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) = \frac{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 z_{0} ∈ D; set A(D, z_{0}) = {|f′(z_{0})| : f holomorphic on D and |f(z)| < 1 for z ∈ D}. Prove that A(D, z_{0}) is a compact subset of R. What is A(C, z_{0})?

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}^{∞} a_{n}z^{n} is holomorphic on the closed unit disc. Prove that

∫_{0}^{2π} |f(e^{iθ})|^{2}dθ = 2π ∑_{n=0}^{∞} |a_{n}|^{2} .

6. Suppose h is holomorphic in a neighborhood of {z : |z| ≤ R}, and that h(z) ≠ 0 for |z| = R.

(a) Use the Theorem of Residues to show that

∮_{|z|=R} \frac{h′(z)}{h(z)}dz = 2πi Z_{R}(h) ,

where Z_{R}(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 Z_{R}(f) = Z_{R}(g).

7. Use the Theorem of Residues for appropriate contours to evaluate

∫_{−∞}^{∞} \frac{\sqrt{x + i}}{1 + x^{2}}dx ,

where on \{Im z > 0\}, we choose the branch of \sqrt{z + i} whose value at 0 is e^{πi/4}. Describe your method carefully, and include verification of all relevant limit statements.

8. Find an \underline{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.