Local analytic solutions of a functional equation

Abstract

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