Local analytic solutions of a functional equation

Abstract

All analytic solutions of the functional equation \[|f(r exp(iθ))|^2 + |f(1)|^2 = |f(r)|^2 + |f(exp(iθ))|^2\]

in the annulus

\[P := {z ∈ C : 1 − ε < |z| < 1 + ε}\]

and in the domain

\[D := {z = re^{iθ} ∈ C : 1 − ε < r < 1 + ε , θ ∈ (−δ, δ)},\]

are found