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 .