Record of daily activities for Math 618, Theory of Functions of a Complex Variable II, Spring 2006

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Tuesday, May 2

We compiled the following list of ten results that will be the basis of the final examination: the Bohr-Mollerup theorem; Cauchy's integral theorem for multiply connected domains (the homology version); the general pole-pushing lemma; Jensen's formula; Hadamard's factorization theorem for entire functions; Hadamard's gap theorem; the monodromy theorem; Picard's little theorem (about the range of entire functions); Runge's approximation theorem; and the Weierstrass factorization theorem for entire functions.

Monday, May 1

We discussed the overview of the proof of the prime number theorem.
Assignment: Finish reading Chapter 16 (namely, section 16.3). In preparation for the final examination, make a list of the major theorems from the course.

Friday, April 28

We discussed properties of Riemann's $\zeta$ function and the statements of the prime number theorem and the Riemann hypothesis.
Assignment: Read in Chapter 16 through the proof of Proposition 16.2.2.

Wednesday, April 26

We discussed characterizations of the gamma function and proved Wielandt's uniqueness theorem. ChangChun presented a solution to exercise 2 on page 465.
Assignment: Read the first five pages of Chapter 16.

Monday, April 24

We discussed the solution to the third problem from last Wednesday, and we looked at the integral representation of the gamma function using the Hankel contour.
Assignment: Finish reading section 15.2, and do either exercise 2 or exercise 3 on page 465.

Friday, April 21

ChangChun presented a solution to the first problem from last time, and Brad presented a solution to the second problem. We also computed the volume and the surface area of the unit ball in ${ℝ}^n$ in terms of the gamma function.
Assignment: Read the first part of section 15.2, pages 455-460 (through Proposition 15.2.4), and work on problem 3 from last time.

Wednesday, April 19

We discussed properties of the gamma function.
Assignment: Finish reading section 15.1, and do one of the following three problems.

1. Show that $\Gamma \left(1/2\right)=\sqrt{\pi }$ (compare exercise 5b on page 465 of the textbook).
2. Show that ${\Gamma }^{\prime }\left(1\right)=-\gamma$ (exercise 6 on page 466).
3. Show that ${\displaystyle {\prod }_{n=1}^{\infty }\frac{n\left(n+1\right)}{{\left(n+\frac{1}{2}\right)}^2}=\frac{\pi }{4}}$.

Challenge problem: show that ${\displaystyle {\int }_0^{\infty }\frac{sin\left(t\right)}{t}\hspace{0.17em}log\left(t\right)\hspace{0.17em}dt}=-\frac{1}{2}\pi \gamma$.

Monday, April 17

We discussed the proof of Mergelyan's theorem.
Assignment: Read the first part of section 15.1, pages 447-451 (through Corollary 15.1.7).

Wednesday, April 12

We worked on an exercise on Swiss cheese.
Assignment: This Friday is a reading day, and classes do not meet. Finish reading Chapter 12, and finish the exercise from class.

Monday, April 10

Jared presented a construction of an entire function, not identically equal to zero, that tends to zero along every line.

Friday, April 7

We discussed the exercise from last time.
Assignment: Next time Jared will preview his Master's exam presentation.

Wednesday, April 5

We worked on an exercise on the $\bar{\partial }$ equation.
Assignment: Finish the exercise, and read Lemmas 12.2.4 and 12.2.5 (and the proofs) on pages 371-372.

Monday, April 3

We worked on an exercise on Runge's theorem.
Assignment: Read the proof of the inhomogeneous Cauchy integral formula in Appendix A, pages 488-489 (the section labelled “Green's theorem”).

Friday, March 31

We discussed the exercise on overconvergence from last time.
Assignment: Do exercise 6 on page 380 (to hand in).

Wednesday, March 29

We worked on an exercise on overconvergence.
Assignment: Finish the exercise, and read the proof of Runge's theorem in section 12.1, pages 364-367.

