In our last class meeting of the semester, we discussed how some of the results of the course change in higher dimension. In particular, we looked at what happens in higher dimension to the Riemann mapping theorem, Runge’s approximation theorem, and the connection between analytic functions and harmonic functions. Two open problems mentioned were the Jacobian conjecture about polynomial mappings of \(\C^2\) and Eva Kallin’s question about the polynomial convexity of four balls in \(\C^2\).
The final exam takes place 8:00–10:00 in the morning of Tuesday, May 7.
April 23
We established Montel’s fundamental normality criterion and applied it to prove Picard’s big theorem.
April 18
We proved Jensen’s formula and a corollary relating the growth of entire functions to the number of zeros. And we applied Hadamard’s factorization theorem to deduce that the function \( e^z + z\) has infinitely many zeros.
April 16
The assignment is posted. The due date is not next class but rather a week from today.
In class, we discussed the notion of the order of an entire function and the statement of Hadamard’s factorization theorem.
April 11
Class was canceled because I had a medical emergency.
April 9
In class, we completed two-thirds of the solution of the Dirichlet problem via Perron’s method.
We proved that an upper semi-continuous function on a compact set attains a maximum and that subharmonic functions are locally integrable and are integrable on circles. Hence the set where a subharmonic function takes the value \(-\infty\) has Lebesgue measure zero.
The assignment is to show that if \( u(z) = \sum_{n=1}^\infty \frac{1}{2^n} \log\left\lvert z-\frac{1}{2^n}\right \rvert\), then \(u\) is subharmonic on \(\C\) and discontinuous at \(0\) (but finite-valued at \(0\)).
April 2
We continued the discussion of subharmonic functions and Perron’s method for solving the Dirichlet problem.
The assignment is to prove a generalization of Liouville’s theorem in the setting of subharmonic functions, namely, if a function is subharmonic on the entire plane and bounded above, then the function necessarily reduces to a constant function. The suggestion is to use comparison harmonic functions of the form \(A+B\log\abs{z}\) on annuli centered at the origin.
March 28
We discussed Harnack’s inequality and Harnack’s theorem about harmonic functions and began a discussion of subharmonic functions in preparation for solving the Dirichlet problem via Perron’s method.
The assignment due next time is as follows.
Show that there exists a harmonic function \(u\) in the unit disk such that \(u(0)=1\) and \(u(1/2)=5\).
Show that there does not exist a positive harmonic function \(u\) in the unit disk such that \(u(0)=1\) and \(u(1/2)=5\).
Show that there exists a positive subharmonic function \(u\) in the unit disk such that \(u(0)=1\) and \(u(1/2)=5\).
Let \(\mathcal{U}\) be the class of positive harmonic functions in the unit disk. Show that \(\mathcal{U}\) is a normal family in the extended sense that every sequence in \(\mathcal{U}\) admits a subsequence that converges uniformly on compact sets either to a harmonic function or to the constant \(+\infty\).
March 26
We solved the Dirichlet problem for the unit disk, completed the proof of the equivalence theorem for various properties of harmonic functions, and saw an example of an unsolvable Dirichlet problem for the punctured disk.
The assignment due next time is as follows.
Prove a removable-singularities theorem for harmonic functions: If a function is harmonic in a punctured disk and bounded, then the function extends to be harmonic on the whole disk.
Prove a version of Montel’s theorem for harmonic functions: If a family of harmonic functions is locally bounded, then the family is normal in the sense that every sequence of functions in the family admits a subsequence that converges uniformly on compact sets to a harmonic function.
March 21
We continued the discussion of the Poisson kernel, recast the integral as the formula of Schwarz exhibiting an analytic function having prescribed real part, observed that the averaging property of the Poisson kernel implies a maximum principle for harmonic functions, and introduced the Dirichlet problem for the unit disk.
The assignment due next time is as follows.
Solve problem 3 on the August 2013 qualifying examination. Hint: observe that \( \dfrac{1-\left( \frac{1}{2}\right)^2} { \left\lvert e^{i\theta}- \frac{1}{2} \right\rvert^2} \) is an instance of the Poisson kernel.
Solve Exercise 8 in Section 1 of Chapter X (pages 255–256) about the Laplace operator and harmonic functions in polar coordinates.
March 19
Welcome back from Spring Break. We began a new topic—harmonic functions. We proved most of an equivalence theorem stating that
solutions of Laplace’s equation are identical to functions that locally match the real part of an analytic function and also are identical to functions that satisfy a local mean-value property.
The assignment due next time is as follows.
Show that the function \(\log\abs{z}^2\) is harmonic on \( \C \setminus\{0\}\) but is not (globally) the real part of an analytic function.
Show that if \(u\) is harmonic and \(g\) is analytic, then \(u\circ g\) is harmonic (assuming suitable hypotheses that make the composition sensible).
