In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling’s formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica.

中文翻译：

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斯特林公式中伽马函数的精确值

在这项工作中，斯特林公式的完整版本由标准项和一个无限渐近级数组成，用于获得复平面所有分支上伽马函数对数的精确值。确切值只能通过正则化获得。引入了两种方法：Borel求和和Mellin-Barnes（MB）正则化。Borel求和余数由指数积分和不连续对数项的无限收敛总和组成，它们出现在特定扇区和称为斯托克斯扇区和线的线上，而MB规范化的余数减少为一个具有相似对数项的复杂MB积分。由于收敛域重叠，经常可以使用两种MB正规化的渐近形式来评估γ函数的对数。尽管必须将Borel总和的余数截短，但发现将这两个余数与（1）截断的渐近级数，（2）斯特林公式和（3）由复平面的较高分支产生的对数项求和伽玛函数的对数的值相同。在可能的情况下，他们也同意Mathematica的结果。