Gamma Function: An Analytical Study of Integral Properties and Asymptotic Expansions

Abstract

The Gamma function Γ(z) is addressed as one of the most important special functions in complex analysis and mathematical physics. The aim of the research is to provide a comprehensive exposition that links integral representations with analytic continuation across the complex plane. Emphasis is placed on Poincaré-type asymptotic expansions and Stirling's formula, and the study is supported by numerical analysis using Python to compute large values of the Gamma function. The investigation concludes that logarithmic convexity and Hankel's contour representation constitute the most rigorous tools for guaranteeing the uniqueness and continuity of the function away from its simple poles.