On some integrals involving the Hurwitz zeta function: Part 1

Ramanujan Journal

Volumn 6, Issue 2, 2002, Pages 159-188

  Espinosa, Olivier ^a   Moll, Victor H ^b  

^a DEPARTAMENTO DE FÍSICA   (Chile)

^b Tulane University   (United States)

Author keywords

Hurwitz zeta function; Integrals; Loggamma; Polylogarithms

Indexed keywords

EID: 0036092871     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1015706300169     Document Type: Article

References (33)

1. V. Adamchik, "32 representations for Catalan's constant," Available at http://www.members.wri.com/victor/articles/catalan/catalan.html.
2. V. Adamchik, "Zeta series and polygammas of the negative order," ISAAC'97, June 3-7, 1997.
3. V. Adamchik, "Polygamma functions of negative order," Journal of Comp. and Applied Math. 100 (1998), 191-198.
4. J. Andersson, "Mean value properties of the Hurwitz zeta-function," Math. Scand. 7 (1992), 295-300.
5. T. Apostol, "Introduction to Analytic Number Theory," Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976.
6. A.K. Arora, S. Goel, and D. Rodriguez, "Special integration techniques for trigonometric integrals," Amer. Math. Monthly 95 (1988), 126-130.
7. H. Bateman, Higher Transcendental Functions, vol. I. Compiled by the Staff of the Bateman Manuscript Project. McGraw-Hill, New York, 1953.
8. B. Berndt, "On the Hurwitz zeta-function," Rocky Mountain Journal 2 (1972), 151-157.
9. B. Berndt, Ramanujan's Notebooks, Part I, Springer Verlag, New York, 1985.
10. O. Espinosa and V. Moll, "On some integrals involving the Hurwitz zeta function, Part 2," Ramanujan J., to appear.
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.
12. D. Bradley, Representations of Catalan's constant. Reference 14 in http://germain.umemat.maine.edu/faculty/bradley/papers/pub.html.
13. C. Dib and O. Espinosa, "The magnetized electron gas in terms of Hurwitz zeta functions," Nucl. Phys. B 612 [FS] (2001), 492-518.
14. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, Lecture Notes in Physics, Springer-Verlag, Berlin, 1995.
15. m/6 ln Γ(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.
16. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 5th edn. (Alan Jeffrey, ed.), Academic Press, New York, 1994.
17. S. Kogan, "A note on definite integrals involving trigonometric functions," Available in http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.
18. m tdt," Math. Comp. 41 (1983), 171-182.
19. K.S. Kolbig, "On three trigonometric integrals of 1n Γ (x) or its derivative," CERN/Computing and Networks Division; CN/94/7, May 1994.
20. L. Lewin, Polylogarithms and Associated Functions, North Holland, New York, 1981.
21. L. Lewin, (ed.), Structural Properties of Polylogarithms, Mathematical Surveys and Monographs, American Mathematical Society, Vol. 37, 1991.
22. H. Mckean and V. Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, Cambridge, 1997.
23. M. Mikolas, "Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz," Publ. Math. Debrecen 5 (1957), 44-53.
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.
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.
26. J.P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics, Vol. 7, Springer Verlag, New York, 1973.
27. J. Spanier and K.B. Oldham, An Atlas of Functions, Hemisphere Publishing Corp., 1987.
28. I. Vardi, "Determinants of Laplacians and multiple gamma functions," SIAM Jour. Math. Anal. 19 (1988), 493-507.
29. I. Vardi, "Integrals, an introduction to analytic number theory," Amer. Math. Monthly 95 (1988), 308-315.
30. E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, 1999.
31. E. Whittaker and G. Watson, A Course of Modern Analysis, Cambridge University Press, 4th edn. reprinted, 1963.
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.
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.