Date sent: Thu, 01 Jun 2000 08:11:40 -0400 (EDT) From: zhu ¡kzhu@csc.albany.edu¿ To: ja984@cnsvax.albany.edu Preliminary Exam in Complex Analysis

June 2000

Notation: C denotes the complex plane; R^{−} denotes the negative real axis (including the origin); and D denotes the unit disk.

1. Suppose f(z) is analytic in C − {0} and satisfies

|f(z)| ≤ \frac{1}{\sqrt{|z|}}, z ∈ C − \{0\}.

Show that f is identically zero.

2. Let Ω = C − R^{−}.

1) Define the principal branch of the logarithm, Log(z), in the region Ω.

2) Show that Log(zw) = Log(z) + Log(w) for all z and w in the (open) right halfplane.

3) Show that the identity in 2) does not hold for all z and w in Ω.

3. For each of the following functions find the radius of convergence for its Taylor series at the specified point.

1) f(z) = (cos z)/(3z + 4) at z_{0} = 10.

2) g(z) = 1/(z^{2} + z + 1) at z_{0} = 0.

4. Suppose f(z) and \overline{f(z)} are both analytic in D. Show that f(z) is constant.

5. Evaluate the following integrals.

1) ∫_{C} (sinh z + \frac{z}{2z + 1}) dz, where C is the unit circle traversed once clockwise.

2) ∫_{Γ}(z+3z̄) dz, where Γ is the path from −1 to 1 along the upper semi-circle |z| = 1.

3) ∫_{0}^{π} \frac{dθ}{2 + cos θ}.