We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function $G(z;\tau)$. Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as $z\to \infty$. Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms $C(\tau)$ and $D(\tau)$.
翻译:我们的目标是实现以下三个目标:首先,我们收集巴恩斯双伽马函数$G(z;\toau)$的所有已知定义、转换属性和功能特性。第二,我们为计算双伽马函数的数字得出一个算法,并展示其完整的非现现现扩展为$z\ to\infty$。第三,我们产生一些新属性,包括新产品特性和新表示的伽马模块格式$C(tau)$和$D(\tau)$。