March 7
We constructed “the” modular function by using the Schwarz reflection principle iteratively, proved a lifting theorem for analytic functions omitting two values, and thereby established Picard’s little theorem.
March 5
We discussed the notion of analytic continuation and the monodromy theorem.
We applied the Schwarz reflection principle to prove that a necessary condition for two annuli to be analytically equivalent is that they have the same ratio of radii. You previously showed the sufficiency of the condition by solving Exercise 8 in Section VII.4 of the textbook. Since you just turned in the take-home midterm exam, there is no assignment due next class.
February 26
We discussed the Schwarz reflection principle and variations.
In class, we discussed (i) an example related to Wielandt’s uniqueness theorem and (ii) the proof of Mittag-Leffler’s theorem about prescribed singularities.
February 19
We discussed an example of applying Runge’s approximation theorem to construct a sequence of polynomials converging pointwise to a discontinuous limit, a version of Runge’s theorem for open sets, the generalization of Runge’s theorem due to S. N. Mergelyan, and Alice Roth’s so-called Swiss cheese. There is no assignment to hand in next time, for the take-home exam will be distributed then.
February 14
We discussed the two key ideas in the proof of Runge’s approximation theorem.
The assignment is to prove the duplication formula for the Gamma function (Exercise 3 in Section 7 of Chapter VII, page 185). You may use either the hint in the textbook or Wielandt’s uniqueness theorem, and you may assume the result of Exercise 2, that \(\Gamma(1/2)=\sqrt{\pi}\).
The exam scheduled for next Thursday will be a take-home exam due a week later.
February 12
We constructed the Gamma function as an infinite product and proved Wielandt’s uniqueness theorem (not in the textbook).
The assignment due next time is to establish the following facts in two ways: (i) using the infinite-product definition of \(\Gamma\) and (ii) using the integral representation for \(\Gamma\). The letters \(x\) and \(y\) represent real numbers. In the case of the integral representation, you need consider only the case that \(x>0\).
We proved the formula representing the sine function as an infinite product.
Here is the assignment due next time:
Prove in two ways the following two facts about the sine function (where \(x\) and \(y\) represent real variables).
\(\abs{\sin(x+iy)} \ge \abs{\sin(x)}\)
\(\lim_{y\to\infty} \abs{\sin(x+iy)} =\infty\)
Method 1: Use the definition of the sine function in terms of the complex exponential function to show that \(\abs{\sin(x+iy)}^2 = \abs{\sin(x)}^2 + \abs{\sinh(y)}^2\).
Method 2: Use the infinite-product representation of the sine function.
Solve Problem 10 on the January 2019 qualifying exam.
The hint is to adapt the trickery used in class today to show that an entire function satisfying a certain addition formula must be constant.
February 5
We proved the theorem of Weierstrass for a general planar domain. The assignment due next time is Problem 9 on the August 2012 qualifying exam (easy!) and Problem 9 on the August 2014 qualifying exam (about normal convergence in the unit disk).
January 31
We proved the theorem of Weierstrass stating that the zeros of an entire function can be arbitrarily prescribed subject to the necessary condition of having no accumulation point. The assignment due next time is Exercise 4 in Section 5 of Chapter VII (page 173) about Blaschke products in the unit disk.
January 29
We finished the proof of the Riemann mapping theorem and began a discussion about infinite products.
The assignment due next time is problem 3 on the January 2019 qualifying exam, which asks how big \( \abs{f'(1/7)}\) can be when \(f\) is an analytic function that maps the unit disk into itself.
January 24
We carried out the proof of the Riemann mapping theorem, except for the details of the last step.
The assignment due next time is Exercises 2 and 8 in Section 4 of Chapter VII (pages 163–164). The main point of Exercise 2 is to show that a punctured disk is not biholomorphically equivalent to an annulus. Exercise 8 is an easy verification that two annuli are biholomorphically equivalent if the ratios of radii match; the converse is true too, but you do not yet have the tools to prove the converse.
January 22
We discussed the proof of Montel’s theorem, Hurwitz’s theorem, and the outline of the proof of the Riemann mapping theorem (to be filled in next time).
The assignment to hand in next time is Exercises 6 and 10 in Section 2 of Chapter VII (page 154). In Exercise 10, there is an implicit hypothesis that \(G\) is a region, that is, a connected open set.
January 17
We discussed a metric on the space \(C(G)\) of continuous functions on an open set \(G\), the corresponding version of the Arzelà–Ascoli theorem, and the statement of Montel’s theorem about normal families of analytic functions.
We discussed compactness in the space of continuous functions on a compact set, characterized by the Arzelà–Ascoli theorem. What would have been the content of the slides if the computer had been working is available. The assignment due next time is to show the equivalence of conditions (2) and (3) in the Arzelà–Ascoli theorem.
Sunday, January 6, 2019
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