Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi

  • J. M. Borwein
  • P. B. Borwein
  • D. H. Bailey


The year 1987 was the centenary of Ramanujan’s birth. He died in 1920 Had he not died so young, his presence in modern mathematics might be more immediately felt. Had he lived to have access to powerful algebraic manipulation software. such as MACSYMA, who knows how much more spectacular his already astonishing career might have been.


Fast Fourier Transform Discrete Fourier Transform Galois Group Theta Function Modular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964.MATHGoogle Scholar
  2. 2.
    D. H. Bailey, The Computation of π to 29,360.000 decimal digits using Borweins’ quartically convergent algorithm, Math. Comput., 50 (1988) 283–96.MATHGoogle Scholar
  3. 3.
    D. H. Bailey, Numerical results on the transcendence of constants involving π. e. and Euler s constant. Math. Comput., 50 (1988) 275–81.MATHGoogle Scholar
  4. 4.
    A. Baker, Transcendental Number Theory Cambridge Univ. Press, London, 1975.CrossRefGoogle Scholar
  5. 5.
    P. Beckmann, A History of Pi, 4th ed., Golem Press, Boulder, CO. 1977.Google Scholar
  6. 6.
    R. Bellman, A Brief Introduction to Theta Functions, Holt, Reinhart and Winston, New York. 1961.Google Scholar
  7. 7.
    B. C. Berndt, Modular Equations of Degrees 3. 5, and 7 and Associated Theta Functions Identities, chapter 19 of Ramanujan’s Second Notebook. Springer—to be published.Google Scholar
  8. 8.
    A. Borodin and I. Munro, The Computational Complexity of Algebraic and Numeric Problems, American Elsevier, New York, 1975.MATHGoogle Scholar
  9. 9.
    J. M. Borwein and P. B. Borwein, The arithmetic-geometric mean and fast computation of elementary functions, SIAM Rev., 26 (1984), 351–365.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    J. M. Borwein and P. B. Borwein, An explicit cubic iteration for pi, BIT, 26 (1986) 123–126.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    J. M. Borwein and P. B. Borwein, Pi and the AGM—A Study in Analytic Number Theory and Computational Complexity, Wiley, N.Y., 1987.Google Scholar
  12. 12.
    R. P. Brent, Fast multiple-precision evaluation of elementary functions, J. ACM, 23 (1976) 242–251.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs. N.J., 1974.MATHGoogle Scholar
  14. 14.
    A. Cayley, An Elementary Treatise on Elliptic Functions, Bell and Sons, 1885; reprint Dover, 1961.Google Scholar
  15. 15.
    A. Cayley, A memoir on the transformation of elliptic functions, Phil. Trans. T., 164 (1874) 397–456.CrossRefGoogle Scholar
  16. 16.
    D. V. Chudnovsky and G. V. Chudnovsky, Padé and Rational Approximation to Systems of Functions and Their Arithmetic Applications, Lecture Notes in Mathematics 1052, Springer, Berlin, 1984.Google Scholar
  17. 17.
    H. R. P. Ferguson and R. W. Forcade, Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two, Bull. AMS, 1 (1979) 912–914.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    C. F. Gauss, Werke, Göttingen 1866–1933, Bd 3. pp. 361–403.Google Scholar
  19. 19.
    G. H. Hardy, Ramanujan, Cambridge Univ. Press, London, 1940.Google Scholar
  20. 20.
    L. V. King, On The Direct Numerical Calculation of Elliptic Functions and Integrals, Cambridge Univ. Press, 1924.Google Scholar
  21. 21.
    F. Klein, Development of Mathematics in the 19th Century, 1928, Trans Math Sci. Press, R. Hermann ed., Brookline. MA, 1979.Google Scholar
  22. 22.
    D. Knuth, The Art of Computer Programming. vol. 2: Seminumerical Algorithms, Addison-Wesley, Reading, MA, 1981.MATHGoogle Scholar
  23. 23.
    F. Lindemann, Über die Zahl π, Math. Ann.. 20 (1882) 213–225.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    G. Miel, On calculations past and present: the Archimedean algorithm. Amer. Math. Monthly. 90 (1983) 17–35.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    D. J. Newman, Rational Approximation Versus Fast Computer Methods, in Lectures on Approximation and Value Distribution, Presses de l’Université de Montreal, 1982, pp. 149–174.Google Scholar
  26. 26.
    S. Ramanujan, Modular equations and approximations to π, Quart. J. Math. 45 (1914) 350–72.Google Scholar
  27. 27.
    E. Salamin, Computation of it using arithmetic-geometric mean, Math. Comput., 30 (1976) 565–570.MathSciNetMATHGoogle Scholar
  28. 28.
    B. Schoenberg, Elliptic Modular Functions. Springer, Berlin. 1976.Google Scholar
  29. 29.
    A. Schönhage and V. Strassen, Schnelle Multiplikation Grosser Zahlen, Computing, 7 (1971) 281–292.MATHCrossRefGoogle Scholar
  30. 30.
    D. Shanks, Dihedral quartic approximations and series for π, J. Number Theory. 14 (1982) 397–423.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    D. Shanks and J. W. Wrench, Calculation of it to 100,000 decimals, Math Comput., 16 (1962) 76–79.MathSciNetMATHGoogle Scholar
  32. 32.
    W. Shanks, Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals, G. Bell, London, 1853.Google Scholar
  33. 33.
    Y. Tamura and Y. Kanada. Calculation of it to 4.196.393 decimals based on Gauss-Legendre algorithm, preprint (1983).Google Scholar
  34. 34.
    J. Tannery and J. Molk, Fonctions Elliptiques, vols. 1 and 2, 1893; reprint Chelsea, New York, 1972.Google Scholar
  35. 35.
    S. Wagon, Is TT normal?, The Math Intelligencer, 7 (1985) 65–67.MathSciNetGoogle Scholar
  36. 36.
    G. N. Watson, Some singular moduli (1), Quart. J. Math., 3 (1932) 81–98.CrossRefGoogle Scholar
  37. 37.
    The final problem: an account of the mock theta functions, J. London Math. Soc.,11 (1936) 55–80.Google Scholar
  38. 38.
    H. Weber, Lehrbuch der Algebra, Vol. 3, 1908; reprint Chelsea. New York, 1980.Google Scholar
  39. 39.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed, Cambridge Univ. Press, London, 1927.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • J. M. Borwein
    • 1
  • P. B. Borwein
    • 1
  • D. H. Bailey
    • 2
  1. 1.Mathematics DepartmentDalhousie UniversityHalifaxCanada
  2. 2.NASA Ames Research CenterMoffett FieldUSA

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