Section 5   Conclusions

I propose an explanation based on the "$h$" factorization method for why the series solution to the gamma integral with a Taylor or hypergeometric function is not widely recognized. Failure to recognize the finite property of the infinite upper limit of the gamma integral results in a false conclusion that the series solution is indeterminate. Applying the "$h$" factorization method, I have demonstrated, for any arbitrarily large limit $c$, that an even larger number $N$ always exists, and it defines the number of expansions in the series solution, making the "$h$" function convergent to a value that is associated with the gamma function. This finding not only defines the algebraic meaning of the "$h$" function, but also explicates its elementary quality in generalizing the factorial function to the non-integer domain.

I further study the basic properties of the "$h$" function and extend its algebraic definition to all real domains for the base parameter $s$. For most of the core functions that are used in statistical distributions, the "$h$" function can serve as the minimal denominator and fully specify them exclusively with its own form. Applying this property to those cumulative distribution functions that lack closed-form expressions, we can explicitly specify their functional forms and evaluate them with arbitrary precision. These core functions include the gamma function, the exponential integral function, the error function, the beta function, the hypergeometric function, the Marcum Q-function, and the truncated normal distribution. They cover a full range of the commonly-used distributions. In addition, the "$h$" function can be also applied to the moment-generating function of the truncated normal distribution.

Most of the contemporary mathematicians have reached a consensus that the integrals, such as the gamma, exponential, and Gaussian integrals have no closed-form solutions, and they do not even consider it an unsolved problem. However, regardless of whether the elementary nature of the "$h$" function is recognized, this paper has demonstrated its application in giving the most general solution to those unsolvable integrals. Given the power of many sophisticated numerical methods today, the main contribution of the "$h$" function might not be computational efficiency, but rather analytical clarity of mathematical deductions, which leads to the discovery of great unity among many seemingly-unrelated distributions.

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