Monday, March 27

Chris gave a solution to the problem from last time, and we started discussing approximation theorems.
Assignment: Read the first part of section 12.1 (pages 361-364) and do exercise 1 on page 379 (to hand in).

Friday, March 24

Aaron showed that the Weierstrass ℘ function with periods $1$ and $i$ maps the open subsquare of the fundamental unit square with side $1/2$ one-to-one onto the lower half-plane. Also we discussed the topological properties from the exercise on simple connectivity.
Assignment: Do problem 3 from the January 2004 qualifying exam: namely, exhibit a closed path in the complex plane such that the integral over the path of the function $1/\left(z-1\right)\left(z-14\right)$ is equal to $2004$.

Wednesday, March 22

We discussed the notion of simple connectivity and the question of whether connectedness should be a part of the definition (in connection with exercise 4 on page 352). Also we discussed part 9 of the exercise on simple connectivity and observed that the case of the square would be solved if we knew that the Riemann map from the square to the disc can be approximated uniformly on compact sets by polynomials.
Assignment: Read section 11.5, pages 349-352, and verify that the Weierstrass ℘ function (the doubly periodic meromorphic function defined in the textbook with periods $1$ and $i$) maps the open subsquare of the fundamental unit square with side $1/2$ one-to-one onto an open half-plane.

Monday, March 20

We worked on an exercise on simple connectivity.
Assignment: Prepare to present the exercise on simple connectivity in class next time, and write up a solution to exercise 4 on page 352 to hand in next time.

Friday, March 10

We finished the second part of the exercise on Picard's theorems and recapitulated the key idea in the proof of the homology version of Cauchy's integral formula.
Assignment: Read sections 11.3 and 11.4, pages 343-349.

Wednesday, March 8

We worked out the first part of an exercise on Picard's theorems.
Assignment: Do the second part of the exercise, and read section 11.2, pages 338-343.

Monday, March 6

We discussed the exercise on the Weierstrass ℘ function.
Assignment: Read section 11.1, pages 335-338.

Friday, March 3

We discussed exercises 19, 20, and 21 on pages 332-333 at the end of chapter 10.
Assignment: Do the exercise on the Weierstrass ℘ function.

Wednesday, March 1

We concluded the discussion of the exercises on the modular function.
Assignment: Various subsets of the class will prepare exercises 19, 20, and 21 on pages 332-333 at the end of chapter 10.

Monday, February 27

We discussed the exercises from last time and the geometry involved in the construction of the modular function.
Assignment: Continue working on the exercises from last time, and read section 10.6 in the textbook (pages 323-330).

Friday, February 24

Chris showed that no two points of the fundamental domain are equivalent under an element of the modular group. One difference from the case of the congruence subgroup considered in the textbook is that there exist two transformations in the full modular group that leave a point in the boundary of the fundamental domain fixed. Indeed, the transformation $z\mapsto -1/z$ fixes the point $i$, and the transformation $z\mapsto -1/\left(z-1\right)$ fixes the point $exp\left(i\pi /3\right)$.
Assignment: Prepare to discuss the exercises on the modular function next time.

Wednesday, February 22

Jared presented a solution to the first part of the exercise on the modular group from last time; Aaron will write this up for distribution. For the second part, Brad showed that every point of the upper half-plane is the image of some point of the alleged fundamental domain under some transformation in the modular group; Lidia will write this up for distribution.
Next time Chris will show that no two points of the alleged fundamental domain are equivalent under an element of the modular group.

Monday, February 20

We worked on an exercise on the modular group.
Assignment: Prepare to present the solution to the exercise next time, and also read the rest of section 10.5, pages 320-323.

Friday, February 17

ChangChun presented a solution of exercise 11 on page 331; the function $g$ extends to be a meromorphic function on the whole plane with simple poles at the integers that are congruent to 3 modulo 4. Aaron presented a solution to problem 10 from the May 2005 qualifying examination.
Assignment: Read the first part of section 10.5, pages 314-320, and do exercise 16 on page 332.

