Numerische Mathematik

, Volume 57, Issue 1, pp 737–746 | Cite as

The Weierstrass mean

I. The periods of ℘(z|e1,e2,e3)
  • John Todd
Article

Summary

Theorem.Let the sequences {ei(n)},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 theei(0)'s will be interpreted as the values of a ℘-function at its half periods and the successive ei(n)′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
$$y^{'2} = 4(y - e_1^{(0)} )(y - e_2^{(0)} )(y - e_3^{(0)} ) if found from W = \frac{1}{{12}}\left( {\frac{\pi }{\omega }} \right)^2 $$
.

Subject Classification

AMS(MOS): 65 D 15 CR: G 1.2 

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: (eds.) Handbook of functions. NBS Applied Math. Series, vol. 55, Washington, D. C., U.S. Government Printing Office. 1964Google Scholar
  2. 2.
    Borwein, J.M., Borwein, P.B.: Pi and the AGM. New York: Wiley 1987Google Scholar
  3. 3.
    Cox, D.A.: The arithmetic-geometric mean of Gauss. Enseign. Math., II. Ser.,30, 275–330 (1984)Google Scholar
  4. 4.
    Erdélyi, A.: (ed.) Higher transcendental functions, based, in part, on notes left by Harry Bateman, 3 vols. New York: McGraw-Hill 1953–55Google Scholar
  5. 5.
    Gauss, C.F.: Werke. 12 vols. reprint. Hildesheim: Georg Olms 1981Google Scholar
  6. 6.
    Geppert, H.: Bestimmung der Anziehung eines elliptischen Ringes. Ostwald's Klassiker, vol. 225. Leipzig: Akad. Verlag 1927Google Scholar
  7. 7.
    Geppert, H.: Zur Theorie des arithmetisch-geometrischen Mittels. Math. Ann.99, 162–180 (1928)Google Scholar
  8. 8.
    Chih-Bing, Ling: Evaluation at half periods of Weierstrass' elliptic function with rectangular primitive period-parallelogram. Math. Comput.14, 67–70 (1960)Google Scholar
  9. 9.
    Schwarz, H.A.: Formeln und Lehrsätze zum Gebrauch der elliptischen Functionen, ed. 2. Berlin Heidelberg New York: Springer 1893Google Scholar
  10. 10.
    Tannery, J., Molk, J.: Eléments de la théorie des functions elliptiques, 4 vols. Paris: Gauthier-Villars, 1893–1902Google Scholar
  11. 11.
    Krafft, M., Tricomi, F.G.: Elliptische Funktionen. Leipzig Akad. Verlag 1968Google Scholar
  12. 12.
    Weierstrass, K.: Mathematische Werke, 7 vols. Berlin: Mayer und Müller 1913Google Scholar
  13. 13.
    Whittaker, E.T., Watson, G.N.: A course of modern analysis, 4th. ed. Cambridge: University Press 1927Google Scholar
  14. 14.
    Carlson, B.C.: Landen transformation of integrals. In: Wong, R. (ed.), Asymptotic and computational analysis. Conference in honor of F.W.J. Olver's 65th birthday. Lecture notes in Pure and Applied Math. Series, vol. 124. New York: Marcel Dekker 1990Google Scholar
  15. 15.
    Grayson, D.R.: The arithmo-geometric mean. Arch. Math.52, 507–512 (1989)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • John Todd
    • 1
  1. 1.253-37 CaltechPasadenaUSA

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