Construction of conjugate functions (Q963258)

By taking into account the proof of a Kerr Theorem [\textit{P. Nurowski} and \textit{A. Trautman}, Differ. Geom. Appl. 17, No.~2--3, 175--195 (2002; Zbl 1035.53067)], here are found all pairs (f,g) of conjugate functions (i.e. analytic functions in \(\mathbb{R}^n\), \(n > 2\), such that \(|\nabla f|=|\nabla g|\) and \((\nabla f) (\nabla g) = 0\)). An application of this construction lies in the fact that a pair of harmonic functions \((f,g)\) is equivalent with a harmonic morphism \(F = (f,g)\) [\textit{P. Baird} and \textit{J. C. Wood}, Harmonic morphisms between Riemannian manifolds. Oxford: Oxford University Press (2003; Zbl 1055.53049)].