MATH 227 Final, Spring 2001
American University of Beirut · Introduction to Complex Analysis · 1 page
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AMEMICAN PNIVERSIT YL
LIBRARY |
OF BEIRUT
Math 227-Final Exam (Spring 01)
1. (32 pts) Evaluate
2. (32 pts) Let 1 € & and y > 0. Find
Oo
S- "cos ra.
n=
(Hint: Ree’**.)
3. (32 pts) Show that all zeroes of p(z) = 32% — 22? + 2iz — 8 hie in the ring 1 < fz| < 2
4. (32 pts) Let f(z) be holomorphic in the ring 4 < fz| < B and suppose [f(<)| = |<|* there.
Show that jis an integer.
5. Let OS R<1,A>0,andQ=c- : Uf], o¢)). - > €Q, define
(a) (16 pts) Prove that
so that F € O(Q).
(b) (16 pts) Prove that
VIR) a eyed , Vt = Ree) > VTP
for 1.0 €
(c) (25 pts) Prove that
[ |F (te ‘Vat < are i i |F (Re!) dos AR? Tay
? AK) _
(Hint: [2 ec" du = /r/2.)
(d) (15 pts) Prove that
lene
+ a :
ARV V1 — =)