Complex Analysis Qualifying Examination
January 12, 2011
1. Suppose f .´/ Dtan.!1´/ tan.!2´/ tan.!3´/, where !1,!2, and !3denote the three cube
roots of 1. Show that the nth coefficient in the Maclaurin series of fis equal to 0when
nis not a multiple of 3.
2. The Schwarzian derivative of fis the expression f00
f00
1
2f00
f02
. Show that if fis
a linear fractional transformation (M¨
obius transformation), then the Schwarzian derivative
of fis identically equal to 0.
3. Suppose that fis an entire function such that jf .´/j> 1=.1 C j´j/for all ´. Prove that
fmust be a constant function.
4. Suppose ˝is a connected open subset of C, and uW˝!Ris a nonconstant harmonic
function. Prove that u.˝/, the image of u, is an open subset of R.
5. Use the residue theorem to prove that Z1
0
1
1Cx2011 dx D=2011
sin.=2011/ .
6. Suppose fis a holomorphic function (not necessarily bounded) on f´2CW j´j< 1 g, the
open unit disk, such that f .0/ D0. Prove that the infinite series P1
nD1f .´n/converges
uniformly on compact subsets of the open unit disk.
7. Either give an example of a holomorphic function (not necessarily one-to-one) that maps
the punctured unit disk f´2CW0 < j´j< 1 gsurjectively onto the unit disk f´2CW
j´j< 1 gor prove that no such function exists.
8. Suppose f .´/ D
1
X
nD1
´n
n2when j´j< 1. Show that this f(called the dilogarithm function)
admits an analytic continuation to CnŒ1; 1/, the complex plane with a slit along the
positive real axis from 1to 1.
9. An exponential polynomial is a function of the form Pn
kD1pk.´/eak´, where each pkis
a polynomial (not identically equal to 0), and a1, ..., anare distinct complex numbers.
Characterize the exponential polynomials that have no zeroes.
10. State and prove one of the following theorems: the Riemann mapping theorem, Rouch´
e’s
theorem, or Runge’s theorem about approximating holomorphic functions by polynomials.