A simple example of a new class of Landen transformation

TL;DR

This paper introduces a basic example of a new class of Landen transformations, which are rational maps that preserve integral values and relate to classical elliptic integral transformations, highlighting their fundamental properties.

Contribution

It presents the simplest case of a rational Landen transformation, illustrating its effect and connection to classical elliptic integral transformations.

Findings

The transformation preserves the value of the integral.

It provides a rational analog to classical Landen transformations.

The simplest case demonstrates fundamental properties of the new class.

Abstract

The rational Landen transformation is a map on the coefficients of a rational integrand that preserves the value of the integral. This is the rational analog of the classical Landen transformations for elliptic integrals that leads to the arithmetic-geometric mean of Legendre and Gauss. We present the effect of this transformation in the simplest possible case.