# LOG-SINE AND LOG-COSINE INTEGRALS

Choi, Junesang

• Accepted : 2013.03.28
• Published : 2013.06.25
• 41 10

#### 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. 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.

#### Keywords

Beta function;Gamma function;Log-sine and log-cosine integrals;Harmonic numbers;Generalized harmonic numbers;Odd harmonic numbers;Generalized odd harmonic numbers;Riemann Zeta function;Hurwitz (or generalized) Zeta function;Psi (or Digamma) function;Polygamma functions;Euler-Mascheroni constant

#### References

1. 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
2. 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
3. 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
4. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1989.
5. L. Lewin, Polylogarithms and Associated Functions, Elsevier (North-Holland), New York, London and Amsterdam, 1981.
6. 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
7. L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ${\zeta}$(n), Trans. Amer. Math. Soc. 347 (1995), 1391-1399.
8. 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
9. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
10. 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.
11. 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
12. 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
13. 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
14. V. S. Adamchik and H. M. Srivastava, Some series of the Zeta and related functions, Analysis 18 (1998), 131-144.
15. 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
16. M. G. Beumer, Some special integrals, Amer. Math. Monthly 68 (1961), 645-647. https://doi.org/10.2307/2311513
17. 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 https://doi.org/10.1016/j.amc.2011.01.062
18. 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
19. 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
20. 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
21. 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

#### Cited by

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