Integral representations of quantities associated with Gamma function

Abstract

We study a series of issues related with integral representations of Gamma functions and its quotients. The base of our study is two classical results in the theory of functions. One of them is a well-known first Binet formula, the other is a less known Malmsten formula. These special formulae express the values of the Gamma function in an open right half-plane via corresponding improper integrals. In this work we show that both results can be extended to the imaginary axis except for the point $z=0$. Under such extension we apply various methods of real and complex analysis. In particular, we obtain integral representations for the argument of the complex quantity being the value of the Gamma function in a pure imaginary point. On the base of the mentioned Malmsten formula at the points $z\neq0$ in the closed right half-plane, we provide a detailed derivation of the integral representation for a special quotient expressed via the Gamma function: $D(z)\equiv\Gamma(z+\frac{1}{2})/\Gamma(z+1)$. This fact on the positive semi-axis was mentioned without the proof in a small note by Dušan Slavić in 1975. In the same work he provided two-sided estimates for the quantity $D(x)$ as $x>0$ and at the natural points $D(x)$ coincided with the normalized central binomial coefficient. These estimates mean that $D(x)$ is enveloped on the positive semi-axis by its asymptotic series. In the present paper we briefly discuss the issue on the presence of this property on the asymptotic series $D(z)$ in a closed angle $|\arg z|\leqslant\pi/4$ with a punctured vertex. By the new formula representing $D(z)$ on the imaginary axis we obtain explicit expressions for the quantity $|D(iy)|^2$ and for the set $\mathrm{Arg}\, D(iy)$ as $y>0$. We indicate a way of proving the second Binet formula employing the technique of simple fractions.