A new Pythagorean functional equation (Q755920)

The author proves that the only harmonic functions p(x,y), q(x,y), r(x,y) satisfying the following Pythagorean functional equation \[ (e^{p(x,y)})^ 2=q(x,y)^ 2+r(x,y)^ 2,\quad (x,y)\in {\mathbb{R}}^ 2, \] are of the form: (i) \(p(x,y)=\log | E(z)|\), \(q(x,y)=Re(E(z))\), \(r(x,y)=Im(E(z))\) or (ii) \(p(x,y)=\log | E(z)|\), \(q(x,y)=Re(E(z))\), \(r(x,y)=-Im(E(z))\), where E(z), \(z=x+iy\), is a never vanishing entire function.