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The take-home final exam for Complex Analysis Math 220B. The exam consists of five problems that require proving statements about polynomials and harmonic functions, finding the largest open set where a product converges normally, and determining the number of roots of a polynomial in an annulus. Students are given 2 hours to complete the exam, which must be scanned and emailed to the instructor by a certain deadline.
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Final Exam
This is a take home open book exam. Please, choose 2 hours of your time, and work on this exam at home. You should time it, and stop working on the exam after 2 hours. You can use the textbook and/or your notes, but you cannot use any online sources or help of other people. After that please scan the exam and send it to the instructor (Anton Gorodetski) via email ([email protected]) as an attachment. The deadline for the submission is 11am on Wednesday, March 18, 2020.
Problem 1 2 3 4 5 Σ
Points
Student’s name:
For each of the following statements explain whether it is TRUE or FALSE (i.e. prove or give a counterexample):
a) Any sequence of polynomials restricted to the unit disc D = {|z| < 1 } has a normally convergent subsequence;
b) Any sequence of polynomials restricted to the unit disc D has a nor- mally convergent subsequence if there is a uniform upper bound on their degrees;
c) Any sequence of polynomials restricted to the unit disc D has a normally convergent subsequence if there is a uniform upper bound on their values in the unit disc;
d) If a sequence of polynomials uniformly converges on the unit disc D, then there is a uniform upper bound on their degrees.
Reminder: normal convergence on D ⇔ uniform convergence on compact sub- sets of D to a function (not to ∞)
Find the largest open set U ⊂ C where the product
∏^ ∞
n=
n^2
exp
nz z − 1
converges normally to a holomorphic function.
Let r < 1 < R. Prove that for all sufficiently small ε > 0 the polynomial
p(z) = εz^10 + z^5 + 1
has exactly five roots (counted with their multiplicities) inside the annulus
rε−^
(^15) < |z| < Rε−^
(^15) .
