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Math 618
Spring 2016 Journal

Tuesday, May 10
The final examination was given, and solutions are available.
Friday, April 29
I posted updated lecture notes.
Thursday, April 28
We discussed the statements of Bloch’s theorem and Schottky’s theorem, the deduction of Montel’s fundamental normality criterion from Schottky’s theorem, and the deduction of Picard’s great theorem from Montel’s criterion.
Tuesday, April 26
We discussed Norbert Wiener’s necessary and sufficient condition (in terms of logarithmic capacity) for a boundary point of a domain to be a regular point for the Dirichlet problem, and we discussed three-lines and three-circles theorems for subharmonic functions and holomorphic functions.
Thursday, April 21
We finished the solution of the Dirichlet problem modulo the existence of subharmonic peak functions (barriers), and we proved that a sufficient condition for the existence of a subharmonic peak function at a boundary point \(z_0\) is the existence of a line segment \([z_0,z_1]\) lying in the complement of the domain.
Tuesday, April 19
We proved Harnack’s inequality and Harnack’s principle about convergence of monotonic sequences of harmonic functions and then worked on the second part of the solution of the Dirichlet problem on domains that admit subharmonic peaking functions.
Thursday, April 14
We proved, modulo establishing Harnack’s principle, that the upper envelope of the Perron family is necessarily harmonic.
Tuesday, April 12
We discussed the unsolvability of the Dirichlet problem on the punctured disk, the notion of subharmonic peak function, and the set-up of Perron’s method for solving the Dirichlet problem on bounded domains that admit peak functions.
I posted an exercise on subharmonic functions and Perron’s method for solving the Dirichlet problem to hand in on Tuesday, April 19.
Thursday, April 7
We discussed Carathéodory’s theorem about boundary continuity of the Riemann mapping theorem, along with some applications; we looked at three equivalent definitions of upper semicontinuous functions; and we saw three equivalent definitions of subharmonic functions.
Tuesday, April 5
We completed the solution of the Dirichlet problem for the unit disk by proving that the limit of the Poisson integral at the boundary equals the given boundary function at points of continuity of that function.
Thursday, March 31
We derived the Poisson integral representation for harmonic functions on the unit disk and the related Schwarz formula that exhibits an analytic function on the unit disk whose real part is a prescribed harmonic function. The next goal is to show that the Poisson integral solves the Dirichlet problem on the unit disk.
Tuesday, March 29
In class, we discussed several equivalent ways to define harmonic functions: locally the real parts of analytic functions (globally on simply connected open sets), solutions of Laplace’s equation, and continuous functions satisfying the small-circle mean-value property. We ended with the statement of the Poisson integral representation of harmonic functions on the unit disk.
Here is the assignment to hand in on Tuesday, April 5.
  • Solve problem 3 (about entire functions) on the January 2014 qualifying exam.
  • 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 \(\log\left|z\right|^2\).
  • 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.
  • Solve problem 9 on the January 2012 qualifying exam.
  • Solve problem 6 on the January 2010 qualifying exam.
Thursday, March 24
We finished the proof of Hadamard’s factorization theorem (a different proof from the one in the textbook).
The next topic to be discussed is harmonic functions.
Tuesday, March 22
We proved Carathéodory’s inequality and applied the result to prove a special case of the remaining part of Hadamard’s factorization theorem.
Monday, March 14
For Pi Day, I posted updated lecture notes.
Thursday, March 10
In class, we discussed Jensen’s formula and the application to the proof that an entire function of finite order has finite rank no larger than the order.
I posted an exercise on the order of an entire function to hand in on Thursday, March 24 (after Spring Break).
Tuesday, March 8
We further discussed the statement of Hadamard’s factorization theorem and proved that the order of an entire function is no larger than the genus plus one.
Thursday, March 3
In class, we discussed the notion of the order of an entire function, the statement of Hadamard’s factorization theorem, and some applications.
