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Math 618
Spring 2013 Journal

Wednesday, May 8
The final examination took place.
Thursday, April 25
In our final class meeting for the semester, we finished the proof of Montel's fundamental normality criterion and deduced Picard's great theorem as a corollary.
Tuesday, April 23
We did most of the proof of Montel’s fundamental normality criterion using the modular function and the monodromy theorem. Also we compiled the following list of theorems to know for the final exam.
  • the Arzelà–Ascoli theorem
  • the theory of automorphisms of special planar regions
  • Hurwitz’s theorem
  • Marty’s criterion for normality
  • Mergelyan’s theorem
  • Mittag-Leffler’s theorem
  • the theory of Möbius transformations
  • the monodromy theorem
  • Montel’s theorem
  • the Müntz–Szász theorem
  • Picard’s theorem
  • the Poisson integral formula
  • the technique of pole pushing
  • the Riemann mapping theorem
  • Runge’s theorem
  • the Schwarz lemma
  • the Schwarz reflection principle
  • the Weierstrass factorization theorem
  • Weierstrass’s theorem on the existence of holomorphic functions with prescribed zeroes
Thursday, April 18
We discussed the monodromy theorem.
The assignment for next time (not to hand in) is to compile a list of the major theorems from the course.
Tuesday, April 16
Students presented solutions to the exercises of showing that the alleged fundamental domains of the modular group and the congruence subgroup do indeed have the required properties.
There is no assignment to hand in next week, for next week is the final week of the semester for Tuesday–Thursday classes.
Thursday, April 11
We discussed the construction of the modular function: a holomorphic function in the upper half-plane that is invariant under the congruence subgroup of the modular group and that maps the upper half-plane locally biholomorphically onto the complex plane with punctures at \(0\) and \(1\).
Tuesday, April 9
We discussed the modular group, the congruence subgroup, and the fundamental domains of these groups. Different subsets of the class took on the tasks of showing that the fundamental domains have the alleged properties.
Thursday, April 4
We discussed the notion of symmetry in relation to linear fractional transformations and (finally!) completed the proof that the automorphisms of an annulus are generated by rotations and a special inversion.
Tuesday, April 2
Here is the assignment due next Tuesday (April 9). In class, we discussed the orientability of complex manifolds and the invariance under Möbius transformations of the cross ratio.
Thursday, March 28
We continued the discussion of linear fractional transformations, observing that they are generated by translations, rotations, dilations, and inversions; that they are conformal; and that they preserve generalized circles.
Tuesday, March 26
In class, we discussed Möbius transformations in the setting of projective space.
I posted solutions to the midterm exam.
Here is the assignment due next Tuesday (April 2).
Thursday, March 21
We proved that the local averaging property on circles characterizes harmonic functions, and we verified that the Poisson integral solves the Dirichlet problem for a disk.
Tuesday, March 19
We worked on the proof of the Schwarz reflection principle, arriving at the observation that we needed some sort of analogue of Morera's theorem in the setting of harmonic functions.
Thursday, March 7
We continued discussing automorphism groups of planar domains and reached the statement of the Schwarz reflection principle.
Tuesday, March 5
We determined the holomorphic automorphism groups of the unit disk, the complex plane, the punctured disk, and the punctured plane. In particular, we observed that an injective holomorphic function cannot have an essential singularity.
Thursday, February 28
The take-home midterm exam is available.
In class we discussed the Poisson integral representation for harmonic functions in the unit disk.
Tuesday, February 26
We completed the proof of the Riemann mapping, discussed the failure of the analogous statement in higher dimension, and looked at some ways to recover information about a holomorphic function from its real part.
Thursday, February 21
We progressed with the proof of the Riemann mapping theorem, reducing the problem to a calculation on the unit disk (to be completed next time).
Tuesday, February 19
In class, we discussed Montel's fundamental normality criterion and some examples.
Here is the assignment due next Tuesday (February 26).
Thursday, February 14
We discussed the Riemann mapping theorem and started the proof.
