The Intuitive Logarithm

TL;DR

This paper introduces an intuitive method for selecting Abel functions of analytic functions, demonstrates its convergence for simple cases, and derives a new polynomial approximation sequence for the logarithm.

Contribution

It presents a novel intuitive approach for Abel functions, proves its convergence in specific cases, and introduces a new polynomial approximation for the logarithm.

Findings

Convergence shown for f(x)=bx case

Derived a new polynomial approximation sequence

Confirmed the Abel function as the logarithm to base b

Abstract

We introduce the intuitive method to select an analytic Abel function of an analytic function f at a non-fixpoint. Due to the complexity of this method by involving matrix inversion of increasing size there is little known about its convergence. We show its convergence in the simplest but still complicated case f(x)=bx. We show that the obtained Abel function is, as expected, the logarithm to base b, independent on its development point. As a by-product we obtain a new polynomial approximation sequence for the logarithm to base b.