Ph.D. Prelim in Complex Analysis January 18, 1994

1. Let f be analytic in the unit disk D. Use Cauchy’s integral formula to establish the power series representation of f in D. Obtain both an integral formula and a derivative formula for the n-th coefficient.

2. Let Ω be a region and let F = {f : f is analytic in Ω and |f(z)| ≤ 1, ∀z ∈ Ω}. Fix z_{0} ∈ Ω and show that ∃ g ∈ F such that Re g′(z_{0}) ≥ Re f′(z_{0}), ∀f ∈ F.

3. Let f be analytic and nonconstant in a region Ω with μ = Ref and v = Imf f. (a) Show that |f′(z)|^{2} = u_{x}^{2} + u_{y}^{2} = v_{x}^{2} + v_{y}^{2}.

(b) Determine all real numbers a and b such that au^{2} + bv^{2} is harmonic in Ω.

4. Let Ω = \{z : |z − i| < 1\} and H = \{z : Im z > 0\}. Map H\\overline{Ω} conformally onto Ω.

5. If p is a polynomial, prove that the series ∑_{n=0}^{∞} p(n)z^{n} defines a rational function. HINT: Note that any linear combination of rational functions is a rational function.

6. (a) Let f be analytic in the unit disk D with

lim_{|z|→1−} f(z) = 0 .

Prove f ≡ 0.

(b) Let g be analytic in D. Prove that the statement

lim_{|z|→1−} g(z) = ∞

is impossible.

7. Let f be meromorphic in C and bounded outside of some circle. Determine the form of f as completely as possible.

8. Let Γ = {z : |z| = 1}.

(a) Show that the mapping z ⟼ (z + 1)^{2} takes Γ onto the cardioid r = 2(1 + cos θ). Sketch this cardioid.

(b) Let g(w) = ∫_{Γ} \frac{z(z + 1)}{z^{2} + 2z − w} dz (Γ traversed once counterclockwise). Use the result of part (a) to sketch a domain containing 0 on which g is analytic.

(c) Determine g(0) and g′(0).