10. Let Ω ⊆ C be open and such that A(0; r, ∞) ⊆ Ω. Suppose that f ∈ H(Ω). We say that ∞ isa removable singularity, pole of order m ∈ N, or essential singularity of f if 0 is the same for the function...

10. Let Ω
⊆
C be open and such that
A(0;
r,


∞)
⊆
Ω. Suppose that
f

∈

H(Ω). We say that
∞
isa removable singularity, pole of order
m

∈
N, or essential singularity of
f
if 0 is the same for the function
g(z) =
f
(1/z) defined on Ω,
=
{
z

∈
C
\

{0}
: 1/z

∈
Ω}. If
f
is entire, classify the singularity of
f
at
∞
by giving conditions on the power series of
f
based at 0. What are the only entire functions
f
such that limz→∞

f
(z) =
∞?