Wednesday, February 15

Jared presented a solution to exercise 9 on page 331 about analytic continuation of the power series ${\displaystyle {\sum }_{j=1}^{\infty }\frac{z^j}{j^2}}$. Also we discussed the Vivanti-Pringsheim theorem.
Assignment: Do problem 10 from the May 2005 complex analysis qualifying examination: namely, if $f$ and $g$ are entire functions such that ${f\left(z\right)}^2+{g\left(z\right)}^2=1$ for all $z$, then there exists an entire function $h$ such that $f\left(z\right)=cos\left(h\left(z\right)\right)$ and $g\left(z\right)=sin\left(h\left(z\right)\right)$ for all $z$. (What does this have to do with the monodromy theorem?)

Monday, February 13

We looked at an example of an entire function for which $\lambda =\mu +1$ (the order attains its maximal value of 1 plus the genus): namely, the infinite product ${\displaystyle {\prod }_{n=2}^{\infty }\left(1-\frac{z}{n{\left(logn\right)}^2}\right)}$, which has genus 0 and order 1. Then we worked in groups on exercises 9 and 11 on page 331.
Assignment: Read sections 10.3 and 10.4. Each group will present its exercise in class next time.

Friday, February 10

Aaron presented a solution to the second part of the exercise from last time. Also we discussed a solution to exercise 12 on page 297 (the second half of Hadamard's theorem).
Assignment: Read sections 10.1 and 10.2, pages 299-307.

Wednesday, February 8

We discussed the solution of exercise 3 on page 296 about genus, and ChangChun presented a solution of exercises 1 and 2 on page 296 (proofs of the Poisson-Jensen formula).
Assignment: Do the exercise on the order of an entire function (to hand in next class).

Monday, February 6

We discussed the construction of a gap power series that has the unit circle as natural boundary, yet derivatives of all orders extend continuously to the boundary. Also we discussed the definition of order of an entire function and some examples.
Assignment: Read the rest of section 9.3 and do exercise 3 on page 296 (about genus).

Friday, February 3

Aaron presented a proof that Jensen's formula carries over unchanged when there are (finitely many) zeroes on the boundary of the disc; the integral is then a convergent improper integral.
Assignment:

• Read part of section 9.3 (pages 288-290 and pages 293-296), skipping the technical lemmas for the moment.
• Use Hadamard's gap theorem to construct a holomorphic function in the unit disc $D\left(0,1\right)$ such that no point on the boundary of the disc is regular for the function (in the sense of the definition on page 270), yet the function and its derivatives of all orders extend continuously to the closed disc $\bar{D}\left(0,1\right)$.

Wednesday, February 1

Students presented proofs of some of the generalizations of Jensen's formula from last time.
Assignment: Read section 9.2 about the Hadamard gap theorem.

Monday, January 30

We discussed some possible generalizations of Jensen's formula (Theorem 9.1.2 on page 280): (A) removing the hypothesis that $f\left(0\right)\ne 0$; (B) moving $0$ to a different point by a linear fractional transformation (the Poisson-Jensen formula, which is exercise 1 on page 296 at the end of the chapter); (C) extending the formula to cover the case of meromorphic functions; (D) relaxing the hypothesis that the function is holomorphic in a neighborhood of the closure to allow the function to be merely continuous on the closure and holomorphic on the interior; (E) allowing the function to have zeroes on the boundary.
Assignment: Everyone will prepare one of these cases to present next time (except Chris, who will do exercise 8 on page 296 at the end of the chapter).

Friday, January 27

We discussed an improvement of the version of Mittag-Leffler's theorem stated in the book (Theorem 8.3.8): namely, in addition to prescribing finite chunks of the Laurent series of a meromorphic function at prescribed points, one can guarantee that the function has no extraneous zeroes.
Aaron presented an elegant solution of exercise 12 on page 276 based on periodicity and a computation showing that both sides of the equation tend to $0$ when $\left|Imz\right|\to \infty$. Chris will write up a solution for distribution. The exercise is essentially the same as starred exercise 68 on page 155.
Assignment: Read section 9.1 and do exercise 6 on page 296 (to hand in).

