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Let $\Omega$ be a non-empty region of $\mathbb{C}$ and suppose $f$ is a holomorphic function on $\Omega$.

How can one show that a meromorphic continuation of $f$ to all of $\mathbb{C}$ is unique, if it exists?

By a meromorphic function $f$ on $\mathbb{C}$ I mean a function with a sequence of points $S=\{z_1,z_2,...\}$ with no limit points in $\mathbb{C}$ such that $f$ has poles at $S$ and is holomorphic in $\mathbb{C}-S$.

Can this be proved along the same lines as showing that an analytic continuation is unique (if it exists)? The problem I have is that two meromorphic continuations might have different sets of poles and even if they were the same, the poles might have different orders.

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If you have two meromorphic continuations $m_1$ and $m_2$ with pole sets (contained in) the countable sets $S_1$ and $S_2$ respectively, then $U = \mathbb{C}\setminus (S_1 \cup S_2)$ is connected, and the restrictions of $m_1$ and $m_2$ to $U$ are holomorphic continuations of $f$, hence

$$m_1\lvert_U \equiv m_2\lvert_U.$$

From that, it follows that $m_1$ and $m_2$ have identical singularities in each point of $S_1 \cup S_2$.

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  • $\begingroup$ I was just about to post another question asking if the complement of the poles of a meromorphic function is a connected set. I think I'll post it anyhow although you've answered it partially for me. One question I have is why should $m_1$ and $m_2$ have identical singularities? $\endgroup$ – Mark Rodriguez Jul 19 '13 at 17:13
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    $\begingroup$ $m_1$ and $m_2$ are the same function (at least outside the union of the singular sets). The Laurent expansion around an isolated singularity is determined by the values of the function on an arbitrarily small punctured neighbourhood. That means $m_1$ and $m_2$ have the same Laurent expansion around each $z \in S_1 \cup S_2$. In other words, a meromorphic continuation is unique modulo gratuitous introduction of removable singularities. $\endgroup$ – Daniel Fischer Jul 19 '13 at 17:26
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Consider 2 different meromorphic extensions, g, and h of your function f.

Consider the function $g-h$ on the set $C /\ (S_1 \cup S_2)$. This function is holomorphic, and is zero in a open region, and thus is zero everywhere. g-h must be a meromorphic function. The only meromorphic continuation of 0 on $C \\ (S_1 \cup S_2)$ is that with removable singularities at $S_1 \cup S_2$.

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