Math 534 PRACTICE PROBLEMS FOR THE FINAL Autumn 2007
1. Find
Z∞
0
cos x−1
x2dx, Z∞
0
log2x
1 + x2dx, Z∞
0
dx
1 + xn,Z∞
0sin kt
t2
dt, k > 0.
2. (a) Let aand bbe complex numbers such that 0 <|a|<|b|. Write down all Taylor and
Laurent series of
f(z) = 1
(z−a)(z−b)
centered at 0, and state where they converge.
(b) Find the expansions of the function fgiven by f(z) = 1
1−z2+1
1−2zin powers of z. Say
explicitly where each converges.
3. Prove that any function fwhich is analytic in C\([0,1] ∪[2,3]) can be written as
f=g1+g2
where g1is analytic in C\[0,1] and g2is analytic in C\[2,3].
4. Let fbe analytic from Dto {z:<(z)>0}such that f(0) = 1. Prove that
|f(z)| ≤ 1 + |z|
1− |z|∀z∈D.
5. Prove that there does not exist an entire function h(z) such that
|h(z)| ≥ eA|z|
for some A > 0 and all sufficiently large |z|.
6. Let fbe an analytic function defined on Dsuch that |f(z)|<1 for all z∈D. Suppose
there are two points z1and z2in Dwith z16=z2such that f(z1) = z1, f(z2) = z2. Then prove
that f(z) = zfor all z∈D.
7. Suppose fand gare meromorphic functions on Csuch that g(z) = f(1/z) for z6= 0.
Show that fis a rational function.
8. Is there a one-to-one, analytic map from the annulus Ω1={z:1
2<|z|<1}onto the
punctured disk Ω2={z: 0 <|z|<1}? (Justify your answer; and you can only use the tools
we have studied so far.)
9. Show that there is no entire function fwith f(i
n) = 1
n+1 for all n∈N.