# definition of meromorphic function : it may have removable singularities

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:

1. $P(f)=D\setminus U$ is a set of poles
2. $P(f)$ is discrete in $D$
3. $f$ 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.

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 $D$.

I was gonna write an answer to my similar question but I got closed so I'll answer this question instead. Most authors say that a function $f$ is meromorphic on $D$ if it's analytic except in a discrete set of poles. However that may lead to some inconveniences, as pointed out in my question Can a meromorphic function have removable singularities?.

So it's reasonable to consider removable singularities in the definition of meromorphic function. A possible fix to the definition of meromorphic function (one without considering too many cases) is the following:

Let $D$ be an open connected subset of $\mathbb{C}$. A function $f$ is meromorphic on $D$ if for all $a\in D$ there is $\delta>0$ such that either $f$ is holomorphic in $B(a,\delta)$ (the open ball of center $a$ and radius $\delta$) or $f$ is holomorphic in $B(a,\delta)-\{a\}$, undefined in $a$ but there exists a positive integer $m$ such that

\begin{equation} \lim_{z\to a}(z-a)^mf(z)=0 \end{equation}

It's not so trivial to prove that the usual definition and this one are equivalent.

Let's imagine the second situation occurs and $m_0$ is the least of such integers $m$ satisfying the limit. If $m_0=1$ then $a$ is a removable singularity. If $m_0\ge 2$ then $a$ is a pole. This is also not so easy to prove.

This stuff might be useful when proving, for example, that the quotient of two meromorphic functions is again meromorphic.