The multiple gamma-functions and the log-gamma integrals (Q456517)

Summary: In this paper, which is a companion paper to [the first author, ``The Barnes \(G\)-function and the Catalan constant,'' Kyushu J. Math. (in press)], starting from the Euler integral which appears in a generalization of Jensen's formula, we give a closed form for the integral of \(\log \Gamma(1 \pm t)\). This enables us to locate the genesis of two new functions \(A_{1/a}\) and \(C_{1/a}\) considered by Srivastava and Choi. We consider the closely related function \(A(a)\) and the Hurwitz zeta function, which render the task easier than working with the functions themselves. We also give a direct proof of Theorem 4.1, which is a consequence of [\textit{K. Chakraborty, S. Kanemitsu} and \textit{T. Kuzumaki}, ``On the Barnes multiple zeta- and gamma function,'' (in press), Corollary 1.1], though.