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1.3 Unsolved Puzzle

A puzzle comes up immediately concerning why (1.2) or (1.4) was not widely recognized as a general solution for the gamma integral. Two related accounts can shed some light on this question. First, while both $g(s,-1,u)$ and $M\left( s,s+1,-u \right)$ have an infinite radius of convergence, they require a massive computational capacity to achieve a minimum level of accuracy when $u$ is large. Second, especially for the gamma function, since the upper limit of the integral is infinity, all the terms of $g(s,-1,u)$ and $M\left( s,s+1,-u \right)$, except the constant of $M$, approach infinity and result in an indefinite outcome (Borwein & Borwein, 2011). Apparently, we cannot use (1.2) or (1.4) to evaluate a gamma integral when it is sepcified as an improper integral.

The essence of the first problem is numerical precision. A previous study shows that we need at least 60 digits of precision to get one digit of accuracy in evaluating $M\left( a,b,z \right)$, where $a=b$ and $z=0+140i$ (Nardin et al., 1992, p.194). Many computational methods are developed to solve this problem, especially when $a$ and $b$ are small but $z$ is larger than $50$ (Muller, 2001, p.50). However, a more difficult and fundamental problem remains unsolved, that is, how can we evaluate the asymptotic properties of the gamma integral when its upper limit approaches infinity? If we can answer this question, theoretical ground would exist to recognize (1.2) or (1.4) as a solution to the gamma integral.

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