A Partial Manuscript on Fourier and Laplace Transforms

  • George E. Andrews
  • Bruce C. Berndt


Most of the results in the partial manuscript on integral transforms discussed in this chapter are classical. However, the partial manuscript contains one of the highlights of the book, a beautiful new transformation formula involving the logarithmic derivative of the gamma function. An extremely clever device used to prove this transformation formula harkens back to Ramanujan’s paper, New expressions for Riemann’s functions \(\xi(s)\,and\,\Xi(s) \)


Zeta Function Summation Formula Riemann Zeta Function Transformation Formula Quarterly Report 
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. 35.
    B.C. Berndt, The quarterly reports of S. Ramanujan, Amer. Math. Monthly 90 (1983), 505–516.CrossRefMathSciNetzbMATHGoogle Scholar
  3. 37.
    B.C. Berndt, Ramanujan’s Notebooks, Part I, Springer-Verlag, New York, 1985.CrossRefzbMATHGoogle Scholar
  4. 51.
    B.C. Berndt and A. Dixit, A transformation formula involving the gamma and Riemann zeta functions in Ramanujan’s lost notebook, in The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, K. Alladi, J. Klauder, and C.R. Rao, eds., Springer, New York, 2010, pp. 199–210.CrossRefGoogle Scholar
  5. 64.
    B.C. Berndt and R.A. Rankin, Ramanujan: Letters and Commentary, American Mathematical Society, Providence, RI, 1995; London Mathematical Society, London, 1995.Google Scholar
  6. 86.
    L. Carlitz, Some finite analogues of the Poisson summation formula, Proc. Edinburgh Math. Soc. (2) 12 (1961), 133–138.Google Scholar
  7. 99.
    J.B. Conway, Functions of One Complex Variable, 2nd ed., Springer-Verlag, New York, 1978.CrossRefGoogle Scholar
  8. 107.
    A. Dixit, Series transformations and integrals involving the Riemann Ξ-function, J. Math. Anal. Appl. 368 (2010), 358–373.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 108.
    A. Dixit, Analogues of a transformation formula of Ramanujan, Internat. J. Number Thy. 7 (2011), 1151–1172.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 109.
    A. Dixit, Transformation formulas associated with integrals involving the Riemann Ξ-function, Monatsh. für Math. 164 (2011), 133–156.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 111.
    A. Dixit, Analogues of the general theta transformation formula, Proc. Royal Soc. Edinburgh, Sect. A 143 (2013), 371–399.Google Scholar
  12. 126.
    I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, San Diego, 1994.Google Scholar
  13. 132.
    A.P. Guinand, On Poisson’s summation formula, Ann. Math. (2) 42 (1941), 591–603.Google Scholar
  14. 133.
    A.P. Guinand, Some formulae for the Riemann zeta-function, J. London Math. Soc. 22 (1947), 14–18.CrossRefMathSciNetzbMATHGoogle Scholar
  15. 135.
    A.P. Guinand, A note on the logarithmic derivative of the gamma function, Edinburgh Math. Notes 38 (1952), 1–4.CrossRefMathSciNetzbMATHGoogle Scholar
  16. 137.
    A.P. Guinand, Some finite identities connected with Poisson’s summation formula, Proc. Edinburgh Math. Soc. (2) 12 (1960), 17–25.Google Scholar
  17. 143.
    G.H. Hardy, Note by G.H. Hardy on the preceding paper, Quart. J. Math. (Oxford) 46 (1915), 260–261.Google Scholar
  18. 190.
    N.S. Koshliakov, On a general summation formula and its applications (in Russian), Comp. Rend. (Doklady) Acad. Sci. URSS 4 (1934), 187–191.Google Scholar
  19. 193.
    N.S. Koshliakov, On a transformation of definite integrals and its application to the theory of Riemann’s function ζ(s), Comp. Rend. (Doklady) Acad. Sci. URSS 15 (1937), 3–8.Google Scholar
  20. 194.
    N.S. Koshliakov (under the name N.S. Sergeev), A study of a class of transcendental functions defined by the generalized Riemann equation (in Russian), Trudy Mat. Inst. Steklov, Moscow, 1949.Google Scholar
  21. 195.
    N.S. Koshliakov, Investigation of some questions of the analytic theory of a rational and quadratic field, II (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 213–260.Google Scholar
  22. 196.
    N.S. Koshliakov, Investigation of some questions of the analytic theory of a rational and quadratic field, III (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 307–326.Google Scholar
  23. 210.
    Y. Lee, Email to B.C. Berndt, August 1, 2008.Google Scholar
  24. 237.
    O. Oloa, On a series of Ramanujan, in Gems in Experimental Mathematics, T. Amdeberhan, L.A. Medina, and V.H. Moll, eds., Contemp. Math. 517, American Mathematical Society, Providence, RI, 2010, pp. 305–311.Google Scholar
  25. 255.
    S. Ramanujan, Some definite integrals, Mess. Math. 44 (1915), 10–18.Google Scholar
  26. 257.
    S. Ramanujan, New expressions for Riemann’s functions ξ(s) and Ξ(s), Quart. J. Math. 46 (1915), 253–260.zbMATHGoogle 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.zbMATHGoogle Scholar
  30. 299.
    H. Tanaka, Multiple gamma functions, multiple sine functions, and Appell’s O-functions, Ramanujan J. 24 (2011), 33–60.CrossRefMathSciNetzbMATHGoogle Scholar
  31. 305.
    E.C. Titchmarsh, Theory of Fourier Integrals, Clarendon Press, Oxford, 1937; 3rd ed., Chelsea, New York, 1986.Google Scholar
  32. 315.
    E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.Google 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