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——- Forwarded message follows ——- From: zhu ¡kzhu@csc.albany.edu¿ Date sent: Mon, 4 Jun 2001 13:18:17 -0400 (EDT) To: richardg@csc.albany.edu Subject: C-Prelim Tex File

Prelim in Complex Analysis, June 2001

1. Evaluate the following integrals.

∫_{|z|=2} tan z dz, ∫_{0}^{π} \frac{dt}{5 − 4 cos t}.

2. Find the Laurent series of the function

f(z) = \frac{1}{(z − 1)(z − 2)}

in the region 1 < |z − 3| < 2.

3. Suppose f(z) is an entire function with Re f(z) > 10 for all z. Show that f is constant.

4. Let F be the family of functions f analytic in |z| < 1 such that ∫_{|z|<1} |f(z)| dA(z) ≤ 1,

where dA is area measure on |z| < 1. Show that F is a normal family.

5. Does there exist an analytic function f in |z| < 1 such that

0 < |f (\frac{1}{n})| < e^{−n}

for n = 2, 3, 4, ⋯? Justify your answer. 6.

(a) Show that

|\frac{1 − 2z}{2 − z} | < 1

for all |z| < 1.

(b) Suppose f is analytic in |z| < 1, f(0.5) = 0, and |f(z)| ≤ 1 for all |z| < 1. Show

that

|f(z)| ≤ |\frac{1 − 2z}{2 − z} |

for all |z| < 1. END