MASTERS COMPREHENSIVE EXAM—COMPLEX VARIABLES—August 2000

1. Find the real part and the imaginary part of the principal value of (1 + i)^{i}.

2. (a) Find the linear fractional transformation w = T (z) which maps the points z_{1} = ∞, z_{2} = i, z_{3} = 0 onto the points w_{1} = 0, w_{2} = i, w_{3} = ∞, respectively.

(b) Determine the image of the unit circle under T.

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. Let C be the unit circle with the counterclockwise orientation. Evaluate the following integrals.

(a) ∫_{C} \overline{z} dz

(b) ∫_{C} e^{3/z} dz

5. Use the Residue Theorem to compute with justification the following integral, for a > 0:

∫_{0}^{∞} \frac{1}{(x^{2} + a^{2})^{2}} dx.