Skip to main content

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

  • Chapter
Pi: A Source Book

Abstract

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.

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

Access this chapter

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. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964.

    MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  3. D. H. Bailey, Numerical results on the transcendence of constants involving π. e. and Euler s constant. Math. Comput., 50 (1988) 275–81.

    MATH  Google Scholar 

  4. A. Baker, Transcendental Number Theory Cambridge Univ. Press, London, 1975.

    Book  Google Scholar 

  5. P. Beckmann, A History of Pi, 4th ed., Golem Press, Boulder, CO. 1977.

    Google Scholar 

  6. R. Bellman, A Brief Introduction to Theta Functions, Holt, Reinhart and Winston, New York. 1961.

    Google Scholar 

  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. A. Borodin and I. Munro, The Computational Complexity of Algebraic and Numeric Problems, American Elsevier, New York, 1975.

    MATH  Google Scholar 

  9. J. M. Borwein and P. B. Borwein, The arithmetic-geometric mean and fast computation of elementary functions, SIAM Rev., 26 (1984), 351–365.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. M. Borwein and P. B. Borwein, An explicit cubic iteration for pi, BIT, 26 (1986) 123–126.

    Article  MathSciNet  MATH  Google Scholar 

  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. R. P. Brent, Fast multiple-precision evaluation of elementary functions, J. ACM, 23 (1976) 242–251.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs. N.J., 1974.

    MATH  Google Scholar 

  14. A. Cayley, An Elementary Treatise on Elliptic Functions, Bell and Sons, 1885; reprint Dover, 1961.

    Google Scholar 

  15. A. Cayley, A memoir on the transformation of elliptic functions, Phil. Trans. T., 164 (1874) 397–456.

    Article  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  18. C. F. Gauss, Werke, Göttingen 1866–1933, Bd 3. pp. 361–403.

    Google Scholar 

  19. G. H. Hardy, Ramanujan, Cambridge Univ. Press, London, 1940.

    Google Scholar 

  20. L. V. King, On The Direct Numerical Calculation of Elliptic Functions and Integrals, Cambridge Univ. Press, 1924.

    Google Scholar 

  21. F. Klein, Development of Mathematics in the 19th Century, 1928, Trans Math Sci. Press, R. Hermann ed., Brookline. MA, 1979.

    Google Scholar 

  22. D. Knuth, The Art of Computer Programming. vol. 2: Seminumerical Algorithms, Addison-Wesley, Reading, MA, 1981.

    MATH  Google Scholar 

  23. F. Lindemann, Über die Zahl π, Math. Ann.. 20 (1882) 213–225.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Miel, On calculations past and present: the Archimedean algorithm. Amer. Math. Monthly. 90 (1983) 17–35.

    Article  MathSciNet  MATH  Google Scholar 

  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. S. Ramanujan, Modular equations and approximations to π, Quart. J. Math. 45 (1914) 350–72.

    Google Scholar 

  27. E. Salamin, Computation of it using arithmetic-geometric mean, Math. Comput., 30 (1976) 565–570.

    MathSciNet  MATH  Google Scholar 

  28. B. Schoenberg, Elliptic Modular Functions. Springer, Berlin. 1976.

    Google Scholar 

  29. A. Schönhage and V. Strassen, Schnelle Multiplikation Grosser Zahlen, Computing, 7 (1971) 281–292.

    Article  MATH  Google Scholar 

  30. D. Shanks, Dihedral quartic approximations and series for π, J. Number Theory. 14 (1982) 397–423.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Shanks and J. W. Wrench, Calculation of it to 100,000 decimals, Math Comput., 16 (1962) 76–79.

    MathSciNet  MATH  Google Scholar 

  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. Y. Tamura and Y. Kanada. Calculation of it to 4.196.393 decimals based on Gauss-Legendre algorithm, preprint (1983).

    Google Scholar 

  34. J. Tannery and J. Molk, Fonctions Elliptiques, vols. 1 and 2, 1893; reprint Chelsea, New York, 1972.

    Google Scholar 

  35. S. Wagon, Is TT normal?, The Math Intelligencer, 7 (1985) 65–67.

    MathSciNet  Google Scholar 

  36. G. N. Watson, Some singular moduli (1), Quart. J. Math., 3 (1932) 81–98.

    Article  Google Scholar 

  37. The final problem: an account of the mock theta functions, J. London Math. Soc.,11 (1936) 55–80.

    Google Scholar 

  38. H. Weber, Lehrbuch der Algebra, Vol. 3, 1908; reprint Chelsea. New York, 1980.

    Google Scholar 

  39. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed, Cambridge Univ. Press, London, 1927.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media New York

About this chapter

Cite this chapter

Borwein, J.M., Borwein, P.B., Bailey, D.H. (1997). Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi. In: Pi: A Source Book. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2736-4_64

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-2736-4_64

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4757-2738-8

  • Online ISBN: 978-1-4757-2736-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics