The Weierstrass mean. I: The periods of \(\wp (z| e_ 1,e_ 2,e_ 3)\) (Q581994)

One of the most important computations ever made was due to Gauss. From his computations of arithmetic-geometric means he was led to the study of \(\vartheta\)-functions, modular functions and the transformation theory of elliptic objects, in particular the Landen-transformation. This is translated into Weierstrassian: given \(e_ 1(0)>e_ 2(0)>e_ 3(0)\) with \(e_ 1(0)+e_ 2(0)+e_ 3(0)=0\), sequences \(\{e_ 1(n)\}\), \(\{e_ 2(n)\}\), \(\{e_ 3(n)\}\) are defined which converge quadratically and monotonically to 2W, -W, -W, where \(W=(\pi /\omega)^ 2/12\), \(\omega\) being the real period of \(\pi (z| e_ 1(0),e_ 2(0),e_ 3(0))\).