MASTERS COMPREHENSIVE EXAM—COMPLEX VARIABLES—January 1995

1. Find the real and imaginary parts of (−\sqrt{12} − 2i)^{6}.

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

a) z sin \frac{1}{z}

b) \frac{1}{z^{3}e^{2z}}

c) \frac{1 − cos z}{z^{2}}

3. In this problem we write γ for the contour defined by γ(t) = e^{it} on [0, 2π]. Evaluate each integral, with explanation.

a) ∫_{γ} sin z^{3} dz

b) ∫_{γ} z^{−2}e^{(z−1)^{2}} dz

4. For fixed r > 0 we write D = {z: 0 < |z| < r}.

a) If f defined by f(z) = 1/z, find the image of D under f.

b) If g is defined by g(z) = e^{1/z}, find the image of D under g.

5. Suppose u is a real-valued function on the complex plane C.

a) If also v is a real-valued function on C, state the Cauchy-Riemann equations for (u, v).

b) Recall that a real-valued function v on C is said to be a harmonic conjugate of u if u + iv is analytic on C. Use the Cauchy-Riemann equations to show that any two harmonic conjugates of u differ by a constant on C.