The principal inverse of the gamma function

Abstract:

Let $\Gamma (x)$ be the gamma function in the real axis and $\alpha$ the maximal zero of $\Gamma ’(x)$. We call the inverse function of $\Gamma (x)|_{(\alpha , \infty )}$ the principal inverse and denote it by $\Gamma ^{-1}(x)$. We show that $\Gamma ^{-1}(x)$ has the holomorphic extension $\Gamma ^{-1}(z)$ to $\textbf {C}\setminus (-\infty , \Gamma (\alpha )]$, which maps the upper half-plane into itself, namely a Pick function, and that $\Gamma (\Gamma ^{-1}(z))= z$ on $\textbf {C}\setminus (-\infty , \Gamma (\alpha )].$