Landen transformations and the integration of rational functions

Abstract

Abstract. We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm for numerical integration of these functions.

Dynamics of the degree six Landen transformation

“…The method described in Section 2 was used in [2] to produce a Landen transformation for the integral of any even rational function, that is, given an even rational function R there is a new one R + such that…”

A Map on the Space of Rational Functions

“…The integral of the even part can be analyzed with the methods described in [1], and the integral of the odd part can be transformed by x → √ x to Here we consider dynamical properties of the map F : R → R. Section 2 characterizes rational functions R for which the orbit Orb(R) := F (k) (R(x)) : k ∈ N (1.6) ends at the fixed point 0 ∈ R. Section 3 describes the dynamics of a special class of functions with all their poles restricted to the unit circle. We establish explicit formulas for the asymptotic behavior of their orbits, expressed in terms of Eulerian polynomials A m (x) defined by the generating function…”

A formula for a quartic integral: a survey of old proofs and some new ones

“…We have called (3.6) a rational Landen transformation. These were extended to all even rational functions in [5] and the convergence of the iterations of these transformation were described in [13]. The extension to arbitrary rational functions on the whole line is described in [14] and [15].…”