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

다국어 초록 (Multilingual Abstract)

The Clausen function Cl2(x) arises in several applications. Alarge number of inde nite integrals of logarithmic or trigonometric functionscan be expressed in closed form in terms of Cl2(x). Very recently,Choi and Srivatava [3] and Choi [1] investigated certain integral formulasassociated with Cl2(x). In this sequel, we present an interesting new definiteintegral formula for the Clausen function Cl2(x) by using a knownrelationship between the Clausen function Cl2(x) and the generalized Zetafunction (s; a). Also an interesting integral representation for the Catalanconstant G is considered as one of two special cases of our main result.

참고문헌 (Reference)

1 H. M. Srivastava, "ZETA AND q-ZETA FUNCTIONS AND ASSOCIATED SERIES AND INTEGRALS" Elsevier 2012

2 T. Clausen, "Uber die function sin φ+ 122 sin 2φ+ 132 sin 3φ+ etc." 8 : 298-300, 1832

3 I. S. Gradshteyn, "Tables of Integrals, Series, and Products" Academic Press 2000

4 H. M. Srivastava, "Series Associated with the Zeta and Related Functions" Kluwer Academic Publishers 2001

5 L. Lewin, "Polylogarithms and Associated Functions" Elsevier (North-Holland) 1981

6 P. J. de Doelder, "On the Clausen integral Cl2(θ)" 11 : 325-330, 1984

7 J. Choi, "Mathieu series and associated sums involving the Zeta functions" 59 : 861-867, 2010

8 A. P. Prudnikov, "Integrals and Series" Gordon and Breach Science Publishers 1986

9 C. C. Grosjean, "Formulae concerning the computation of the Clausen integral Cl2(θ)" 11 : 331-342, 1984

10 V. E. Wood, "Effcient calculation of Clausen's integral" 22 : 883-884, 1968

11 J. Choi, "Clausen function Cl2(x) and its related integrals" 2014

12 K. S. Kolbig, "Chebyshev coeffcients for the Clausen function Cl2(x)" 64 : 295-297, 1995

13 J. Choi, "An integral representation of the Clausen function Cl2(x)" 2014