Rational Landen transformations on the real line

The rational Landen transformation is a map on the space of coefficients of a rational integrand that preserves the value of the integral. We provide a family of these transformations that apply to rational integrands on the whole line. Given an integer m, these transformations produce a numerical scheme to evaluate the integral that is of order m.Comment: 22 page

Càlcul d'integrals usant sistemes dinàmics discrets

Si per a una família d'integrals definides, dependent de paràmetres, el valor de la integral no varia quan es canvien d'una certa manera els valors dels paràmetres es diu que aquest canvi de paràmetres és una transformació de Landen. Equivalentment, en el llenguatge dels sistemes dinàmics, la integral definida és una integral primera del sistema dinàmic associat a la transformació de Landen. Aquestes transformacions existeixen, per exemple, per a determinades famílies d'integrals el.líptiques o per a famílies d'integrals racionals. En aquest treball presentarem diversos exemples de transformacions de Landen i les aplicarem al càlcul d'integrals definides. També recordarem l'algoritme de Brent-Salamin per a calcular , ja que està basat en aquest tipus de transformacions. Com veurem, la dinàmica global d'algunes transformacions de Landen encara està lluny de ser totalment entesa.If for a family of defined integrals, depending on parameters, the value of the integral remains unchanged when the values of the parameters vary in some special way, it is said that this change of parameters is a Landen transformation. Analogously, using dynamical systems terminology, this defined integral is a first integral of the discrete dynamical system associated with the Landen transformation. These transformations exist, for instance, for some families of elliptic integrals or for certain rational integrals. In this paper we present several examples of Landen transformations and we apply them to the computation of defined integrals. We also recall the Brent-Salamin algorithm for computing , because it is based on these types of transformations. As we will see, the global dynamics of certain Landen transformations are far from being fully understood

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

We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.Comment: 10 page

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same $_2F_1$ hypergeometric function with different rational pullbacks. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus $_2F_1$ hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that emerged in a paper by Casale. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to $_3F_2$, hypergeometric functions, and show that one just reduces to the previous $_2F_1$ cases through a Clausen identity. In a $_2F_1$ hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or $_2F_1$ hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.Comment: 43 page