Theorems Featuring the Gamma Function

  • George E. Andrews
  • Bruce C. Berndt


Some integrals of gamma functions are evaluated. A remarkable approximation to the gamma function, with a slightly less precise approximation submitted as a problem by Ramanujan to the Journal of the Indian Mathematical Society, is examined in detail.


Asymptotic Expansion Gamma Function Asymptotic Formula Bernoulli Number Lost Notebook 
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.


  1. 1.
    M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965.Google Scholar
  2. 4.
    H. Alzer, On Ramanujan’s double inequality for the gamma function, Bull. London Math. Soc. 35 (2003), 601–607.CrossRefMathSciNetMATHGoogle Scholar
  3. 5.
    H. Alzer, Sharp upper and lower bounds for the gamma function, Proc. Royal Soc. Edinburgh 139A (2009), 709–718.CrossRefMathSciNetGoogle Scholar
  4. 6.
    H. Alzer, Email to B.C. Berndt, December 19, 2009.Google Scholar
  5. 9.
    G.D. Anderson, M.K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, Wiley, New York, 1997.MATHGoogle Scholar
  6. 37.
    B.C. Berndt, Ramanujan’s Notebooks, Part I, Springer-Verlag, New York, 1985.CrossRefMATHGoogle Scholar
  7. 38.
    B.C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.CrossRefMATHGoogle Scholar
  8. 43.
    B.C. Berndt, Some integrals in Ramanujan’s lost notebook, Proc. Amer. Math. Soc. 132 (2004), 2983–2988.CrossRefMathSciNetMATHGoogle Scholar
  9. 49.
    B.C. Berndt, Y.-S. Choi and S.-Y. Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, in Continued Fractions: From Analytic Number Theory to Constructive Approximation, B.C. Berndt and F. Gesztesy, eds., Contem. Math. 236, American Mathematical Society, Providence, RI, 1999, pp. 15–56; reprinted and updated in [65, pp. 215–258].Google Scholar
  10. 65.
    B.C. Berndt and R.A. Rankin, Ramanujan: Essays and Surveys, American Mathematical Society, Providence, RI, 2001; London Mathematical Society, London, 2001.Google Scholar
  11. 72.
    R.P. Boas and H. Pollard, Continuous analogues of series, Amer. Math. Monthly 80 (1973), 18–25.CrossRefMathSciNetMATHGoogle Scholar
  12. 120.
    P.J. Forrester, Extensions of several summation formulae of Ramanujan using the calculus of residues, Rocky Mt. J. Math. 13 (1983), 557–572.CrossRefMathSciNetMATHGoogle Scholar
  13. 126.
    I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, San Diego, 1994.Google Scholar
  14. 161.
    M.D. Hirschhorn, A new version of Stirling’s formula, Math. Gaz. 90 (2006), 286–292.Google Scholar
  15. 176.
    A.A. Karatsuba and S.M. Voronin, The Riemann Zeta-Function, DeGruyter, Berlin, 1992.CrossRefMATHGoogle Scholar
  16. 177.
    E.A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comp. Appl. Math. 135 (2001), 225–240.CrossRefMathSciNetMATHGoogle Scholar
  17. 202.
    K.S. Krishnan, On the equivalence of certain infinite series and the corresponding integrals, J. Indian Math. Soc. 12 (1948), 79–88.MathSciNetMATHGoogle Scholar
  18. 227.
    C. Mortici, Ramanujan’s estimate for the gamma function via monotonicity arguments, Ramanujan J. 25 (2011), 149–154.CrossRefMathSciNetMATHGoogle Scholar
  19. 228.
    C. Mortici, Improved asymptotic formulas for the gamma function, Comput. Math. Appl. 61 (2011), 3364–3369.CrossRefMathSciNetMATHGoogle Scholar
  20. 229.
    C. Mortici, On Ramanujan’s large argument formula for the gamma function, Ramanujan J. 26 (2011), 185–192.CrossRefMathSciNetMATHGoogle Scholar
  21. 231.
    G. Nemes, New asymptotic expansion for the gamma function, Arch. Math. 95 (2010), 161–169.CrossRefMathSciNetMATHGoogle Scholar
  22. 238.
    F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.Google Scholar
  23. 240.
    S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), 278–301.CrossRefMathSciNetMATHGoogle Scholar
  24. 255.
    S. Ramanujan, Some definite integrals, Mess. Math. 44 (1915), 10–18.Google Scholar
  25. 260.
    S. Ramanujan, Question 754, J. Indian Math. Soc. 8 (1916), 80; solutions in 12 (1920), 101; 13 (1921), 151.Google Scholar
  26. 266.
    S. Ramanujan, A class of definite integrals, Quart. J. Math. 48 (1920), 294–310.Google Scholar
  27. 267.
    S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence, RI, 2000.Google Scholar
  28. 268.
    S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957; second ed, 2012.Google Scholar
  29. 269.
    S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • George E. Andrews
    • 1
  • Bruce C. Berndt
    • 2
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations