I'd like to know about the definition of meromorphic function. Usually I see the definition of meromorphic function as follows: Let $D\subset\mathbb C$ be a connected open set, a function $f$ defined on a subset $U$ of $D$ and with value in $\mathbb C$ is meromorphic on $D$ if the following conditions are satisfied:
- $P(f)=D\setminus U$ is a set of poles
- $P(f)$ is discrete in $D$
- $f$ is holomorphic on $U$.
However, in the book " Complex anlysis for mathematics and engineering" by John H. Mathews and Russel W. Howeell, $P(f)=D\setminus U$ is a set of poles and removable singularities.
I think removable singularities are not real singularities, since we can extend the function to the holomorphic function. Thus, two definitions may be almost same.
I'd like to know how other people think about this question.
Would you give any comments about this question? Thanks in advance!