Real and discrete holomorphy: Introduction to an algebraic approach (Q884003)

A complex-valued (even more generally a ring-valued) function \(\varphi\) defined on a Riemannian manifold or a graph \(X\) is here said to be holomorphic if \(\varphi\) and \(\varphi^2\) are harmonic on \(X\). Some properties of such holomorphic functions (such as holomorphic extensions, holomorphic morphisms, holomorphic classification etc.) are considered extensively on the bi-infinite tree of valency 3, with indications of extensions to other types of graphs. The question of finding a real function \(g\) on \(X\) associated to a known real function \(f\) on \(X\) so that \(\varphi= f+ ig\) is holomorphic on \(X\) is considered, with examples of existence and non-existence of this conjugate function \(g\).