Indefinite Integration of the Gamma Integral
and Related Statistical Applications
1.4 Research Purpose
In this research, I apply a different factorization method to investigate the gamma integral with a new function "$h$", which can be expressed in terms of Kummer's confluent hypergeometric function
\begin{equation*}h_{s}^{c}\equiv\frac{1-M\left( 1,s+1,-c \right)}{c}. \tag{1.5} \end{equation*}
[Definition for (1.5)]
Through the analysis of the "$h$" function, I found that "$h$" directly links to the gamma function, and its asymptotic property is well-defined when the argument $c$ approaches negative infinity. This finding not only gives a general definition of the gamma function that extends the concept of the factorial function to the non-integer domain, but also completes the solution of the gamma integral, which can be further applied to the cumulative distribution functions of many important distributions. Moreover, the closed-form expression of the moment-generating function for the truncated normal distribution can be also derived as an "$h$" function. The overall finding is a significant contribution to general statistical science, and its application radically changes current practices in calculating probability.
Welcome all math lovers!
