# 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.

• Poles are also just removable singularities, in a way. If we consider functions to the Riemann sphere instead of the complex plane, poles are singularities where the continuous extension maps the singular point to the point at infinity. So it makes sense to treat poles and removable singularities the same way. – Vercassivelaunos Feb 1 at 17:24

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.