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2 I added the requirement that $V$ is simply connected

Is the following fact true (I think that I can prove it but I don't trust myself on these matters): let $f(z)$ be an analytic function defined on some open subset $U$ of ${\mathbb C}$. Assume that the function $|f(z)|^2$ extends as a real-analytic function to some bigger simply connected open subset $V$ of ${\mathbb C}$. Then $f$ extends analytically to $V$.

Is there a reference for this fact?

1

# Analytic continuation via square of absolute value

Is the following fact true (I think that I can prove it but I don't trust myself on these matters): let $f(z)$ be an analytic function defined on some open subset $U$ of ${\mathbb C}$. Assume that the function $|f(z)|^2$ extends as a real-analytic function to some bigger open subset $V$ of ${\mathbb C}$. Then $f$ extends analytically to $V$.

Is there a reference for this fact?