MATH 227 Final, Spring 2000
American University of Beirut · Introduction to Complex Analysis · 1 page
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EEE IIE sl
a
—
“Math 227-Final Exam (Spring 00) B. Shayya
1. (30 pts) Evaluate
2. Let 1
_\ _ loge
Plz) = 1423
(a) (10 pts) Show that
1 [xr ne
Res (Fy e'"/8) = — (2 4,7
es (F, 7/3) mF +i)
(b) (15 pts) Find limp 49 Z, and limps. Zz, where
2m /3 ;
i= i F (re™®) ire do (r>0, r#1,)
Sd
(c) (5 pts) Show that for x > 0,
F(a) — 273 (62/8) = 3-iv3 F(a)+ ; V3 +i .
(d) (20 pts) Evaluate the improper integral
~ |
| “* dx.
9 lL+a3
(e) (5 pts) Find all z € C for which (*) (with « replaced by z) holds.
(22 pts) Show that /3z?—1 can be defined to be a single valued holomorphic function for
|z| > 1, and that W5—= can be defined to be a single valued holomorphic function for |z| < 2.
_ 4. (33 pts) Let U = {z = a+ iy: y > 0} be the upper half plane. Let f € O(U) and sup-
pose lim, 49 2eu f(z) = 0. Show that
. y fily\
sie FH) =0
uniformly on every angle |@ — a1 <a<n/2.
5. Let f € O(C — 0) and suppose [f{2)| = 1 for fz) = 1,
(a) (12 pts) For z ¥ 0, define g(=) = (1/2). Prove that g € O(c — 0).
(b) (12 pts) Prove that f(z) £0 for z#0.
(c) (12 pts) Show that Res(f’//. O) EZ.
(d) (24 pts) Prove that there exist an integer m and a function ¢ € O(C ~ 0) such that
f(z) = 2c, zg £0.
Also prove that ¢(z) + 6(1/2) = 0 for z £0.