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Math 618
Spring 2015 Journal

Thursday, May 7
The final examination was given.
Thursday, April 30
In this final class meeting, we established Montel's fundamental normality criterion, thus polishing off the proof of Picard's theorems. I posted updated lecture notes.
Tuesday, April 28
We proved Picard's theorems on the range of entire functions and the range of functions near an essential singularity—modulo proving Montel's fundamental normality criterion.
Thursday, April 23
We proved three-lines theorems for bounded subharmonic functions and for bounded holomorphic functions.
Tuesday, April 21
Subgroups of students presented solutions to the exercises from last time about the modular group and its fundamental domain.
Thursday, April 16
We discussed fundamental domains for the modular group and the congruence subgroup along with the construction of the modular function. The assignment—in subgroups—is to prove analogues for the modular group of three statements proved in the textbook for the congruence subgroup: namely, (1) distinct group elements map the interior of the alleged fundamental domain onto disjoint sets; (2) every point in the upper half-plane can be mapped into the alleged fundamental domain by some element of the group; (3) the translation sending \(z\) to \(z+1\) and the inversion sending \(z\) to \(-1/z\) generate the modular group.
Tuesday, April 14
We looked at the mapping and symmetry properties of the sine function as a warm-up to the construction of the modular function.
Thursday, April 9
In class, we discussed the Schwarz reflection principle for holomorphic functions.
The assignment to hand in a week from today is problem 7 on the January 2009 qualifying exam. The suggestion is to make use of the circular version of the reflection principle.
Tuesday, April 7
We proved that the upper envelope of the Perron family of subharmonic functions solves the Dirichlet problem when every boundary point of the domain admits a subharmonic peak function.
Thursday, April 2
In class, we discussed Harnack's principle and subharmonic peak functions.
I posted an assignment on three-circles theorems to hand in a week from today April 14.
Tuesday, March 31
In class, we worked on proving that the upper envelope of the Perron family of subharmonic functions is harmonic.
Here is the assignment to hand in a week from today. In each of the following settings, determine the upper envelope of the corresponding Perron family of subharmonic functions.
  1. The domain is the punctured disk, and the prescribed function on the boundary is equal to a real number \(b\) on the outer boundary and equal to a real number \(c\) at the center.
  2. The domain is the upper half-plane, and the prescribed function on the boundary (the real axis) is equal to \( -(\Re z)^2\).
  3. The domain is the unit disk, and the prescribed function on the boundary is the discontinuous function equal to a real number \(b\) when \(\left|z\right|=1\) but \(z\ne 1\) and equal to a real number \(c\) when \(z=1\).
Thursday, March 26
In class, we discussed three equivalent definitions of subharmonicity and looked at some examples of subharmonic functions.
Here is the assignment to hand in a week from today.
  1. Find an example of an infinitely differentiable real-valued function on the whole plane that has no local maximum (not even a weak local maximum), yet is not subharmonic on any open set.
  2. Find an example of a continuous subharmonic function on the whole plane that is radial (in other words, a function of \(\left|z\right|\) only), positive, and not a convex function.
  3. Find an example of a continuous subharmonic function on the whole plane that is not bounded below and not harmonic on any open set.
  4. Prove that if a function is subharmonic on the entire plane \(\C\) and is bounded above, then the function must be constant.
    (This statement is a generalization of Liouville’s theorem, because \(\left|f\right|\) is subharmonic when \(f\) is holomorphic. Hint: for arbitrary values of constants \(A\) and \(B\), the function \(A+B\log\left|z\right|\) is an available harmonic comparison function on an arbitrary annulus centered at the origin.)
Tuesday, March 24
We discussed the notions of upper semicontinuous functions and subharmonic functions, in preparation for solving the Dirichlet problem on suitable planar domains.
Thursday, March 12
We discussed the question of whether local boundedness characterizes normality of families of harmonic functions, that is, whether Montel’s theorem generalizes from holomorphic functions to harmonic functions. The answer is yes.
Tuesday, March 10
In class, we discussed variations of the maximum principle for harmonic functions.
Thursday, March 5
Here are three exercises to hand in a week from today.
  1. Observe that when \(a\) is a positive real number less than \(1\), the Poisson kernel \[ \frac{1}{2\pi}\cdot \frac{1-|a|^2} {|e^{i\theta}-a|^2} \] can be written as \[ \frac{1}{2\pi}\cdot \frac{1-a^2} {1-2a\cos\theta+a^2}. \] Now show how to solve problem 3 on the August 2013 qualifying exam in a very easy way.
