Master’s Comprehensive Exam Complex Variables August 1994

1. (10 pts.) Evaluate the following:

(a) sin (\frac{π}{2} − 3i)

(b) Log (\frac{−1 − i\sqrt{3}}{2}), where Log represents the Principal branch of the logarithm.

2. (16 pts.) Let f(z) = zIm(z).

(a) What is the domain of f(z)?

(b) Where is f(z) continuous?

(c) Where is f(z) differentiable and what is its derivative at points where it is differentiable?

(d) Where is f(z) analytic?

3. (14 pts.) Evaluate the integral ∫_{0}^{−1+i} zRe(z^{2}) dz along the path C given in the figure at the right.

4. (15 pts.) Evaluate ∮_{C} \frac{e^{πz}}{z^{2} + iz + 2} dz if C is the following contour taken in a counter-clockwise direction:

(a) C : |z| = \frac{3}{2}  (b) C : |z − 1| = 1  (c) C : |z − i| = 4.

5. (20 pts.) Evaluate the integral ∫_{0}^{π} \frac{dΘ}{5 + 4 cos Θ}. (Hint: Use the subsitution z = e^{iΘ}.)

6. (5 pts. each)

(a) Show that the function u(x, y) = x^{2} − (y − 1)^{2} is harmonic in the plane.

(b) Find the harmonic conjugate of u.

7. (15 pts.) Find a bounded harmonic function u(x, y) in the quadrant x ≤ 0, y ≥ 0 which satisfies the following properties: u(x, 0) = 5 for x < 0 and u(0, y) = 2 for y > 0. Justify that your function is harmonic and satisfies the boundary properties. Also evaluate u(−2, 2).