DOI QR코드

DOI QR Code

SOME INTEGRAL REPRESENTATIONS OF THE CLAUSEN FUNCTION Cl2(x) AND THE CATALAN CONSTANT G

Choi, Junesang

  • Received : 2015.05.16
  • Accepted : 2016.01.14
  • Published : 2016.01.30

Abstract

The Clausen function $Cl_2$(x) arises in several applications. A large number of indefinite integrals of logarithmic or trigonometric functions can be expressed in closed form in terms of $Cl_2$(x). Very recently, Choi and Srivatava [3] and Choi [1] investigated certain integral formulas associated with $Cl_2$(x). In this sequel, we present an interesting new definite integral formula for the Clausen function $Cl_2$(x) by using a known relationship between the Clausen function $Cl_2$(x) and the generalized Zeta function ${\zeta}$(s, a). Also an interesting integral representation for the Catalan constant G is considered as one of two special cases of our main result.

Keywords

Clausen function $Cl_2$(x);Catalan constant;Generalized zeta function

References

  1. J. Choi, An integral representation of the Clausen function $Cl_2$(x), submitted for publication (2014).
  2. J. Choi and H. M. Srivastava, Mathieu series and associated sums involving the Zeta functions, Comput. Math. Appl. 59 (2010), 861-867. https://doi.org/10.1016/j.camwa.2009.10.008
  3. J. Choi and H. M. Srivastava, Clausen function $Cl_2$(x) and its related integrals, submitted for publication (2014).
  4. T. Clausen, Uber die function sin ${\varphi}+\frac{1}{2^2}$ sin $2{\varphi}+\frac{1}{3^2}$ sin $3{\varphi}+$ etc., J. Reine Angew. Math. 8 (1832), 298-300.
  5. P. J. de Doelder, On the Clausen integral $Cl_2({\theta})$, J. Comput. Appl. Math. 11 (1984), 325-330. https://doi.org/10.1016/0377-0427(84)90007-4
  6. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Corrected and Enlarged edition prepared by A. Jeffrey), Academic Press, New York, 1980; Sixth edition, 2000.
  7. C. C. Grosjean, Formulae concerning the computation of the Clausen integral $Cl_2({\theta})$, J. Comput. Appl. Math. 11 (1984), 331-342. https://doi.org/10.1016/0377-0427(84)90008-6
  8. K. S. Kolbig, Chebyshev coeffcients for the Clausen function $Cl_2(x)$, J. Comput. Appl. Math. 64 (1995), 295-297. https://doi.org/10.1016/0377-0427(95)00150-6
  9. L. Lewin, Polylogarithms and Associated Functions, Elsevier (North-Holland), New York, London and Amsterdam, 1981.
  10. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, Gordon and Breach Science Publishers, New York, 1986.
  11. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, and London, 2001.
  12. 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.
  13. V. E. Wood, Effcient calculation of Clausen's integral, Math. Comput. 22 (1968), 883-884.

Cited by

  1. Generalized rational zeta series for ζ(2n) and ζ(2n+1) vol.28, pp.12, 2017, https://doi.org/10.1080/10652469.2017.1393809
  2. Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n ) pp.1572-9303, 2019, https://doi.org/10.1007/s11139-018-0081-0