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 .