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 = r exp(iθ) ∈ C : 1 − ε < r < 1 + ε, θ ∈ (−δ, δ)} are found.