![[Graphics:Images/SingularityZeroPoleMod._gr_331.gif]](../Images/SingularityZeroPoleMod._gr_331.gif){kind=small}
,
then ![[Graphics:Images/SingularityZeroPoleMod._gr_332.gif]](../Images/SingularityZeroPoleMod._gr_332.gif){kind=small}
has a removable singularity at ![[Graphics:Images/SingularityZeroPoleMod._gr_333.gif]](../Images/SingularityZeroPoleMod._gr_333.gif){kind=small}
. If
we define ![[Graphics:Images/SingularityZeroPoleMod._gr_334.gif]](../Images/SingularityZeroPoleMod._gr_334.gif){kind=small}
, then
g(z) has a zero of order
k at ![[Graphics:Images/SingularityZeroPoleMod._gr_335.gif]](../Images/SingularityZeroPoleMod._gr_335.gif){kind=small}
. Corollary 7.6.
If f(z) has a
pole of order k at the point ,
then
has a removable singularity at . If
we define , then
g(z) has a zero of order
k at .
Proof.
Proof of Corollary 7.6 is an exercise in the book.