Skip to main content
SpringerLink
Log in

The Lemniscate Constants

  • Chapter
Pi: A Source Book

  • 1012 Accesses

Abstract

The lemniscate constants, and indeed some of the methods used for actually computing them, have played an enormous part in the development of mathematics. An account is given here of some of the methods used—most of the derivations can be made by elementary methods. This material can be used for teaching purposes, and there is much relevant and interesting historical material. The acceleration methods developed for the purpose of evaluating these constants are useful in other problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Airey, J.R., The “converging factor” in asymptotic series and the calculation of Bessel, Laguerre and other functions. Phil. Mag. 7, 24 (1937), 521–552.

    Google Scholar 

  2. Barna, B., Ein Limessatz aus der Theorie des arithmetisch-geometrischen Mittel. J. Reine Angew. Math. 172 (1934), 86–88.

    Google Scholar 

  3. Bickley, W.G., and Miller, J.C.P., The numerical summation of slowly convergent series of positive terms. Phil. Mag. 7, 22 (1936), 754–767.

    Google Scholar 

  4. Bickley, W.G.. Manuscript, “The numerical summation of series,” 1945.

    Google Scholar 

  5. Brezinski, C., Accélération de la convergence en analyse numérique. Mimeographed lecture notes, Lille, France, 1973.

    Google Scholar 

  6. Carlson, B.C., Algorithms involving arithmetic and geometric means. Amer. Math. Monthly 78 (1971), 496–505.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dahlquist, G., Gustafson, S.A., and Sikleisi, K. Convergence acceleration from the point of view of linear programming. BIT S (1965), 1–16.

    Google Scholar 

  8. von David, L., Arithmetisch-geometrisches Mittel und Modulfunktion. J. Reine Anger. Math. 159 (1928), 154–170.

    MATH  Google Scholar 

  9. Euler, L., Opera Omnia,Vol. 1, 11, 1913, Leipzig.

    Google Scholar 

  10. Euler, L.. Introductio in Analysis Jnfnitorum, I, Ch. 16, 1748.

    Google Scholar 

  11. Euler, L., óutitationes Calculi Inegralis,1748, pp. 228–236.

    Google Scholar 

  12. Ewald, P.P. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. (1921), 253–287.

    Google Scholar 

  13. Fagnano, G.C., Opere Matematiche,3 vols., 1911–12.

    Google Scholar 

  14. Fagnano, G.C.. Produzioni Matematiche, 1750.

    Google Scholar 

  15. Fox, L., Comments on singularities in numerical integration and the solution of differential equations, In Numerical Methods, P. Rósza (Ed.), Coll. Math. Soc.]. Bolyai, 3, 1968, pp. 61–91.

    Google Scholar 

  16. Fox, L., Romberg integration for a ch.: of singular inter_,±s. Computer J. 7 1967 87–93.

    Google Scholar 

  17. Freud. G. Error estimates for Gauss-Jacz, quadrature formulae. pp. 93–112. In Topics in Vtim, r.s.,.. 4n,,/isis. J.J.H \Eller Ed.’. London. 1973.

    Google Scholar 

  18. Gauss, C.F., IhSx4e. 3, Leij1ig. 1576.

    Google Scholar 

  19. Gauss, C.F.. 14’erie 10. Leipzig. 1917.

    Google Scholar 

  20. Geppert. H.. Ed.; Ostwald’s Klassiker,:sr evakten 11 issenwhaften, 225. Leipzig. 1927.

    Google Scholar 

  21. Geppert. H., Wie Gauss zur elliptischen Ntodu!i_-ktion Lam. Deutsche Muth. 5,1940’, ISS-175.

    Google Scholar 

  22. Hardy, G.H., and Wright, E.M. Therm %,m,.’rs. Clarendon Press. Ovford, 1938.

    Google Scholar 

  23. Ki3ek. K., and Schmidt, H. Ausaertung einiger spezieller unendlicher Reihen aus dem Bereich der elliptischen Funktionen. Arch. Math. 18 119671. 438–443.

