Local analytic solutions of a functional equation

Abstract

All analytic solutions of the functional equation
\[
|f(r\exp(i\theta))|^{2}+|f(1)|^{2}=|f(r)|^{2}+|f(\exp(i\theta))|^{2}
\]
in the annulus
\[
P:=\{z\in \mathbb{C}: 1-\epsilon\le|z|\le1+\epsilon\}
\]
and in the domain
\[
D:=\{z=re^{i\theta}\in \mathbb{C}: 1-\epsilon\le r \le 1+\epsilon, \theta\in (-\delta,\delta)\},
\]
are found.