# An extension of Landen transformations

A Landen transformation is a transformation on the parameters of a definite integral that fixes its value. The earliest discovery of such a transformation was the arithmetic-geometric mean iteration, phi: (a, b) ((a + b)/2, ab ). Gauss connected the limit of this iteration to the complete elliptic integral, whose value is fixed under phi. This result both links the study of integrals and Number Theory and provides a quadratically convergent numerical method for approximating an elliptic integral We present a Landen transformation for a rational integral over the real line. This generalizes the work done by Boros and Moll for the even case. Let Rp :=&cubl0;a&ar;e R2p&vbm0;I a&ar;< infinity&cubr0;, where a&ar;:= a0,&ldots;,ap;b0,&ldots;,b p-2and Ia&ar; := RRx dx , with Rx :=BxA x=b0xp -2+&cdots;+bp-2a0xp +a1xp-1+&cdots;+ap . We construct, for all 2 ≤ m, p ∈ N , a rational Landen transformation Lm,p :Rp Rp , and prove its scaled version converges to a limit in R2p with convergence order m. We also prove the invariant property Ia&ar; =I&parl0;Lm,p a&ar; &parr0;, which leads to the formula RR xdx=pLa &d1;, where La&ar; :=lim n→infinitybn 0an 0, Lnm, na =:&parl0;a n0,a n1,&cdots;,a np;b n0,&cdots;b np-2&parr0; is the limit of the Landen iteration. This generates a numerical method for a rational integral which converges to order m