    Google Scholar 

  24. Knopp. K., Thann and.application or h::: ’m•.Series. 2nd ed. Blackie. London. 1948.

    Google Scholar 

  25. Lehmer, D.H., The lemniscate constant. 11T.3C?,194.8 9. 550–551.

    Google Scholar 

  26. Lehmer, D.H., On arccotangent relations for r. Amer. Math. Mon:hln 45 1938i,657–664.

    Google Scholar 

  27. Markoff, A., Diffrenceurecluurng. Tr. of Russian edition of 1889–91. Leipzig. 1896.

    Google Scholar 

  28. Markushevich, A.I., The Remarkable Sine Functions. American Elsevier, New York. 1966.

    MATH  Google Scholar 

  29. Muckenhoupt. B., The norm of a discrete singular transform. Studia Alurh. 25 11964 5 97–102.

    Google Scholar 

  30. Ogigoca, H., Les lettres de Ch. Hermite A. \larkoti-. 1885–1889. Rex d’histoire de.s sciences et de !ears applications, 20 í1967i, 1–32. Letter dated 11 December 1359.

    Google Scholar 

  31. an der Pol, B. Demonstration élémentaire de le relation 8,’ = 8,’ _ 8.? entre les différentes fonctions de Jacobi. Enseignement Muth. 1 119551, 259–262.

    Google Scholar 

  32. Reichardt, H. (Ed.) C.F. Gauss. Gedenkhand, Leipzig. 1957.

    Google Scholar 

  33. Shanks, D., Nonlinear transformations of divergent and slowly convergent series. J. Math. Phys. 36:1955i. I -62.

    Google Scholar 

  34. Shanks, J.A., Romberg tables for singular integrands. Computer J. 15 (1972i,360–361.

    Google Scholar 

  35. Siegel, C.L. Topics in Complex Function Theory, Vol. 1. ( Viler. New York, 1969.

    MATH  Google Scholar 

  36. Siegel, C.L., Transcendental Numbers, Princeton L’. Press, 1949.

    Google Scholar 

  37. Stirling, J., Aletlutdus Difjerentiali.s, London. 1730. English trans. by F. Holliday, London, 1749.

    Google Scholar 

  38. Stprmer, C., Sur un problème curieux de la théorie des nombres concernant les fonctions elliptiques. Arch. Afath. Js’atunid. B47, t 5 (1948),83–85.

    Google Scholar 

  39. Thacher, H.C., Jr. Numerical application of the generalized Euler transformation pp. 627–631 in Information Processing 74,. 1. Rosenfeld (Ed,), North-Holland Pub. Co., Amsterdam, 1974.

    Google Scholar 

  40. Todd, John., Optimal parameters in two programs. Ball. 1.11.9 6 (1970), 31–35.

    Google Scholar 

  41. Todd, John.. A problem on arctangent relations. Amer. Math. Monthly. 56 (1949),517–528.

    Google Scholar 

  42. Todd, John.. Table of arctangents of rational numbers. U.S. Nat. Bur. Standards, Appl. Math. Series 11, 1951, 1965, U.S. Government Printing Office, Washington, D.C.

    Google Scholar 

  43. Todd, John.. Introduction to the Constructive Theory of Functions. Academic Press, New York, 1963.

    MATH  Google Scholar 

  44. Tweedie, C., James Stirling. Oxford, 1922.

    Google Scholar 

  45. Watson, G.N., The marquis and the land-agent-a tale of the eighteenth century. Math. Gazette 17, 1933. 5–16.

    Google Scholar 

  46. Whittaker, E.T., and Watson, G.N. Alodern Analysis. Cambridge U. Press, 1927.

    Google Scholar 

  47. Widder, D.V., The Laplace Transform. Princeton U. Press, 1941.

    Google Scholar 

  48. Wrench, J.W., Jr. Manuscript (1955).

    Google Scholar 

  49. Wrench, J.W.. MTAC 4 ( 1948, 19 ), 201–203.

    Google Scholar 

  50. Wynn, P., Acceleration techniques in numerical analysis, with particular reference to problems in one independent variable. Proc. IFIP Congress, 62, Munich, North-Holland Pub. Co., Amsterdam, pp. 149–156.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. California Institute of Technology, USA

    John Todd

Authors
  1. John Todd
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media New York

About this chapter

Cite this chapter

Todd, J. (2000). The Lemniscate Constants. In: Pi: A Source Book. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3240-5_45

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3240-5_45

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4757-3242-9

  • Online ISBN: 978-1-4757-3240-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation