On the nonlinear Cauchy-Riemann equations of structural transformation and nonlinear Laplace equation

TL;DR

This paper introduces a generalized framework for complex differentiability through a functional transformation, leading to new equations and operators, and explores their applications to nonlinear Laplace equations.

Contribution

It develops a generalized notion of holomorphicity via a functional transformation, deriving simpler Carleman-Bers-Vekua equations and new differential operators applicable to nonlinear Laplace equations.

Findings

Derived a generalized Carleman-Bers-Vekua equation dependent on a structural function.

Established a generalized exterior differential operator and Wirtinger derivatives.

Analyzed second-order nonlinear Laplace equations within this new framework.

Abstract

This paper aims at studying a functional $K$-transformation $w\left( z \right)\to \widetilde{w}\left( z \right)=w\left( z \right)K\left( z \right)$ that is made to reconsider the complex differentiability for a given complex function $w$ and subsequently we obtain structural holomorphic to judge a complex function to be complex structural differentiable. Since $K\left( z \right)$ can be chosen arbitrarily, thus it has greatly generalized the applied practicability. And we particularly consider $K \left( z \right)= 1+\kappa \left( z \right)$, then we found an unique Carleman-Bers-Vekua equations which is more simpler that all coefficients are dependent to the structural function $\kappa \left( z \right)$. The generalized exterior differential operator and the generalized Wirtinger derivatives are simultaneously obtained as well. As a discussion, second-order nonlinear Laplace equation is studied.