Skip to main content

A Few Theta-Function Identities and Some of Ramanujan's Modular Equations


We prove three modular equations of Ramanujan using theta-function identities. Proofs via methods known to Ramanujan were not available hitherto. One had previously been proved by classical methods, and two had been proved using the theory of modular forms.

This is a preview of subscription content, access via your institution.


  1. B.C. Berndt, “Introduction to Ramanujan's Modular Equations,” in Proceedings of the Ramanujan Centennial International Conference, Annamalainagar, 15–18 December 1987. The Ramanujan Mathematical Society, Madras, 1988, pp. 15–20.

    Google Scholar 

  2. B.C. Berndt, “Ramanujan's modular equations,” in Ramanujan Revisited, Academic Press, Boston, 1988, pp. 313–333.

    Google Scholar 

  3. B.C. Berndt, Ramanujan's Notebooks, Part II, Springer Verlag, New York, 1989.

    Google Scholar 

  4. B.C. Berndt, Ramanujan's Notebooks, Part III, Springer Verlag, New York, 1991.

    Google Scholar 

  5. B.C. Berndt, Ramanujan's Notebooks, Part IV, Springer Verlag, New York, 1994.

  6. B.C. Berndt, “Ramanujan's theory of theta-functions,” in Theta-Functions, From, the Classical to the Modern (M. Ram Murty, ed.), American Mathematical Society, Providence, RI, 1993, pp. 1–63. Centre de Recherches Mathe0matiques, Proceedings and Lecture Notes, Vol. 1.

    Google Scholar 

  7. B.C. Berndt, Ramanujan's Notebooks, Part V, Springer Verlag, New York, 1997.

    Google Scholar 

  8. B.C. Berndt and R.A. Rankin, Ramanujan: Letters and Commentary, Affiliated East-West Press Pvt. Ltd., New Delhi, 1997.

    Google Scholar 

  9. R.J. Evans, “Theta-functions identities,” J. Math. Anal. Appl. 147 (1990) 97–121.

    Google Scholar 

  10. K.G. Ramanathan, “Ramanujan's modular equations,” Acta. Arith. 53 (1990) 403–420.

    Google Scholar 

  11. S. Ramanujan, Notebooks (2 vols.), Tata Institute of Fundamental Research, Bombay, 1957.

    Google Scholar 

  12. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

    Google Scholar 

  13. L.C. Shen, “On some modular equations of degree 5,” Proc. Amer. Math. Soc. 123 (1995) 1521–1526.

    Google Scholar 

  14. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Universal Book Stall, New Delhi, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baruah, N.D. A Few Theta-Function Identities and Some of Ramanujan's Modular Equations. The Ramanujan Journal 4, 239–250 (2000).

Download citation

  • Issue Date:

  • DOI:

  • theta-function
  • elliptic integral
  • modulus
  • modular equation
  • multiplier