Skip to main content
Log in

Ramanujan’s Master Theorem

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

S. Ramanujan introduced a technique, known as Ramanujan’s Master Theorem, which provides an explicit expression for the Mellin transform of a function in terms of the analytic continuation of its Taylor coefficients. The history and proof of this result are reviewed, and a variety of applications is presented. Finally, a multi-dimensional extension of Ramanujan’s Master Theorem is discussed.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amdeberhan, T., Moll, V.: A formula for a quartic integral: a survey of old proofs and some new ones. Ramanujan J. 18, 91–102 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berndt, B.: Ramanujan’s Notebooks, Part I. Springer, New York (1985)

    Book  MATH  Google Scholar 

  3. Boros, G., Moll, V.: The double square root, Jacobi polynomials and Ramanujan’s master theorem. J. Comput. Appl. Math. 130, 337–344 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boros, G., Espinosa, O., Moll, V.: On some families of integrals solvable in terms of polygamma and negapolygamma functions. Integral Transforms Spec. Funct. 14, 187–203 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borwein, J.M., Nuyens, D., Straub, A., Wan, J.: Random walk integrals. Ramanujan J. (2011). doi:10.1007/s11139-011-9325-y

    MathSciNet  Google Scholar 

  6. Borwein, J.M., Straub, A., Wan, J.: Three-step and four-step random walk integrals (2010, submitted)

  7. Broadhurst, D.: Bessel moments, random walks and Calabi-Yau equations. Preprint (2009)

  8. Edwards, J.: A Treatise on the Integral Calculus, vol. 2. MacMillan, New York (1922)

    Google Scholar 

  9. Glaisher, J.W.L.: Letter to the editors: On a new formula in definite integrals. Philos. Mag. 48(319), 400 (1874)

    Google Scholar 

  10. Glaisher, J.W.L.: A new formula in definite integrals. Philos. Mag. 48(315), 53–55 (1874)

    MATH  Google Scholar 

  11. Gonzalez, I., Moll, V.: Definite integrals by the method of brackets. Part 1. Adv. Appl. Math. (2009). doi:10.1016/j.aam.2009.11.003

    Google Scholar 

  12. Gonzalez, I., Schmidt, I.: Optimized negative dimensional integration method (NDIM) and multiloop Feynman diagram calculation. Nucl. Phys. B 769, 124–173 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, New York (2007). Edited by A. Jeffrey and D. Zwillinger

    MATH  Google Scholar 

  14. Hardy, G.H.: Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work, 3rd edn. Chelsea, New York (1978)

    Google Scholar 

  15. O’Kinealy, J.: On a new formula in definite integrals. Philos. Mag. 48(318), 295–296 (1874)

    Google Scholar 

  16. Pearson, K.: The random walk. Nature 72, 294 (1905)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Armin Straub.

Additional information

O. Espinosa is deceased.

The fifth author wishes to thank the partial support of NSF-DMS 0070757. The work of the last author was partially supported, as a graduate student, by the same grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Amdeberhan, T., Espinosa, O., Gonzalez, I. et al. Ramanujan’s Master Theorem. Ramanujan J 29, 103–120 (2012). https://doi.org/10.1007/s11139-011-9333-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-011-9333-y

Keywords

Mathematics Subject Classification

Navigation