Searching the (really) real general solution of 2D Laplace differential equation

TL;DR

This paper presents a method to find the analytical expression of the general real solution to the 2D Laplace equation, addressing a gap in existing literature and clarifying misconceptions about complex solutions.

Contribution

It introduces a novel approach to derive the real general solution of the 2D Laplace equation without relying solely on complex-analytic functions.

Findings

Provides a systematic method for real solutions

Clarifies the role of complex functions in Laplace solutions

Enhances understanding of real solutions in science and engineering

Abstract

This is not a new result. Purpose of this work is to describe a method to search the analytical expression of the general real solution of the two-dimensional Laplace differential equation. This thing is not easy to find in scientific literature and, if present, often it is justified with the assertion that an arbitrary analytic complex function is a solution of Laplace equation, so introducing the condition of complex-differentiability which is not really necessary for the existence of a real solution. The question of the knowledge of real exact solutions to Laplace equation is of great importance in science and engineering.