August 12, 2011
2011 Complex Prelim
∆ is the open unit disk; ∂D is the boundary of the domain D;Cis the complex plane and ˆ
Cis the
Riemann sphere (i.e. the extended complex plane); O=O(D) is the set of function holomorphic
(or equivalently analytic) in the domain D.
Justify your reasoning in all problems.
(1) Evaluate the following integral:
Z∞
−∞
1
x4+ 1 dx
This problem requires more than just a calculation. Justify all steps, including al l the
estimates and taking limits that are needed.
(2) Suppose
D:= {z∈C:|z−i|<√2 and |z+i|<√2}.
(a) Prove that there is no conformal map of Donto C.
(b) If there is one, find a conformal map of Dto the unit disk ∆. If none exists, prove it.
(3) Prove that the Fundamental Theorem of Algebra is a direct consequence of Rouch´e’s The-
orem.
(4) (a) Suppose fis holomorphic in ∆ \ {0}. What consequence would follow if |z2f(z)|is
bounded?
(b) Consider a family of functions that are holomorphic on ∆ \{0}and for which there is
a uniform bound for |z2f(z)|on ∆ \ {0}. Is this family normal?
(5) Consider all functions fwhich are holomorphic for Re(z)>0, take values in ∆, and vanish
at z= 1. What is the least upper bound for |f(2)|? Is it achieved?