and Related Statistical Applications

Liouville's theorem proves that certain integrals cannot be evaluated with elementary functions. It demonstrates why the gamma, exponential and Gaussian integrals lack antiderivatives. However, by applying the "$h$" factorization method, I present an analytical solution to the antiderivative of the gamma integral. This solution applies to all integrals that can be transformed into a gamma integral, including the exponential and Gaussian integrals. I further provide a thorough discussion of the algebraic properties of the ''$h$" function. The major contribution to statistical science is that ''$h$" can serve as the most fundamental function which unifies many cumulative distribution functions, such as the gamma function, the exponential integral function, the error function, the beta function, the hypergeometric function, and the Marcum Q-function, and the truncated normal distribution. The closed-form expression of the moment-generating function for the truncated normal distribution can be also derived as an ''$h$" function.