The Gamma function converges for positive values of x and also for negative non-integer values, though this aspect is often overlooked in literature. To analyze convergence for negative non-integer values, the recursive formula for the Gamma function is recommended, as the integral representation does not apply in this case. The concept of analytic continuation allows the extension of the Gamma function beyond its initial domain, providing a broader understanding of its behavior. This approach clarifies the limitations of the integral representation for negative non-integer inputs. Overall, understanding these nuances is essential for accurate applications of the Gamma function in various mathematical contexts.