We continue with a discussion about meromorphic functions and the properties of analytic functions. Later notes will consider the Riemann mapping theorem, harmonic functions and the Dirichlet problem among other topics.
Definition 2.1 A function on an open set
is meromorphic if there exists a discrete set of points
such that
is holomorphic on
and has poles at each
. Furthermore,
is meromorphic in the extended complex plane if
is either meromorphic or holomorphic at
. In this case we say that
has a pole or is holomorphic at infinity.
By collecting results from the previous section, we are immediately led to the following proposition regarding the Laurent expansions of complex valued functions.
Proposition 2.2 Let be the discrete set of singularities of a complex function
where
is an open set in
. For a fixed
, suppose the Laurent expansion for
in an annulus about
is given by
. Then,
- The function
has a removable singularity at
if and only if
for all
.
- The function
has a pole at
if and only if there exists
with
such that
for all
; that is, the Laurent expansion of
about
has only finitely many negative terms.
- The function
has an essential singularity at
if and only if the Laurent expansion of
about
has infinitely many negative terms.
- Furthermore,
is meromorphic on the extended complex plane if and only if there exists
such that
for
.