Complex Analysis rules!

In the first three weeks of the class we have studied several versions of the Maximum Modulus Theorem, Caratheadory's Theorem, Lindelof's Theorem, the Phragmen-Lindelof technique and Theorem, and the Schwarz Lemma (describing Aut (D)). Today we considered linear fractional transformations and the group of automorphisms of the upper half plane, which is isomorphic to SL(2,R) mod {I,-I}. It's a basic example of a Lie group. We have 5 Maximum Modulus Theorem proofs and one proof concerning Phragmen-Lindelof due on Wednesday (Feb 4).

The Maximum Modulus Theorem exercises were fairly straightforward, but I couldn't figure out the Phragmén-Lindelöf problem. I am just now beginning to see the light on it--it's a difficult concept. During the last week we continued to study linear fractional transformations. We looked at the decomposition Aut(Upper half plane)=NAK where the subgroups are N={horizontal translations}, A={dilations}, and K={automorphisms that fix i}. We had one easy exercise on this involving an alternate way of constructing linear fractional transformations. Next we verified that there exists a linear fractional transformation sending any three points of the extended plane to any other three points, and we looked at a way of generating all linear fractional transformations of the extended plane. Finally, we showed that l.f.t.'s send lines and circles to lines and circles. Yesterday we started a new topic--infinite products, Weierstrass Elementary Factors, and finding f in H(C) with a given zero set. More on this later.

February 17, 1998

As I was saying, last week we talked about convergence and absolute convergence of infinite products, how they can diverge to zero and other neat stuff. Given a sequence going to infinity, we used infinite products of Weierstrass Elementary Factors to come with an entire function that has that sequence as its zero set (with multiplicity). Modifying this a bit, we had the Weierstrass Factorization Theorem, which gives an expression of any entire function in a revealing form. We marvelled at the genius of Weierstrass, especially in coming up with an amazing estimate on the rate of convergence of the elementary factors on D. We received a rather large assignment on the W.F.T. and convergence of infinite products. (I finally finished the Phragmén-Lindelöf exercise last week.) We spent Friday factorizing the sine function. (Cool!) Later on we'll see an extension of the W.F.T. that applies on any domain. This week we started a new topic--Runge's Theorem. It says that any holomorphic function on certain domains is the limit of rational functions.

February 27, 1998

For over a week we have been focusing on Runge's Theorem in its various versions, along with applications. Runge's Theorem states that if V is an open subset of the plane and A is a subset of the extended complement which intersects every component of the extended complement, then there exists a sequence of rational functions which approach a function holomorphic on V in H(V) and have poles contained in A. A particularly useful special case of this is when the extended complement is connected, then every f in H(V) can be uniformly approximated by polynomials (rational functions with poles at infinity). It turns out that connectedness of the extended complement of V is equivalent to simple connectedness of V. This is especially neat since it is a purely topological formulation of "simply connected," and jives with the intuitive "no holes" definition. We did some pole pushing--given a function with poles at certain points outside a compact set K, we can approximate it on the compact set by another function on the compact set which has these poles moved to other points outside K. This definitely comes in handy. I have fallen behind the whirlwind pace of the course, but in vague summary, we proved some other lemmas needed for Runge's Theorem including a technical lemma about covering an open set with a nested sequence of compact sets with a certain intersection property and a lemma about approximating complex integrals with Riemann sums. We then did a one-compact-set-version and an open-set-version of Runge's Theorem. In a slight generalization, we showed the equivalence of i) a & b are in the same component of the complement, and ii) a rational function with poles at a can be approximated by rational functions with poles at b (ie. a pole can be pushed from a to b). Some of the previous homework problems were difficult, and there is still one that I haven't turned in yet. Today we were assigned two exercises on applications of Runge's Theorem.

March 18, 1998

Before Spring Break we proved the Mittag-Leffler Theorem using Runge's Theorem. It says that a meromorphic function can be made to have any given set (discrete, of course) of poles. We then worked up to the Wierstrass Factorization Theorem for an arbitrary region by reviewing some basics from last semester and proving a proposition and a lemma. The proposition states the existence of log((z-a)/(z-b)) holomorphic in V if a & b are in the same component of the extended complement of V. (If b=infinity, leave off the denominator above.) The lemma eventually said that if K is a compact subset of V with every component of C-infinity\K intersecting every component of C-infinity\V and {a₁, . . . ,a_n} is a finite set of points in V\K, then given epsilon, there exists f in H(V) such that Z(f)={a₁, . . . ,a_n} and |1-f(z)| < epsilon for z in K. We were assigned an easy exercise of extending the W.F.T. to show there exists a meromorphic function in V with arbitrary zeros and poles.

After Spring Break we covered the Schwarz Reflection Principle (reflecting about R) and got an exercise concerning reflection about the unit circle. Today we started a new topic--the Dirichlet Problem and subharmonic functions. Last semester we solved the Problem for a disk; now we will look at other regions.

April 1, 1998

We looked at two versions of the Maximum Principle for subharmonic functions. Some corollaries of these include a subharmonic function being less than a harmonic function in V if it is less on the boundary of V (although in some component they may be equal), the equivalence of two definitions of subharmonic (a local one and one concerning any closed disk in the set), the maximum of two subharmonic functions being harmonic, and that a subharmonic function altered to be harmonic in a disk is still subharmonic and is at least as big as the original. Next we considered the genius of Perron in coming up with Perron families and Perron functions to solve the Dirichlet Problem. If a solution exists, it is the Perron function, u_p, which is the supremum of all subharmonic functions having small enough values approaching the boundary. u_p is always harmonic, but sometimes it doesn't approach the right boundary values. It turns out the Dirichlet Problem is solvable in a domain if and only if there exists a barrier at every boundary point. A barrier is a family of functions which are superharmonic in a neighborhood of the boundary point and approach 0 at the boundary point. They also approach 1 on the boundary of the neighborhood. Then we proved, conveniently, that if every component of the boundary of V contains more than one point, then there is a barrier at every boundary point of V. Thus for most domains it is easy to tell whether or not the Dirichlet Problem can be solved.