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General Information
Announcements
I will be in the classroom at our usual meeting time on Friday 4/28 to return exams and homework. In the meantime, please fill out the SIRS form for this course:
In particular, I very much appreciate constructive comments on how the course could be improved as well as specific comments regarding what you think is working well.
Two old qualifying exams may be downloaded from the resources section of this Piazza page
From Sarason Chapter X: 12.2, 12.3, 12.7, 12.8, 16.1, 16.5, 19.3, 19.4
From Sarason Chapter X: 10.3, 10.4, 10.6
Additional problems:
Problem 1: Let
and . Find a formula for
and prove that it is correct.
Problem 2: Compute
for
. Prove that your answer is correct.
From Sarason: IX.5.3, IX.17.1 (Hint: fix and consider the set of all points that can be reached from
along a polygonal path. Prove that this set is open and closed.)
Also work the following three problems:
Problem 1: Let
Suppose that is holomorphic on and that exists. Prove that
for any
. Problem 2: Let
denote the slit plane. Let be a holomorphic function on . Suppose that
, and . Suppose further that for each
the two limits
and
exist, with the convergence being locally uniform in
. Prove that
for any
Problem 3: Use the previous problem to derive the formula
where
denotes the principle branch of the square root.
In yesterday's lecture I state the last theorem incorrectly. The correct statement is as follows:
Thm. Let
be a piecewise closed curve such that for some we have for all . Then the interior of is non-empty.
The restriction that
If you are curious, a proof of the Jordan Curve theorem for piecewise smooth curves can be found here:
https://www.jstor.org/stable/2316660
The general theorem, for continuous curves, is more difficult to prove.
Please turn in revised solutions to problems 2 and 6 on the exam along with your original exam in class tomorrow, March 3.
From Sarason Ch. VIII: 1.1, 2.1, 4.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.2, 12.1, 12.2, 12.3
Name | Office Hours | |
---|---|---|
Jeffrey Schenker | When? Where? | |
Rodrigo Bezerra de Matos | When? Where? |