Journal for Math 618 Spring
2011
- May 6
- The final exam was given, and solutions are posted at the TAMU eLearning site.
- April 28
- Class today was devoted to examples in which subharmonic peak functions exist, a list of theorems for the final exam, and the course evaluations.
- April 26
- In class, we proved that the upper envelope of the Perron family solves the Dirichlet problem if there exist subharmonic peak functions.
I posted solutions to Assignment 12 at the TAMU eLearning site.
- April 23
- I posted solutions to Assignments 8–11 at the TAMU eLearning site.
- April 21
- We proved that the upper envelope of a Perron family of
subharmonic functions is harmonic.
- April 19
- We discussed the Poisson integral and showed that cutting out
the values of a subharmonic function in a disk and replacing them
with the Poisson integral of the boundary values creates a new
function that is at least as large as the original function (even
when the original function is only upper semicontinuous).
I posted the twelfth assignment,
which is due on April 26 (Tuesday).
- April 14
- We looked at additional examples of subharmonic functions.
- April 12
- I posted the eleventh assignment,
which is due on April 19 (Tuesday).
In class, we discussed upper semicontinuous functions and
subharmonic functions, and we saw a statement of Perron’s method
for solving the Dirichlet problem (when the Dirichlet problem
admits a solution).
- April 7
- We finished the proof of Montel’s fundamental normality criterion for a general connected open set. Then we looked at some examples of solvability and unsolvability of the Dirichlet problem in doubly connected domains in preparation for the study of Perron’s method for solving the Dirichlet problem.
- April 5
- In class, we proved Schottky’s theorem.
I posted the tenth assignment,
which is due on April 12 (Tuesday).
- March 31
- In class, we proved Carathéodory’s inequality, stated Schottky’s theorem, and observed that Schottky’s theorem implies Montel’s fundamental normality criterion for the unit disk.
- March 29
- In class, we finished the proof of Hadamard’s factorization theorem, modulo the proof of Carathéodory’s inequality.
I posted the ninth assignment,
which is due on April 5 (Tuesday).
- March 24
- Class today covered the statement and the proof of Jensen’s formula and part of the proof of Hadamard’s factorization theorem.
- March 22
- In class today, we defined the order and the genus of an entire
function and stated Hadamard’s factorization theorem.
I posted the eighth assignment,
which is due on March 29 (Tuesday).
- March 17
- I posted solutions to the midterm exam at the TAMU eLearning site.
- March 11
- I posted posted solutions to Assignment 7 at the TAMU eLearning site.
In class yesterday, we proved Mittag-Leffler’s theorem about prescribing singular parts of functions on arbitrary open sets in the plane, and we applied Bloch’s theorem from Assignment 7 to prove Picard’s theorem (the range of a nonconstant entire function cannot omit two values).
- March 8
- In class today, we proved the general form of Weierstrass’s theorem about prescribing zeroes of holomorphic functions, and we stated Mittag-Leffler’s theorem about prescribing principal parts at isolated singularities.
- March 3
- I posted the seventh assignment,
which is due on March 10 (Thursday). In class, the midterm exam
was given.
- March 1
- We proved the Weierstrass factorization theorem for entire functions, stated the theorem about prescribing zeroes of a holomorphic function on an arbitrary open subset of the plane, and applied that theorem to construct a noncontinuable holomorphic function on an arbitrary open subset of the plane.
- February 26
- I posted solutions to Assignments 5 and 6 at the TAMU eLearning site.
- February 24
- I posted solutions to Assignment 4 at the TAMU eLearning site. In class, we continued the discussion of infinite products and stated a theorem about the Weierstrass product construction of an entire function having prescribed zeroes. There is no assignment to hand in next week because the midterm exam is scheduled for Thursday, March 3.
- February 22
- We discussed the notion of convergence of infinite products and examined some examples.
- February 17
- I posted the sixth assignment,
which is due on February 24 (Thursday). In class, we discussed the general version of Runge’s approximation theorem as well as the improvement due to S. N. Mergelyan. We also looked at Alice Roth’s Swiss cheese.
- February 15
- The topics today were (i) the statement (but not the proof) of Carathéodory’s theorem about boundary behavior of the Riemann mapping function and (ii) Runge’s theorem about polynomial approximation of analytic functions.
- February 11
- I posted the fifth assignment,
which is due on February 17 (Thursday).
- February 10
- Class today was devoted to a computation-free version of the Fejér–Riesz proof of the Riemann mapping theorem.
- February 8
- We discussed various properties of normally convergent sequences of analytic functions, such as Hurwitz’s theorem and Vitali’s theorem.
- February 4
- The university is closed today because of the weather. I posted
solutions to the first three assignments at the TAMU eLearning site.
- February 3
- We discussed the metric on the space of continuous functions on an
open set in the plane, the Arzelà–Ascoli theorem, and
Montel’s theorem about normal families of analytic functions.
Later I posted the fourth assignment,
which is due on February 10 (Thursday).
- February 1
- In class today, we discussed the notion of analytic continuation along a curve and the monodromy theorem.
- January 27
- The third assignment, due on February 3 (Thursday), is available. In class today, we proved Hadamard’s gap theorem.
- January 25
- We proved Pringsheim’s lemma, stated Hadamard’s gap theorem, and analyzed the example of \(\sum_{n=1}^\infty z^{n!}/2^n.\)
- January 21
- The second assignment, due on
January 27 (Thursday), is available.
- January 20
- In class today, we proved the Schwarz reflection
principle. There is no assignment to turn in next class.
- January 18
- In class today, we discussed the general version of the Schwarz reflection principle and applied it to determine the general form of a biholomorphic automorphism of the unit disk. The proof of the Schwarz reflection principle is held over for next class.
- January 17
- The first assignment, due on January 20, is available. Although the assignment looks long, it is essentially one problem with lots of hints.
- January 16
- Welcome to Math 618. I will be regularly updating this page
with homework assignments, brief summaries of class activities,
and other information.
Harold P. Boas