  2. Solve problem 9 on the January 2012 qualifying exam.
  3. Solve problem 6 on the January 2010 qualifying exam.
Tuesday, March 3
In class, we used disk automorphisms to derive the Poisson integral representation for harmonic functions in the unit disk and observed that the formula also shows how to represent a harmonic function in the disk as the real part of a holomorphic function.
Here is an exercise to hand in a week from today: Show that the function \(\log\left|z\right|^2\) is harmonic on \(\C\setminus\{0\}\), the punctured plane, but there is no holomorphic function \(f\) on the punctured plane such that \(\Re f(z)\) equals \(\log\left|z\right|^2\).
Thursday, February 26
This was the day of the midterm exam.
Tuesday, February 24
We made a list of some of the main theorems in the course so far, we solved the question about changing the extremal problem in the proof of the Riemann mapping theorem, and we looked at the problems on normal families due today. The midterm exam takes place next class.
Sunday, February 22
I posted lecture notes for the course so far.
Thursday, February 19
In class, we discussed the preservation of symmetry under linear fractional transformations and concluded the proof of the Riemann mapping theorem.
Here is an exercise to hand in a week from today: Suppose, in the proof of the Riemann mapping theorem, the extremal problem is solved within the class of holomorphic functions that are not necessarily injective (but that satisfy all the other conditions). Is the solution to the extremal problem for this larger class of functions again the Riemann mapping function, or is the extremal function something else? Explain. Update: In view of the midterm exam taking place on February 26, this problem is no longer assigned to hand in. We solved the problem in class on February 24.
Tuesday, February 17
In class, we worked on a proof of the Riemann mapping theorem.
Here are some exercises on normal families to hand in a week from today.
Thursday, February 12
I posted an exercise on convergence and disk automorphisms to hand in a week from today.
Tuesday, February 10
In class, we concluded the discussion of the Weierstrass factorization theorem and discussed the Arzelà–Ascoli theorem.
An exercise to hand in a week from today is to prove that if a family of functions is equicontinuous at each point of a compact set, then the family is automatically uniformly equicontinuous on that compact set.
Thursday, February 5
I posted an exercise on combining the theorems of Weierstrass and Mittag-Leffler to hand in a week from today.
Tuesday, February 3
In class, we discussed the notion of convergence of infinite products.
An exercise to hand in next class is number 6.5 on page 102 of the textbook: a proof that \[ \frac{\pi}{2} = \left( \frac{2}{1}\right) \left( \frac{2}{3}\right) \left( \frac{4}{3}\right) \left( \frac{4}{5}\right) \cdots \] (known as Wallis’s formula). In particular, you need to check that the infinite product on the right-hand side converges.
Here are two exercises to hand in a week from today.
  1. Exercise 13.3 on page 256 of the textbook.
  2. Prove the following product representation of the cosine function: \[ \cos(\pi z) = \prod_{n=1}^\infty \left( 1 - \frac{z^2}{\left(n-\frac{1}{2}\right)^2} \right) = \prod_{n=1}^\infty \left( 1- \frac{4z^2}{(2n-1)^2} \right). \]
Thursday, January 29
In class, we discussed Hadamard’s gap theorem and Mordell’s proof.
I posted an exercise on Runge’s theorem.
Tuesday, January 27
In class, we discussed an important application of Runge’s approximation theorem: namely, Mittag-Leffler’s theorem about functions with prescribed singularities and prescribed principal parts.
I posted an exercise on approximation.
Thursday, January 22
In class, we discussed Runge’s approximation theorem and Mergelyan’s generalization.
I posted an exercise on Alice Roth’s Swiss cheese.
Tuesday, January 20, 2015
In class, we discussed the simplest version of Runge’s approximation theorem.
The group assignment due next Tuesday is to prove the existence of a universal power series \(\sum_{n=0}^\infty c_n z^n\) that has radius of convergence equal to \(1\) and has the additional property that for every closed disk \(D\) disjoint from the closed unit disk and for every polynomial \(p\), there exists a subsequence of partial sums of the series that converges uniformly on \(D\) to \(p(z)\).
(A series that has a subsequence of partial sums converging outside the disk of convergence is called “overconvergent.” The series you are to construct is overconvergent in an extremely strong way.)
Sunday, December 7, 2014
Welcome to Math 618, Spring 2015. This site went live today. I will post regular updates during the semester.