Computed multivalues of AGM reveal periodicities of inverse functions

TL;DR

This paper explores how multiple values arise in the computation of the AGM and related elliptic integrals, revealing periodicities in inverse functions through multivalued analysis.

Contribution

It introduces a method to compute multiple AGM values based on different choice sequences, extending to elliptic integrals and revealing underlying periodicities.

Findings

Multiple AGM values can be generated with 2^N different choices.

Repetition of AGM computations reveals periodicities in inverse functions.

The approach applies to both complete and incomplete elliptic integrals.

Abstract

The article shows how two choices are possible whenever computing the geometric mean, and the repetition of this process can in general yield 2-to-the power N different values when the choices are compounded in the first N steps of evaluation of the arithmetic-geometric mean. This happens not only in the simple AGM involved in the computation of the complete elliptic integral of the first kind, but also in analogous methods for the computation of the complete and incomplete elliptic integrals of the first and second kind.