MASTERS COMPREHENSIVE EXAM—COMPLEX VARIABLES—August 1999

1. (a) Find all values of log (2 − 2\sqrt{3} i).

(b) Find all values of (2 − 2\sqrt{3} i)^{i}.

2. (a) Let f(z) = z^{2} + z̄. Prove that f(z) is nowhere analytic.

(b) Indicate whether the given function is analytic on the set where it is defined. No proof is required.

(i) e^{2z} − z
(ii) |z|^{2}
(iii) sin(z̄)
(iv) Arg(z) (the principal value of the argument function)
(v) cos (z^{2} − e^{z})

3. Let u(x, y) = x^{3} + x − 3xy^{2}.

(a) Show that u is harmonic on the complex plane.

(b) Find all the harmonic conjugates of u.

(c) Find an analytic function f so that Ref = u, and find the Taylor series of f about the point 0.

4. Suppose that f is an entire function with period 1 and period i. (That is, f is analytic in the entire plane and f(z + 1) = f(z + i) = f(z) for all z.) Show that f is constant.

5. Use the Residue Theorem to compute with justification the following integral:

∫_{−∞}^{∞} \frac{cos x dx}{x^{2} + 2x + 2}.