Suppose f has an isolated singularity at α. Show the singularity is essential if and
only if the principal part of the Laurent series of f has infinitely many terms. That
is, show f
(z − α)
k + ∑k=1∞
where infinitely many of the
bk are nonzero.
Suppose Ω is a bounded open set and fn is analytic on Ω and continuous on Ω.
Suppose also that fn→ f uniformly on Ω and that f≠0 on ∂Ω. Show that for all n
large enough, fn and f have the same number of zeros on Ω provided the zeros are
counted according to multiplicity.