17.2 Meromorphic on Extended Complex Plane
Definition 17.2.1 We say f ∈ℳ
if it is meromorphic on ℂ and either
which it is said that f has a removable singularity at ∞ or
∞ in which case we
say f has a pole at ∞. That is, f
has a pole at 0.
Observation 17.2.2 If f ∈ℳ
, then it has only finitely many poles.
If f has a pole at ∞, this implies there cannot be a sequence of poles converging to ∞. In terms of what
this means, this says that there exists r > 0 such that all poles of f are in B
. Since there is no limit
point for the poles, there are only finitely many.
If f has a removable singularity at ∞, then f
for small z.
Hence if there exists a
sequence of poles
then one could get βn
and for large n,
for large n, but this is not possible because the right converges to a0 while the left is unbounded.