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Suppose $f(z)$ and $g(z)$ are defined for some open subset $U$ of the complex plane, and that they are holomorphic on that subset. We then know that their pointwise quotient $(f/g)(z)$ is meromorphic on that subset.

Assume furthermore that both functions admit meromorphic continuations the entire complex plane, which I will denote as $\hat f(z)$ and $\hat g(z)$. Likewise, assume their pointwise quotient has a meromorphic continuation to the entire complex plane, denoted as $\hat {(f/g)}(z)$.

My question: if you analytically continue $f$ and $g$ separately, then take the quotient of the two functions, do you get the same thing as if you analytically continued $(f/g)$ directly?

Or, expressed directly, do we have

$\frac{\hat f(z)}{\hat g(z)} = \hat{(f/g)}(z)$

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Yes, upon further thought, this is trivially true. If the functions $\hat{f/g}(z)$ and $\frac{\hat f(z)}{\hat g(z)}$ agree on some open subset $U$, then they have the same analytic continuation.

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