Isolated singularities.
The function
has an isolated singularity at
if it is analytic on a deleted open neighborhood of
, but is not analytic at
.
Example 8.3.1
-
has an isolated singularity at 0.
-
has an isolated singularity at 2.
Example 8.3.4
Let
. This function has a singular point at 1, but:
Therefore
and the singularity is removable.
Example 8.3.5
Let
. This function has a singularity at 2. But:
and the singularity is not removable.
Recall that in Calculus, you would have proven that the real valued
function of the real variable
given by
has two infinite one-sided limits at 2.
Proposition 8.3.6
Suppose that
has an isolated singularity at
. If there exists a natural number
such that
and
, then
has a pole of order
at
.
Example 8.3.8
has a pole of order 1 at 0 as a Laurent series expansion of
about 0 is precisely
.
Example 8.3.9
has a pole of order 2 at 3. To prove this according to Proposition
3.7,
we need some work:
A Taylor expansion of
about 3 is:
Thus
Example 8.3.10
has an essential singularity at 0. The MacLaurin expansion of
is
By substitution we get a Laurent expansion of
about 0:
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