Complex Analysis: Final Exam
TIFR-CAM Bengaluru
Dec 2, 2012
(A) Question 1, parts C and D have negative marking. BE CAREFUL.
(B) For the objective type questions, write down only the answers. Do not show your work.
(C) For questions 2 and 3 show your work as usual, giving full justification at each stage.
(D) As far as possible, answer the questions in the order they appear in the question paper.
GOOD LUCK!
1. OBJECTIVE TYPE QUESTIONS: For the following questions, write down the answers. Justification is
not necessary:
(A) (2 marks ×3) Find all holomorphic functions fon the unit disc such that for each integer n≥2 we have
(a)
f1
n
≤1
2n.
(b)
f1
n
≤1
n2.
(c) f1
n=(−1)n
n2.
(B) (2 marks ×4) Find all entire functions fsuch that for each z∈Cwe have
(a) |f(z)|≤p|z|.
(b) |f(z)|≤|z|2.
(c) |f(z)|≥|z|2.
(d) Im(f(z)) ≥Re(f(z))2
(C) Give an example of each of the following objects, or say that such an object does not exist: ( 2 marks ×
7; If you give a wrong example, or state that the object does not exist when it does, 1 mark is taken away.)
(a) A domain with a nontrivial compact automorphism group (write down the group also.)
(b) An even elliptic function which cannot be represented as a polynomial in℘.
(c) An elliptic function which is not a rational function of ℘.
(d) An automorphism of the unit disc with no fixed points.
(e) A domain Ω, and a holomorphic fon Ω, such that 1 <|f(z)|<2 for all z∈Ω, but fdoes not have a
logarithm on Ω.
(f) A domain Ωand a harmonic function on Ωwith no harmonic conjugate.
(g) A harmonic function hon Csuch that h(z)>0 if |z|>1 and h(0) = 0.
(D) State if the following assertions are true or false ( 2 marks ×7; 1 mark is taken off for a wrong answer.):
(a) Let {an}be a given sequence of complex numbers. There is a holomorphic function fon the unit
disc such that f(n)(0) = an.
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