The Weierstrass mean. I: The periods of \(\wp (z| e_ 1,e_ 2,e_ 3)\). (English) Zbl 0689.65005

Todd, John

Numer. Math. 57, No. 8, 737-746 (1990).

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 \(\pi (z| e_ 1(0),e_ 2(0),e_ 3(0))\).

Reviewer: J.Todd

MSC:

65D20	Computation of special functions and constants, construction of tables
65B10	Numerical summation of series
40A25	Approximation to limiting values (summation of series, etc.)

Keywords:

Weierstrass mean; theta function; Weierstrass P function; quadratic convergence; arithmetic-geometric means; modular functions; elliptic objects; Landen-transformation; period

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