Ph.D. Comprehensive Exam Complex Analysis June 1993

Solve the following problems and justify your answers. Below D = {z : |z| < 1}. Four essentially complete solutions will constitute a pass.

1. Let P (x, y) be a polynomial with complex coefficients in the real variables x and y. Prove that there exists a polynomial Q(z) such that P (x, y) = Q(x + iy) for all x, y ∈ ℝ, if and only if P_{y} = iP_{x} on ℝ^{2}. (Here P_{y} and P_{x} denote the partials of P with respect to x and y, respectively.)

2. Suppose that {z_{n}}_{n=0}^{∞} is a sequence of distinct points in D such that lim z_{n} = z_{0} = 0. Let F be analytic in D \ {z_{n}}_{n=0}^{∞}. a. If there exists an integer N such that z^{N}f(z) can be defined on {z_{n}}_{n=0}^{∞} so as

to be analytic in a neighborhood of z_{0}, show that the points z_{n} are removable singularities or poles of order at most N.

b. Prove or disprove: If for every k ∈ ℕ, z^{k}f(z) cannot be defined on {z_{n}}_{n=0}^{∞} so as to be analytic in a neighborhood of z_{0}, then for every N ∈ ℕ there exists n such that z_{n} is a pole of order at least N or an essential singularity.

3. Suppose that f is entire and for all z ∈ ℂ, if Re z = 0 then Re f(z) = 0 and if Im z = 0 then Im f(z) = 0. Prove that f(z) = −f(−z) for all z.

4. Determine the number of zeroes of z^{6} − 5z^{4} + 3z^{2} − 1 in {z : |z| ≤ 1}.

5. Let C_{ρ,θ} be the positively oriented contour {z_{0} + ρe^{it} : 0 ≤ t ≤ θ}.

a. If f is analytic in the punctured disk {z : R > |z −z_{0}| > 0} and f has a simple pole at z_{0}, then

lim_{ρ→0^{+}} ∫_{C_{ρ,θ}} f(z)dz = θiRes{f; z_{0}}.

b. Find the principal value of

∫_{−∞}^{∞} \frac{x}{1 − x^{3}} dx.

6. Suppose that f(z) is a non-constant analytic function in a disk about z_{0} and z_{0} is a zero of order 2. Prove or disprove: There exists a neighborhood N of z_{0} and an analytic function g such that g^{2} = f in N .

7. Let h(z) be a (complex valued) continuous function on the boundary of D and

let

g(z) = ∫_{|w|=1} \frac{h(w)}{w − z} dw

for z ∈ D. a. Prove that g(z) is analytic in D. b. Compute g(z) if h(z) = \overline{z}^{n} for some n ∈ ℕ. c. Compute g(z) if h(z) = f(\overline{z}) for some f analytic on \overline{D}.

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