Tuesday, February 12
We continued the discussion of compactness in function spaces and arrived at Montel's theorem about normal families of holomorphic functions. The next assignment is available as a pdf file.
Thursday, February 7
We discussed Weierstrass's theorem about every planar domain supporting a holomorphic function that is singular at every boundary point. Then we turned to the concept of the space of continuous functions as a metric space, the coming application being to the metric space of holomorphic functions. We saw the statement of the Arzelà–Ascoli theorem (proved in Section 28B of the textbook).
Tuesday, February 5
We proved Mittag-Leffler's theorem about the existence of meromorphic functions with prescribed singularities and principal parts. The next assignment is available as a pdf file.
Thursday, January 31
We proved (modulo the homework exercise) the theorem of Weierstrass about the existence of entire functions with prescribed zeroes. And we discussed the statements of the Weierstrass factorization theorem, the Weierstrass theorem for general regions, and the theorem of Mittag-Leffler.
Tuesday, January 29
We continued the discussion of infinite products and stated the theorem of Weierstrass about the existence of entire functions with prescribed zeroes of prescribed orders.
Here is the assignment due next Tuesday (February 5).
  1. Adapt the method of Section 31B in the textbook to prove that \begin{equation*} \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). \end{equation*}
  2. Prove that if \(|z|\le 1\), and \(p\) is a positive integer, then \begin{equation*} \left| 1 - (1-z) \exp\left(z+\tfrac{1}{2}z^2+\tfrac{1}{3}z^3+\cdots+\tfrac{1}{p}z^{p}\right)\right| \le |z|^{p+1}. \end{equation*} Hint: If \(a_n\ge 0\) for every \(n\), and \(|z|\le1\), then \(\bigl| \sum_{n=0}^\infty a_n z^n\bigr| \le \sum_{n=0}^\infty a_n\).
Thursday, January 24
We discussed how the classification of isolated singularities can be interpreted in terms of Laurent series, introduced the problem of whether every meromorphic function can be expressed globally as the quotient of two holomorphic functions (solved affirmatively by Weierstrass), and began a study of infinite products (with the aim of eventually proving the indicated theorem of Weierstrass).
Tuesday, January 22
Topics today included pole pushing (in the proof of Runge's theorem); an aside on polynomial convexity in \(\mathbb{C}^2\), Eva Kallin's theorem, and the open problem about polynomial convexity of four balls; and Laurent series.
The individual assignment due next Tuesday (January 29) is to use Bernoulli numbers to find the Laurent series for \(1/\sin(\pi z)\) in powers of \(z\) and \(1/z\)
  • valid when \(0\lt |z|\lt 1\), and more generally
  • valid in the annulus where \((k-1)\lt |z|\lt k\), the number \(k\) being a positive integer.
[Section 15C in the textbook shows how to determine the Maclaurin series for \(\tan(z)\) in terms of Bernoulli numbers.]
Thursday, January 17
Topics today included some generalizations of the approximation theorems of Weierstrass and Runge (the Müntz–Szász theorem and one of Mergelyan's theorems) and a sketch of the two main ideas in the proof of Runge's theorem.
Tuesday, January 15
Topics today included the Weierstrass approximation theorem from real analysis (as motivation for Runge's approximation theorem in complex analysis), a first version of Runge's theorem, and an example of how Runge's theorem produces a sequence of polynomials with a strange pointwise limit.
The group assignment due next Tuesday (January 22) is to prove the existence of a “universal” power series \(\sum_{n=0}^\infty a_n z^n\) with radius of convergence 1 and with the property that for every closed disk \(D\) disjoint from the closed unit disk and every function \(f\) holomorphic in an open neighborhood of \(D\) and every positive \(\varepsilon\), there exists a value of \(N\) such that the partial sum \(\sum_{n=0}^N a_n z^n\) approximates \(f(z)\) uniformly on \(D\) within \(\varepsilon\).
Remark: Every universal power series is overconvergent in the sense of Section 17D of the textbook, but not every overconvergent series is universal.
Sunday, January 13
Welcome to Math 618. This site went live today. I will post regular updates during the semester.