The assignment is to read sections 20A and 20B and to do exercises 19.6 and 20.2.
Tuesday, February 20
We discussed harmonic functions and harmonic conjugates. The assignment for next time is to do exercise 19.3 and parts (a) and (d) of exercise 19.4.
Sunday, February 18
I have posted a pdf file of Chapter 4 of the textbook to WebCT.
Friday, February 9
The take-home exam is available.
Thursday, February 8
We continued the discussion of boundary behavior of power series, including Pringsheim's lemma and the theorems of Abel and Tauber. I will post the take-home examination before departing on my trip.
Tuesday, February 6
I will be out of town next week, so our class will not meet on February 13 and February 15, but we will have a take-home examination next week. In class today, we discussed topics from the reading: the Schwarz reflection principle, overconvergence, boundary singularities, the Hadamard gap theorem, and examples. The assignment for next time is to do exercises 17.2 and 17.5 from the textbook and problem 3 from the January 2003 qualifying examination.
Monday, February 5
I have uploaded to WebCT a revised pdf file of Chapter 3.
Thursday, February 1
Students presented the problems assigned last time. The assignment for next class is to read section 17 of the textbook.
Tuesday, January 30
We discussed analytic continuation and the monodromy theorem. For next time, different groups will prepare to present the following problems from past qualifying examinations: number 2 from January 2005, number 7 from January 2005, number 10 from May 2005, and number 8 from May 2006.
Thursday, January 25
We discussed some consequences of the maximum principle and the Schwarz lemma, in particular, Carathéodory's inequality (in the textbook) and a theorem of Rogosinski (not in the textbook): namely, if f is an analytic function mapping the unit disk into itself and omitting the value 0, then the modulus of the derivative f'(0) does not exceed 2/e (sharp).
The assignment for next time is to do problem 2 from the May 2001 complex analysis qualifying examination (a variation on the Schwarz lemma) and problem 6 from the January 2002 complex analysis qualifying examination (a variation on the maximum principle).
Tuesday, January 23
We discussed the maximum principle and the Schwarz lemma. The assignment for next time is to do Exercise 16.2 (page 149) and Supplementary Exercises 2 and 6 at the end of section 16A (page 150).
Thursday, January 18
We discussed the representation of analytic functions in annuli by Laurent series. The assignment for next time is to do Exercise 14.8 on page 137, Exercise 14.10 on page 138, Supplementary Exercise 5 on page 143 (at the end of section 15C), and Supplementary Exercise 2 on page 147 (at the end of section 15F).
Tuesday, January 16
The university was closed today because of icy conditions, so classes did not meet. I uploaded a revised Chapter 2 to WebCT.
Monday, January 15
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 pdf files of the first three chapters of the textbook at the TAMU WebCT site.