The Clausen function ${\sf Cl}_2(x)$ and its Related Integrals

Junesang Choi, Hari M. Srivastava

Abstract

The Clausen function ${\sf Cl}_2(x)$ arises in several applications. A large number of indefinite integrals of logarithmic or trigonometric functions can be expressed in closed form in terms of ${\sf Cl}_2(x)$.The $G$-function introduced and systematically investigated by Barnes in about $1900$, which was revived in the middle of the1980s in connection with the study of the determinants of the Laplacians, also has several useful and widely-spread applications.Here, in this paper, we aim at presenting some interesting definite integral formulas by using a known relationship between the Clausen function ${\sf Cl}_2(x)$ and the Barnes $G$-function.