Math 609 Exam 1 Jerry L. Kazdan
March 18, 2004 1:30 — 2:50
Directions This exam has two parts, the first is short answer (6 points each) while the second
has traditional problems (12 points each). Closed book, no calculators – but you may use one
3”×5” card with notes.
Part A: Short Answer Problems (5 problems, 6 points each)
A–1. Find all complex values of 1iin the form a+ib.
A–2. a) For which values of the constant c is u(x, y ) := 2y+e3ysin cx the real part of an analytic
function f=u+iv ?
b) For these values of c, find the corresponding function f(z).
A–3. If Panznis the power series expansion of 1
cos(z+ 1) about z= 0 , what is its radius of
convergence?
A–4. Compute IC
ez
z2−2zdz , where Cis the ellipse x2/25 + y2/9 = 1 (counterclockwise).
A–5. Let f(z) be holomorphic for 0 <|z|<∞. If fhas no zeroes and |f(z)| ≥ |f(2)|in the disk
|z−2|<1, what can you conclude about f(z) ? Justify your assertions.
Part B: Traditional Problems (6 problems, 12 points each)
B–1. Let f(z) be holomorphic in {|z| ≤ 1}except for a simple pole at z=i/2. If falso satisfies
f(1
2) = 0 as well as |f(z)| ≤ 1 on |z|= 1, show that |f(0)| ≤ 1 .
B–2. Find a conformal map f(z) = u+iv from the unit disk {|z|<1}to the first quadrant,
{u > 0, v > 0}.
B–3. Give a complete clear proof of the fundamental theorem of algebra: Every nontrivial complex
polynomial has at least one root.
B–4. Evaluate Z∞
0
cos x
1 + x2dx.
B–5. Let g(z) be holomorphic in the closed unit disk D={|z| ≤ 1}and assume that |g(z)| ≤ 2
for |z|= 1. How many roots does h(z) := g(z) + 5z3−2 have in D? As usual, justify your
assertions.
B–6. Let h(z) =
∞
X
n=1
an
nz, where z=x+iy , and assume the sequence anis bounded, say |an| ≤ M.
Show that h(z) is holomorphic in the half-plane {x > 1}.