Explicit Ramanujan-type approximations to pi of high order

  • J. M. Borwein
  • P. B. Borwein


We combine previously developed work with a variety of Ramanujan’s higher order modular equations to make explicit, in very simple form, algebraic approximations to π which converge with orders including 7, 11, 15 and 23.


Ramanujan-type approximations recursive approximation to pi 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Berndt B C,Chapters 19 and 20 of Ramanujan’s second notebook (Springer Verlag) (to appear)Google Scholar
  2. [2]
    Borwein J M and Borwein P B,Pi and the AGM: A study in analytic number theory and computational complexity (New York: John Wiley) (1987)zbMATHGoogle Scholar
  3. [3]
    Borwein J M, Borwein P B and Bailey D H, Ramanujan, modular equations and pi: or how to compute a billion digits of pi,M A A Monthly (in press)Google Scholar
  4. [4]
    Hardy G H,Ramanujan’s collected papers (New York: Chelsea Publishing) (1962)Google Scholar
  5. [5]
    Magnus W, Formulae and theorems for the special functions of mathematical physics (Berlin: Springer Verlag) (1966)Google Scholar
  6. [6]
    Ramanujan S, Modular equations and approximations to π,Q. J. Math. Oxford 45 (1914) 350–372 (New York: Chelsea) (1980)Google Scholar
  7. [7]
    Weber H,Lehrbuch der algebra (Baruschweig) vol. 3 (1908)Google Scholar
  8. [8]
    Whittaker E T and Watson G N,Modern analysis (Cambridge: University Press) 4th edn (1927)zbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 1987

Authors and Affiliations

  • J. M. Borwein
    • 1
  • P. B. Borwein
    • 1
  1. 1.Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifaxCanada

Personalised recommendations