If $f$ and $g$ are analytic functions in a region $D$ and $|f|^2 + |g|^2$ is a constant.
show that $f$ and $g$ are constant functions in a region $D$
$f$ and $g$ are complex functions
So far my attempt is a mess.
From writing $|f|^2$ as $f\overline{f}$, $|g|^2$ as $g\overline{g}$ and differentiating it
to using Cauchy-Reimann equations.
None of which is fruitful, hence any help or insights is deeply appreciated.
