Analytic Implicit Functions

TL;DR

This paper presents a method to locally convert implicit equations into explicit functions using power series, even without differentiability, supported by theoretical analysis and numerical demonstrations.

Contribution

It introduces a novel approach to represent analytic implicit functions as power series in non-differentiable contexts, expanding the applicability of implicit function analysis.

Findings

Implicit functions can be represented as power series in the weak-star sense.

The method works for systems with continuous functions and unique analytic solutions.

Numerical examples demonstrate practical effectiveness of the approach.

Abstract

In this paper, we introduce a method of converting implicit equations to the usual forms of functions locally without differentiability. For a system of implicit equations which are equipped with continuous functions, if there are unique analytic implicit functions, that satisfies the system in some rectangle, then each analytic function is represented as a power series which is the weak-star limit of partial sums in the space of essentially bounded functions. We also provide numerical examples in order to demonstrate how the theoretical results in this article can be applied in practice and to show the effectiveness of the suggested approaches.