Wednesday, January 25

We completed the discussion of the approximation problem held over from last time. Lidia observed that one can simplify the proof by using an exhaustion of the open set $U$ by an increasing sequence of compact sets (as on page 194 of the textbook in the proof of Montel's theorem). Also she observed that the problem is a corollary of the generalized pole-pushing lemma 12.1.6 (which we will get to later). We also discussed how to generalize problem 4 on page 275 of the textbook to remove the hypothesis that $b_n>1$.
Assignment: Solve exercises 10 and 12 on page 276 of the textbook (to discuss on Friday). These exercises can be viewed as concrete instances of the theorems of Weierstrass and Mittag-Leffler.

Monday, January 23

We discussed solutions of the two problems assigned last class. Brad will write up a solution of the first problem for distribution. Some technical topological issues in the second problem are held over for next class.
Assignment: Write up a solution (to hand in) of problem 4 on page 275 of the textbook. The hypothesis “$b_n>1$” can be substantially weakened; prove the most general statement that you can.
(The key technical issue is whether one can choose a consistent branch of the logarithm for all the terms.)

Friday, January 20

We discussed some of the ideas in the proofs of the Weierstrass and Mittag-Leffler theorems.
Assignment: Prepare solutions to the following exercises to present at the next class meeting.

1. Suppose $U$ is an open set in $ℂ$, and $\left\{a_j\right\}$ is a sequence of points in $U$ with no interior accumulation point. Assume (after making a translation, if necessary) that $0\in U\backslash \left\{a_j\right\}$. Let $b_j$ be a point of $\widehat{ℂ}\backslash U$ that minimizes $d\left(a_j,b_j\right)$, where $\widehat{ℂ}$ denotes the extended complex numbers $ℂ\cup \left\{\infty \right\}$, and $d$ denotes the spherical metric described in exercise 34 on page 150 of the textbook.
   Define a linear fractional transformation $L_j$ such that ${\displaystyle L_j\left(z\right)=\left(\frac{z}{a_j}\right)\left(\frac{a_j-b_j}{z-b_j}\right)}$ if $b_j\ne \infty$ and ${\displaystyle L_j\left(z\right)=\frac{z}{a_j}}$ if $b_j=\infty$. Prove that the infinite product $\prod _{j=1}^{\infty }{E}_j\left(L_j\left(z\right)\right)$ converges normally on $U$ to a holomorphic function whose zero set is $\left\{a_j\right\}$ (counted according to multiplicity). Here the functions $E_j$ are the Weierstrass elementary factors defined on page 263 of the textbook.
   (This exercise is a variation on the proof of the Weierstrass factorization theorem. Notice that if the set $U$ happens to be the whole complex plane, then this proof is exactly the proof given for the case of entire functions in section 8.2 of the textbook.)
2. Let $U$ be an open set such that the complement $ℂ\backslash U$ is both connected and unbounded. (A weaker hypothesis that still suffices is that every connected component of $ℂ\backslash U$ is unbounded. A statement that is equivalent to this weaker hypothesis is that $U$ is a simply-connected open set that is not the whole plane, but we do not yet officially know that equivalence.)
   Suppose that $f\left(z\right)$ is a rational function whose poles lie in the complement $ℂ\backslash U$. Prove that there is a sequence $\left\{p_j\left(z\right)\right\}$ of polynomials converging normally to $f\left(z\right)$ on $U$.
   Suggestion: use the pole-pushing lemma 8.3.5 (and/or its proof) to push the poles off to infinity.

Wednesday, January 18

We discussed reasons why convergence of an infinite product is not defined to mean simply convergence of the partial products, and we reviewed the statement of the Weierstrass factorization theorem for entire functions.
Assignment: Read section 8.3 in the textbook concerning the general Weierstrass and Mittag-Leffler theorems about construction of meromorphic functions with prescribed principal parts.
Do exercise 1 on page 275 at the end of chapter 8 (to hand in next class).