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Application of the Hurwitz Zeta Function to the Evaluation of Certain Integrals

Published online by Cambridge University Press:  20 November 2018

Zhang Nan Yue  and
Kenneth S. Williams
Zhang Nan Yue
Affiliation:
Information Department The People's University of China Beijing 100872 People 5 Republic of China
Kenneth S. Williams
Affiliation:
Department of Mathematics and Statistics Carleton University Ottawa, Ontario KIS 5B6
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Abstract

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The Hurwitz zeta function ζ(s, a) is defined by the series

for 0 < a ≤ 1 and σ = Re(s) > 1, and can be continued analytically to the whole complex plane except for a simple pole at s = 1 with residue 1. The integral functions C(s, a) and S(s, a) are defined in terms of the Hurwitz zeta function as follows:

Using integral representations of C(s, a) and S(s, a), we evaluate explicitly a class of improper integrals. For example if 0 < a < 1 we show that

Keywords

11M35Hurwitz zeta functionintegral representationevaluation of improper integralsrecurrence relations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 3 , 01 September 1993 , pp. 373 - 384
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Actor, A., £- and n-function resummation of infinite series: general case, J. Phys. A: Math. Gen. 24(1991), 3741–3759.Google Scholar
2. Apostol, T. M., Remark on the Hurwitz zeta junction, Proc. Amer. Math. Soc. 2(1951), 690–693.Google Scholar
3. Apostol, T. M., Introduction to Analytic Number Tlieory, Springer-Verlag, New York, Heidelberg, Berlin, 1976.Google Scholar
4. Bendt, B. C., On the Hurwitz zeta-function, Rocky Mountain J. Math. 2(1972), 151–157.Google Scholar
5. Elizalde, E., An asymptotic expansion for the first derivative of the generalized zeta function, Math. Comp. 47(1986), 347–350.Google Scholar
6. Elizalde, E. and Romeo, A., An integral involving the generalized zeta function, Internat. J. Math. & Math. Sci. 13(1990), 453–460.Google Scholar
7. Elizalde, E., A simple recurrence for the higher derivatives of the Hurwitz zeta function, Penn. State University, (1991), preprint.Google Scholar
8. Erdélyi, A., Higher Transcendental Functions, Vol. 1, McGraw-Hill Book Company, Inc., 1953.Google Scholar
9. Fine, N. J., Note on the Hurwitz zeta function, Proc. Amer. Math. Soc. 2(1951), 361–364.Google Scholar
10. Hansen, E. R. and Patrick, M. L., Some relations and values of the generalized zeta function, Math. Comp. 16(1962), 265–274.Google Scholar
11. Kummer, E. E., Beitrag zur Théorie der Function F(x) — j£° e∼vvx∼ldv, J. Reine Angew. Math. 35(1847), 1–4.Google Scholar
12. Lang, S., Complex Analysis (Second Edition), Springer-Verlag, New York, 1987.Google Scholar
13. Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, Academic Press, New York, 1980.Google Scholar
14. Mikolâs, M., Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz, Publ. Math. Debrecen 5(1957), 44–53.Google Scholar
15. Powell, E. O., A table of the generalized Riemann zeta Junction in a particular case, Quart. J. Mech. Appl. Math. 5(1952), 116–123.Google Scholar
16. Stoica, G., A recurrence formula in the study of the Riemann zeta function, Stud. Cere. Mat. (3) 39(1987), 261–264.Google Scholar
17. Titchmarsh, E. C., The Theory of the Riemann Zeta-Function (Second Edition), Clarendon Press, Oxford, 1986.Google Scholar
18. Vardi, I., Integrals, an introduction to analytic number theory, Amer. Math. Monthly 95(1988), 308–315.Google Scholar
19. Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis (Fourth Edition), Cambridge at the University Press, 1963.Google Scholar