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The Ramanujan Journal

, Volume 10, Issue 3, pp 305–324 | Cite as

A New Class of Infinite Products Generalizing Viète's Product Formula for π

  • Aaron Levin
Article

Abstract

We show how functions F(z) which satisfy an identity of the form Fz) = g(F(z)) for some complex number α and some function g(z) give rise to infinite product formulas that generalize Viète's product formula for π. Specifically, using elliptic and trigonometric functions we derive closed form expressions for some of these infinite products. By evaluating the expressions at certain points we obtain formulas expressing infinite products involving nested radicals in terms of well-known constants. In particular, simple infinite products for π and the lemniscate constant are obtained.

Key Words

infinite products nested radicals elliptic functions 

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References

  1. 1.
    M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th Printing, Dover, New York, 1972.MATHGoogle Scholar
  2. 2.
    P. Beckmann, A History of Pi, 5th edn. The Golem Press, Boulder Colorado, 1982.Google Scholar
  3. 3.
    L. Berggren, J. Borwein, and P. Borwein, Pi: A Source Book, 2nd edn., Springer Verlag, New York, 2000.MATHGoogle Scholar
  4. 4.
    B.C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, New York, 1989.MATHGoogle Scholar
  5. 5.
    B.C. Berndt and R.A. Rankin (eds.), Ramanujan: Essays and Surveys, American Mathematical Society, 2001.Google Scholar
  6. 6.
    J. Borwein and P. Borwein, Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts 4,John Wiley and Sons, Inc., New York, 1998.Google Scholar
  7. 7.
    B.C. Carlson, “The logarithmic mean,” Amer. Math.Monthly 79 (1972), 615–618.MathSciNetMATHGoogle Scholar
  8. 8.
    A. Herschfeld, “On infinite radicals,” Amer. Math.Monthly 42 (1935), 419–429.MathSciNetMATHGoogle Scholar
  9. 9.
    K.E. Morrison, “Cosine Products, Fourier Transforms, and Random Sums,” Amer. Math. Monthly 102 (8) (1995),716–724.MathSciNetMATHGoogle Scholar
  10. 10.
    T.J. Osler, “The union of Vieta's and Wallis's products forpi,” Amer. Math. Monthly 106 (8) (1999), 774–776.MathSciNetMATHGoogle Scholar
  11. 11.
    T.J. Osler, “Interesting finite and infinite products fromsimple algebraic identities,” Math. Gaz. (to appear).Google Scholar
  12. 12.
    J.F. Ritt, “Periodic functions with a multiplication theorem,”Trans. Amer. Math. Soc. 23(1) (1922), 16–25.MathSciNetMATHGoogle Scholar
  13. 13.
    J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer-Verlag, New York, 1994.Google Scholar
  14. 14.
    K.B. Stolarsky, “Mapping properties, growth, uniqueness of Vieta(infinite cosine) products,” Pacific J. Math. 89(1)(1980), 209–227.MathSciNetMATHGoogle Scholar
  15. 15.
    J. Todd, “The lemniscate constants,” Comm. ACM 18 (1975), 14–19; corrigendum, ibid. 18(8), 462.Google Scholar
  16. 16.
    I.J. Zucker, “The evaluation in terms of Gamma-functions ofthe periods of elliptic curves admitting complexmultiplication,” Math. Proc. Camb. Phil. Soc. 82(1977), 111–118.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidence

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