On Functions with a Conjugate

TL;DR

This paper explores the concept of conjugates in harmonic functions and extends the idea to certain three-variable functions governed by specific partial differential equations.

Contribution

It introduces a new class of three-variable functions that admit conjugates, extending the classical harmonic function theory.

Findings

Harmonic functions in two variables have conjugates characterized by gradient orthogonality.

Certain three-variable functions are shown to admit conjugates via PDEs.

The paper broadens the understanding of conjugate functions beyond harmonic cases.

Abstract

Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial differential equations controlling the functions of three variables that admit a conjugate.