The Inverse of the Complex Gamma Function

We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case

Properties and Computation of the Inverse of the Gamma function

We explore the approximation formulas for the inverse function of Γ. The inverse function of Γ is a multivalued function and must be computed branch by branch. We compare three approximations for the principal branch Γ̌ 0 . Plots and numerical values show that the choice of the approximation depends on the domain of the arguments, specially for small arguments. We also investigate some iterative schemes and find that the Inverse Quadratic Interpolation scheme is better than Newton’s scheme for improving the initial approximation. We introduce the contours technique for extending a real-valued function into the complex plane using two examples from the elementary functions: the log and the arcsin functions. We show that, using the contours technique, the principal branch Γ̌ 0 (x) has the extension Γ̌ 0 (z) to the branch cut C\] − ∞, Γ(ψ 0 )] and the branch Γ̌ −1 (x) has the extension Γ̌ −1 (z) to the branch cut C\]0, Γ(ψ 0 )], where ψ 0 is the positive zero of Γ 0 (x)