The Weierstrass mean

Summary

Theorem.Let the sequences {e ⁽ⁿ⁾ _i },i=1, 2, 3,n=0, 1, 2, ...be defined by

where the e ^(0)′ s satisfy

and where all square roots are taken positive. Then

where the convergence is quadratic and monotone and where

The discussions of convergence are entirely elementary. However, although the determination of the limits can be made in an elementary way, an acquaintance with elliptic objects is desirable for real understanding.

In fact thee ⁽⁰⁾ _i 's will be interpreted as the values of a ℘-function at its half periods and the successive e ^(n)′ _i s will be the corresponding values for a ℘-function with the same real period and with the imaginary period doubled at each stage-this is the Landen transformation. Ultimately the ℘-function will degenerate into a trigonometrical function.

The subtitle is explained by the fact that the real half-period, ω, of the ℘-function defined by

$$y^{'2} = 4(y - e_1^{(0)} )(y - e_2^{(0)} )(y - e_3^{(0)} ) if found from W = \frac{1}{{12}}\left( {\frac{\pi }{\omega }} \right)^2 $$

.