호남수학학술지 (Honam Mathematical Journal)

  • 제35권2호
  • /
  • Pages.137-146
  • /
  • 2013
  • /
  • 1225-293X(pISSN)
  • /
  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

LOG-SINE AND LOG-COSINE INTEGRALS

  • 투고 : 2013.03.04
  • 심사 : 2013.03.28
  • 발행 : 2013.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

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. The main object of this paper is to present explicit evaluations of some families of log-sine and log-cosine integrals by making use of the familiar Beta function.

키워드

참고문헌

  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 generalized 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 coefficients, harmonic numbers and generalized harmonic numbers, J. Inequ. Appl. 2013, 2013:49. http://www.journalofinequalitiesandapplications.com/content/2013/1/49 https://doi.org/10.1186/1029-242X-2013-49
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1989.
  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 coefficients, 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.
  19. K. S. Williams and N.-Y. Zhang, Special values of the Lerch Zeta function and the evaluation of certain integrals, Proc. Amer. Math. Soc. 119 (1993), 35-49. https://doi.org/10.1090/S0002-9939-1993-1172963-7
  20. N.-Y. Zhang and K. S. Williams, Values of the Riemann Zeta function and integrals involving log (2 sinh $\frac{\theta}{2}$) and log (2 sin $\frac{\theta}{2}$), Pacific J. Math. 168 (1995), 271-289. https://doi.org/10.2140/pjm.1995.168.271
  21. I. J. Zucker, On the series ${\Sigma}^{\infty}_{k=1}(^{2k}_k)^{-1}k^{-n}$ and related sums, J. Number Theory 20 (1985), 92-102. https://doi.org/10.1016/0022-314X(85)90019-8

피인용 문헌

  1. Series representations for special functions and mathematical constants vol.40, pp.2, 2016, https://doi.org/10.1007/s11139-015-9679-7
  2. A family of polylog-trigonometric integrals 2017, https://doi.org/10.1007/s11139-017-9917-2
  3. Evaluation of log-tangent integrals by series involving ζ(2n+1) vol.28, pp.6, 2017, https://doi.org/10.1080/10652469.2017.1312366
  4. HARMONIC NUMBERS AT HALF INTEGER AND BINOMIAL SQUARED SUMS vol.38, pp.2, 2016, https://doi.org/10.5831/HMJ.2016.38.2.279
  5. FURTHER LOG-SINE AND LOG-COSINE INTEGRALS vol.26, pp.4, 2013, https://doi.org/10.14403/jcms.2013.26.4.769
  6. Families of Integrals of Polylogarithmic Functions vol.7, pp.2, 2019, https://doi.org/10.3390/math7020143