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!
Best Answer
Indeed, removable singularities can be removed. I guess the point the authors are trying to make is this. Suppose you are initially presented with a function defined, let's say, by a formula $f(z) = A(z)/B(z)$ where $A$ and $B$ are holomorphic in $D$. This is not defined at the zeros of $B$, forming a discrete set in $D$. These may be removable singularities or poles, and it may take some work to figure out which is which. But you can still say that the function is meromorphic in $D$.