Uniqueness of meromorphic continuation

complex-analysis analysis

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 $S_1 \cup S_2$.


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 $S_1 \cup S_2$.

Related

Differential of Dirac Delta Function
Efficient Algorithm for Generalized Sylvester's Equation
P-adic integers and roots of unity
Showing that the "left-hand limit function" is left continuous
Direct products in the category Rel
Are non-degenerate bilinear forms equivalent to isomorphisms $V \cong V^*$?
Possible values of $\int \frac{dz}{\sqrt{1-z^2}}$ over a closed curve in a region?
Fundamental group of the n-holed torus
Inverse Function for a Taylor Series?
Green's Function / Impulse Response Confusion
When $\delta$ decreases should $\epsilon$ decrease? (In the definition of a limit when x approaches $a$ should $f(x)$ approach its limit $L$? )
Evaluate $\lim_{x\to \infty} \cos (\sqrt {x+1})-\cos (\sqrt x)$