The principal inverse of the gamma function
Mitsuru Uchiyama · Proceedings of the American Mathematical Society · 2011
Let Γ ( x ) \Gamma (x) be the gamma function in the real axis and α \alpha the maximal zero of Γ ′ ( x ) \Gamma ’(x) . We call the inverse function of Γ ( x ) | ( α , ∞ ) \Gamma (x)|_{(\alpha , \infty )} the principal inverse and denote it by Γ − 1 ( x ) \Gamma ^{-1}(x) . We show that Γ − 1 ( x ) \Gamma ^{-1}(x) has the holomorphic extension Γ − 1 ( z ) \Gamma ^{-1}(z) to C ∖ ( − ∞ , Γ ( α