The Ramanujan Journal

, Volume 6, Issue 2, pp 159–188

On Some Integrals Involving the Hurwitz Zeta Function: Part 1

  • Olivier Espinosa
  • Victor H. Moll

Abstract

We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, ln Γ(q) and ln sin(q).

Hurwitz zeta function polylogarithms loggamma integrals 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Adamchik, “32 representations for Catalan's constant,” Available at http://www.members.wri.com/victor/articles/catalan/catalan.html.Google Scholar
  2. 2.
    V. Adamchik, “Zeta series and polygammas of the negative order,” ISAAC'97, June 3-7, 1997.Google Scholar
  3. 3.
    V. Adamchik, “Polygamma functions of negative order,” Journal of Comp. and Applied Math. 100 (1998), 191-198.Google Scholar
  4. 4.
    J. Andersson, “Mean value properties of the Hurwitz zeta-function,” Math. Scand. 7 (1992), 295-300.Google Scholar
  5. 5.
    T. Apostol, “Introduction to Analytic Number Theory,” Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976.Google Scholar
  6. 6.
    A.K. Arora, S. Goel, and D. Rodriguez, “Special integration techniques for trigonometric integrals,” Amer. Math. Monthly 95 (1988), 126-130.Google Scholar
  7. 7.
    H. Bateman, Higher Transcendental Functions, vol. I. Compiled by the Staff of the Bateman Manuscript Project. McGraw-Hill, New York, 1953.Google Scholar
  8. 8.
    B. Berndt, “On the Hurwitz zeta-function,” Rocky Mountain Journal 2 (1972), 151-157.Google Scholar
  9. 9.
    B. Berndt, Ramanujan's Notebooks, Part I, Springer Verlag, New York, 1985.Google Scholar
  10. 10.
    O. Espinosa and V. Moll, “On some integrals involving the Hurwitz zeta function, Part 2,” Ramanujan J., to appear.Google Scholar
  11. 11.
    J. Borwein and P. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley and Sons, New York, 1987.Google Scholar
  12. 12.
    D. Bradley, Representations of Catalan's constant. Reference 14 in http://germain.umemat.maine.edu/faculty/bradley/papers/pub.html.Google Scholar
  13. 13.
    C. Dib and O. Espinosa, “The magnetized electron gas in terms of Hurwitz zeta functions,” Nucl. Phys. B 612 [FS] (2001), 492-518.Google Scholar
  14. 14.
    E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, Lecture Notes in Physics, Springer-Verlag, Berlin, 1995.Google Scholar
  15. 15.
    R.Wm. Gosper, Jr. “∫n/4m/6 ln Г (z) dz,” in Special Functions, q-Series and Related Topics (M. Ismail, D. Masson, and M. Rahman, eds.), The Fields Institute Communications, AMS, 1997, pp. 71-76.Google Scholar
  16. 16.
    I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 5th edn. (Alan Jeffrey, ed.), Academic Press, New York, 1994.Google Scholar
  17. 17.
    S. Kogan, “A note on definite integrals involving trigonometric functions,” Available in http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.Google Scholar
  18. 18.
    K.S. Kolbig, “On the integral ∫0 eµt t v−1 logm tdt,” Math. Comp. 41 (1983), 171-182.Google Scholar
  19. 19.
    K.S. Kolbig, “On three trigonometric integrals of ln Г(x) or its derivative,” CERN/Computing and Networks Division; CN/94/7, May 1994.Google Scholar
  20. 20.
    L. Lewin, Polylogarithms and Associated Functions, North Holland, New York, 1981.Google Scholar
  21. 21.
    L. Lewin, (ed.), Structural Properties of Polylogarithms, Mathematical Surveys and Monographs, American Mathematical Society, Vol. 37, 1991.Google Scholar
  22. 22.
    H. Mckean and V. Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, Cambridge, 1997.Google Scholar
  23. 23.
    M. Mikolas, “Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz,” Publ. Math. Debrecen 5 (1957), 44-53.Google Scholar
  24. 24.
    J. Miller and V. Adamchik, “Derivatives of the Hurwitz zeta function for rational arguments,” Journal of Comp. and Applied Math. 100 (1999), 201-206.Google Scholar
  25. 25.
    A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Translated from the Russian by G.G. Gould), Gordon and Breach Science Publishers, New York, 1990.Google Scholar
  26. 26.
    J.P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics, Vol. 7, Springer Verlag, New York, 1973.Google Scholar
  27. 27.
    J. Spanier and K.B. Oldham, An Atlas of Functions, Hemisphere Publishing Corp., 1987.Google Scholar
  28. 28.
    I. Vardi, “Determinants of Laplacians and multiple gamma functions,” SIAM Jour. Math. Anal. 19 (1988), 493-507.Google Scholar
  29. 29.
    I. Vardi, “Integrals, an introduction to analytic number theory,” Amer. Math. Monthly 95 (1988), 308-315.Google Scholar
  30. 30.
    E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, 1999.Google Scholar
  31. 31.
    E. Whittaker and G. Watson, A Course of Modern Analysis, Cambridge University Press, 4th edn. reprinted, 1963.Google Scholar
  32. 32.
    K.S. Williams and Z.Y. Yue, “Special values of the Lerch zeta function and the evaluation of certain integrals,” Proc. Amer. Math. Soc. 119 (1993), 35-49.Google Scholar
  33. 33.
    Z.Y. Yue and K.S. Williams, “Application of the Hurwitz zeta function to the evaluation of certain integrals,” Canad. Math. Bull. 36 (1993), 373-384.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Olivier Espinosa
    • 1
  • Victor H. Moll
    • 2
  1. 1.Departamento de FísicaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.Department of MathematicsTulane UniversityNew Orleans

Personalised recommendations