A new Pythagorean functional equation

Summary

The purpose of this paper is to solve the following Pythagorean functional equation:(e ^p(x,y) ) ² ) = q(x,y) ² + r(x, y) ², where each ofp(x,y), q(x, y) andr(x, y) is a real-valued unknown harmonic function of the real variablesx, y on the wholexy-planeR ².

The result is as follows.

Theorem. Suppose that each of p(x, y), q(x, y) and r(x, y) is a real-valued unknown harmonic function on R ².The only systems of harmonic solutions of (1) are

$$\left\{ {\begin{array}{*{20}c} {p(x,y) = \log \left| {E(z)} \right|} \\ {q(x,y) = \operatorname{Re} (E(z))} \\ {r(x,y) = \operatorname{Im} (E(z))} \\ \end{array} } \right.$$

((i))

and

$$\left\{ {\begin{array}{*{20}c} {p(x,y) = \log \left| {E(z)} \right|} \\ {q(x,y) = \operatorname{Re} (E(z)){\mathbf{ }},} \\ {r(x,y) = - \operatorname{Im} (E(z))} \\ \end{array} } \right.$$

((ii))

In other words, there exists an entire function E(z) such that p(x, y) = log|E(z)|, q(x, y) = Re(E(z))and either r(x, y) = Im(E(z))or r(x, y) = −Im(E(z))and p(x, y), q(x, y) and r(x, y) satisfy (1).