Math 534 Homework #4
Autumn 2008
Let D={z:|z|<1}.
1. Prove the uniqueness of the Laurent series expansion, that is if
∞
X
n=−∞
anzn=
∞
X
n=−∞
bnzn
for r < |z|< R then an=bnfor all n. Convergence of the series on the region is part of the
assumption.
2. Prove the following version of Weierstrass’s theorem: Suppose {Un}is an increasing sequence
of regions and suppose fnis defined and analytic on Un. If the sequence fnconverges uniformly
on compact subsets of U≡ ∪nUnto a function f, then fis analytic on Uand the sequence f0
n
converges to f0uniformly on compact subsets of U(even though perhaps none of the fnare defined
on all of U).
3. Suppose Ω is a bounded region whose boundary consists of finitely many disjoint piecewise
differentiable curves. Orient ∂Ω so that the region lies on the left for each b oundary component
(the inner normal is i times the unit tangent vector). Prove n(∂Ω, a) = 1 if a∈Ω and n(∂Ω, a) = 0
if a∈C\Ω.
4. Let Ube an open set in C. Define an equivalence relation on the points of Uby: a∼bif and
only if there is a polygonal arc contained in Uwith edges parallel to the axes and with endpoints a
and b. Show that each equivalence class is open and closed in Uand connected and that there are
at most countably many equivalence classes. The equivalence classes are called the components of
U.
5. Prove that if a sequence of analytic p olynomials converges uniformly on a region Ω then the
sequence converges uniformly on a simply connected region containing Ω.
6. Find all entire functions fsatisfying f(f(z)) = f(z) for all z∈C.
7. Find all expansions in powers of zfor
z
(z2+ 4)(z−3)2(z−4)
