Master’s Comprehensive Exam Complex Variables May 1995

1. (10 pts.) Compute i^{i}.

2. (10 pts.) Compute ∫_{C} z^{2}dz, where C is the line segment from the origin to the point (1 + i) in the complex plane.

3. (20 pts.) In each case, find the residue of the function at 0 and indicate whether 0 is a removable singular point, an essential point or a pole (in which case, give the order of the pole).

(a) e^{1/z};

(b) \frac{cos z − 1}{z^{2}}.

4. (15 pts.) Using the Residue Theorem, compute the following integral:

∫_{0}^{+∞} \frac{dx}{1 + x^{2}} .

5. (15 pts.) Find the Laurent series expansions in the domain 0 < |z| < 1 for the function

f(z) = \frac{1}{z^{2}(1 − z)}

6. (30 pts.) Let u(x, y) = x^{2} − y^{2} + x where x and y are real variables.

(a) Show that u(x, y) is harmonic in the plane.

(b) Find all the harmonic conjugates of u(x, y).

(c) Find an analytic function f such that Re(f) = u(x, y).