Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

FURTHER LOG-SINE AND LOG-COSINE INTEGRALS

  • Received : 2013.07.24
  • Accepted : 2013.10.11
  • Published : 2013.11.15
Download PDF
⟨ Previous Next ⟩

Abstract

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine and log-cosine integrals have been evaluated, in the existing literature on the subject, in many different ways. Very recently, Choi [6] presented explicit evaluations of some families of log-sine and log-cosine integrals by making use of the familiar Beta function. In the present sequel to the investigation [6], we evaluate the log-sine and log-cosine integrals involved in more complicated integrands than those in [6], by also using the Beta function.

Keywords

References

  1. V. S. Adamchik and H. M. Srivastava, Some series of the Zeta and related functions, Analysis 18 (1998), 131-144.
  2. N. Batir, Integral representations of some series involving $(^{2k}_{k})^{-1}$ $k^{-n}$ and some related series, Appl. Math. Comput. 147 (2004), 645-667. https://doi.org/10.1016/S0096-3003(02)00802-0
  3. M. G. Beumer, Some special integrals, Amer. Math. Monthly 68 (1961), 645-647. https://doi.org/10.2307/2311513
  4. J. Choi, Certain summation formulas involving harmonic numbers and gen-eralized harmonic numbers, Appl. Math. Comput. 218 (2011), 734-740; doi: 10.1016/j.amc.2011.01.062.
  5. J. Choi, Finite summation formulas involving binomial coeffcients, harmonic numbers and generalized harmonic numbers, J. Inequ. Appl. 2013, 2013:49. http://www.journalofinequalitiesandapplications.com/content/2013/1/49
  6. J. Choi, Log-sine and log-cosine integrals, Honam Math. J. 35 (2013), no. 2, 137-146. https://doi.org/10.5831/HMJ.2013.35.2.137
  7. J. Choi, Y. J. Cho, and H. M. Srivastava, Log-sine integrals involving series associated with the Zeta function and Polylogarithms, Math. Scand. 105 (2009), 199-217. https://doi.org/10.7146/math.scand.a-15115
  8. J. Choi and H. M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005), 51-70. https://doi.org/10.1007/s11139-005-3505-6
  9. J. Choi and H. M. Srivastava, Some applications of the Gamma and Polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log-sine integrals, Math. Nachr. 282 (2009), 1709-1723. https://doi.org/10.1002/mana.200710032
  10. J. Choi and H. M. Srivastava, Explicit evaluations of some families of log-sine and log-cosine integrals, Integral Transforms Spec. Funct. 22 (2011), 767-783. https://doi.org/10.1080/10652469.2011.564375
  11. J. Choi and H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Computer Modelling 54 (2011), 2220-2234. https://doi.org/10.1016/j.mcm.2011.05.032
  12. M. W. Coffey, On some series representations of the Hurwitz zeta function, J. Comput. Appl. Math. 216 (2008), 297-305. https://doi.org/10.1016/j.cam.2007.05.009
  13. L. Lewin, Polylogarithms and Associated Functions, Elsevier (North-Holland), New York, London and Amsterdam, 1981.
  14. Th. M. Rassias and H. M. Srivastava, Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers, Appl. Math. Comput. 131 (2002), 593-605. https://doi.org/10.1016/S0096-3003(01)00172-2
  15. L. C. Shen, Remarks on some integrals and series involving the Stirling numbers and ${\zeta}$(n), Trans. Amer. Math. Soc. 347 (1995), 1391-1399.
  16. A. Sofo and H. M. Srivastava, Identities for the harmonic numbers and binomial coeffcients, Ramanujan J. 25 (2011), 93-113. https://doi.org/10.1007/s11139-010-9228-3
  17. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  18. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London, and New York, 2012.

Cited by

  1. A family of polylog-trigonometric integrals 2017, https://doi.org/10.1007/s11139-017-9917-2