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Prelim in Complex Analysis, June 2001
1. Evaluate the following integrals.
∫
|z|=2
tan z dz,
∫ π
0
dt
5 − 4 cos t
.
2. Find the Laurent series of the function
f(z) =
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 ( 1n
)∣∣∣∣ < e−n
for n = 2, 3, 4, · · ·? Justify your answer.
6.
(a) Show that
∣∣∣∣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)| ≤
∣∣∣∣1 − 2z
2 − z
∣∣∣∣
for all |z| < 1.
END