# Analytic continuation of quotient of analytic functions

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

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

?

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.