Ph.D. Preliminary Examination
Complex Analysis
August 26, 1994
1. Find an explicit conformal map from the region
{z : |z| < 1} − {x ∈ R : x ≤ 0}
onto the upper halfplane {Im z > 0}.
2. Find the explicit Laurent series of the function
f(z) =
1
z(z − 3)
on the annulus {z : 1 < |z − 1| < 2} centered at 1.
3. Let D ⊂ C be open and connected, and fix z0 ∈ D; set A(D, z0) = {|f ′(z0)| : f
holomorphic on D and |f(z)| < 1 for z ∈ D}. Prove that A(D, z0) is a compact subset
of R. What is A(C, z0)?
4. Let f be holomorphic in the connected region Ω ⊂ C, and assume that there exists a
nonempty open set U ⊂ Ω, such that |f(z)| = 1 for all z ∈ U . Show that f is constant
in Ω.
5. Suppose f(z) =
∞∑
n=0
anz
n is holomorphic on the closed unit disc. Prove that
∫ 2π
0
|f(eiθ)|2dθ = 2π
∞∑
n=0
|an|2 .
6. Suppose h is holomorphic in a neighborhood of {z : |z| ≤ R}, and that h(z)
6= 0 for
|z| = R.
(a) Use the Theorem of Residues to show that
∮
|z|=R
h′(z)
h(z)
dz = 2πi ZR(h) ,
where ZR(h) is the number of zeroes of h in {|z| < R}, counted with multiplicities.
(b) Use (a) to prove that if f and g satisfy the same hypotheses as h, and if
|f − g| < |f | on {|z| = R} ,
then ZR(f) = ZR(g).
7. Use the Theorem of Residues for appropriate contours to evaluate
∫ ∞
−∞
√
x + i
1 + x2
dx ,
where on {Im z > 0}, we choose the branch of
√
z + i whose value at 0 is eπi/4. De-
scribe your method carefully, and include verification of all relevant limit statements.
8. Find an explicit series representation for a meromorphic function on C, which is
holomorphic on C − {1, 2, 3, . . .}, and which has at each point z = n ∈ N a simple
pole with residue n. Include proofs of all required convergence statements.
9. Prove that all holomorphic automorphisms of C (i.e. holomorphic maps f : C → C
which are one-to-one and onto) are precisely the linear functions f(z) = a + bz for
arbitrary a, b ∈ C.