Approximations for the natural logarithm from solenoid-toroid correspondence

TL;DR

This paper proposes new approximations for the natural logarithm derived from the correspondence between solenoid and toroid inductances, offering simpler expressions than Taylor polynomials with broader applicability.

Contribution

It introduces novel approximants for the natural logarithm based on solenoid-toroid correspondence, expanding the tools for mathematical approximation beyond traditional methods.

Findings

Different averaging methods yield various approximants.

The approximants are simpler than Taylor polynomials.

They are valid over a wider domain.

Abstract

It seems reasonable that a toroid can be thought of approximately as a solenoid bent into a circle. The correspondence of the inductances of these two objects gives an approximation for the natural logarithm in terms of the average of two numbers. Different ways of averaging give different approximants. They are expressions simpler than Taylor polynomials, and are meaningful over a wider domain.