Complex Analysis Preliminary Exam
August 2009
Instructions:
(i) Complete all problems. Give full justifications for all answers in the exam booklet.
(ii) •The complex numbers are denoted C, the real numbers R, and the natural numbers N.
•The letter zis used for complex numbers, the letters xand yare real numbers, and the letter nis a natural
number.
•The open unit disc is denoted D={z:|z|<1}.
1. Evaluate the following integral, justifying all steps. The number ais real and positive.
Z∞
0
cos ax
(1 +x2)2dx
2. (i) Determine, with proof, the number of zeros of f(z)=z3−z+1
10 ezin the disc |z|<2.
(ii) Suppose that {fn(z)}is a sequence of functions, each of which is analytic on the closure of Dand has exactly
one zero in D. If the sequence fnconverges uniformly on the closure of Dto a function f, must it be the case
that fhas exactly one zero in D? Give a proof or a counterexample.
3. (i) Show there is no non-constant bounded analytic function on C\ {0}.
(ii) Show there is no non-constant bounded analytic function on C\N.
(iii) Give an example of a non-constant bounded analytic function on C\[0,∞).
4. Let fbe analytic and satisfy |f(z)| ≤ 1 on D, and suppose that f(0) =f0(0) =0. Prove that |f00(0)| ≤ 2 and
describe the functions having |f00(0)|=2.
5. Show that the following function on R2is harmonic and compute a harmonic conjugate.
u(x,y)=x3+2xy −2x2−3xy2+2y2
6. Let Fbe a set of functions that are analytic on Dand satisfy f(0) =1 for all f∈ F . Let F0={f0(z) : f∈ F }.
Show that Fis a normal family if and only if F0is normal.