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Context: I am searching for a reference for the following result:

Let $f:U\to \mathbb C$ be a meromorphic function. There exists a meromormphic function $g:U\to \mathbb C$ such that $g^2=f$ if and only if all the poles and zeros of $f$ in $U$ have even order.

A proof of this result can be found here, or here for the special case of holomorphic functions, near a single zero. But these answers do not provide a reference.

So far: this result doesn't seem to appear in the following books:

  • Ahlfors, L.V. (1966) Complex Analysis. McGraw-Hill Book Company.
  • Conway, J. (2012). Functions of One Complex Variable. Springer Science & Business Media.
  • Lang, S. (2013). Complex analysis. Springer Science & Business Media.
  • Sveshnikov, A. G., & Tikhonov, A. N. (1978). The Theory of Functions of a Complex Variable.
  • Stein, E. M., & Shakarchi, R. (2010). Complex analysis. Princeton University Press.

I'll try to keep this list up to date. Don't hesitate to give plausible reference, I'll try to go through it myself.

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