I posted an exercise about combining the Weierstrass and Mittag-Leffler theorems to hand in on Thursday, March 10.
Tuesday, March 1
We discussed Mergelyan’s generalization of Runge’s approximation theorem, along with the statement and proof of Mittag-Leffler’s theorem about functions with prescribed singularities.
Monday, February 29
I posted solutions to the midterm examination.
Thursday, February 25
The midterm examination was given.
Tuesday, February 23
We reviewed for the midterm examination to be given next class, and we finished the proof of Runge’s approximation theorem (the construction of the integration path).
Thursday, February 18
We discussed two pieces of the proof of the general version of Runge’s approximation theorem: pole pushing and the approximation of the Cauchy integral by Riemann sums.
There is no assignment to hand in next week, for the midterm examination takes place on Thursday, February 25.
Tuesday, February 16
We proved the Weierstrass theorem about analytic functions with prescribed zeroes on an arbitrary open subset of the plane.
I posted lecture notes for the course to date.
Thursday, February 11
In class, we discussed the theorem of Weierstrass about analytic functions with prescribed zeroes, along with some applications.
The assignment to hand in on Thursday, February 18 is Exercises 4, 10, and 11 in \(\S\)5 of Chapter VII of the textbook.
Tuesday, February 9
I presented a solution of Exercise 9 from the previous assignment.
Thursday, February 4
In class, we discussed the convergence of infinite products and did a warm-up for the proof of the theorem of Weierstrass about the existence of entire functions with prescribed zeroes.
Here is the assignment to hand in on Thursday, February 11.
Tuesday, February 2
We determined the holomorphic automorphisms of the unit disk, finished the proof of the Riemann mapping theorem, and discussed Frédéric Marty’s notion of the spherical derivative and its role in characterizing normal families of meromorphic functions.
Thursday, January 28
We completed the first two steps in the proof of the Riemann mapping theorem by setting up and solving the problem of finding an injective holomorphic function mapping a simply connected region into the unit disk, taking a specified point \(z_0\) to the origin and maximizing the distance from the origin of the image of a second specified point \(z_1\). One deduction was left as part of the assignment (Exercise 10 in \(\S\)2). In preparation for the final step of the proof, we proved the Schwarz lemma.
Here is the assignment to hand in on Thursday, February 4.
  • Exercise 5 in \(\S\)1 of Chapter VII.
  • Exercise 10 in \(\S\)2 of Chapter VII. The open set \(G\) should be assumed to be connected in this problem.
  • Exercise 2 in \(\S\)3 of Chapter VII. This problem asks you to supply the half of the proof of Theorem 3.8 that is “left to the reader” in the textbook. In particular, you need to read \(\S\)3, which I have not discussed in class.
  • Exercises 2 and 9 in \(\S\)4 of Chapter VII.
Tuesday, January 26
We followed up on the concept of normal families by discussing Montel’s fundamental normality criterion; a proof (using the Baire category theorem) that a pointwise bounded family of continuous functions is necessarily locally bounded on a dense open set; an application of Runge’s approximation theorem to construct an example of a pointwise convergent but not locally bounded sequence of holomorphic functions; and the definition of the Julia set.
Thursday, January 21
We discussed Montel’s theorem characterizing normal families of holomorphic functions as being locally bounded families.
Here is the assignment to hand in on Thursday, January 28.
  • Exercise 8 in \(\S\)1 of Chapter VII.
  • Exercise 8 in \(\S\)2 of Chapter VII.
  • Exercise 12 in \(\S\)2 of Chapter VII.
  • Exercise 13 in \(\S\)2 of Chapter VII.
Tuesday, January 19
At the first class meeting, we discussed some machinery in preparation for proving the Riemann mapping theorem: namely, a metric on continuous functions that characterizes uniform convergence on compact subsets of an arbitrary open subset of \(\C\). We ended with the statement of the Arzelà–Ascoli theorem.
The assignment is to read the proof in the textbook of the Arzelà–Ascoli theorem.