Complex Analysis Qualifying Examination
August 2011
1. Suppose u.x; y / is a (real-valued) harmonic function on a simply connected domain in C.
Show that u.x; y / can be written in the form f .x Ciy/ Cg.x iy/, where fand gare
holomorphic functions.
2. An inversion is a function on the extended complex numbers of the form ´7! 1
´´0
,
where ´0is some complex constant. Show that the dilation ´7! 4´ can be obtained by
composing three inversions.
3. Determine, with proof, the set of all biholomorphic self-mappings of Cnf0g, the punctured
plane.
4. Suppose fis a continuous function on f´2CW j´j 1g, the closed unit disk, and fis
holomorphic on the open unit disk. Prove that if f .´/ is real when j´j D 1, then fis a
constant function.
5. Suppose that gis a bounded, continuous function on the real axis. Show that the improper
integral R1
0e´t g.t / dt (the Laplace transform) represents a holomorphic function of ´in
the half-plane where Re ´>0.
6. Use the residue theorem to prove that Z1
0
x2
1Cx5dx D=5
sin.2=5/ .
7. Find the general form of an entire function fsatisfying the property that
f .w/ f .´/
w´Df0wC´
2
for all distinct complex numbers wand ´.
8. Let ffng1
nD1be the sequence of iterates of the sine function: namely, f1.´/ Dsin.´/, and
fnC1.´/ Dsin.fn.´// when n1. Show that this sequence ffngis not locally bounded
in any neighborhood of the origin.
9. Suppose that fis holomorphic on f´2CW0 < j´j< 1 g(the punctured unit disk), and
fhas no zeroes. Show that there exist an integer mand a function gholomorphic on the
punctured disk such that f .´/ D´meg.´/ for all ´in the punctured disk.
10. State and prove one of the following theorems: the Riemann mapping theorem, Runge’s
theorem about polynomial approximation, or the Schwarz reflection principle.