Multiple Mellin-Barnes integrals are often used for perturbative calculations in particle physics. In this context, the evaluation of such objects may be performed through residues calculations which lead to their expression as multiple series in powers and logarithms of the parameters involved in the problem under consideration. However, in most of the cases, several series representations exist for a given integral. They converge in different regions of values of the parameters, and it is not obvious to obtain them. For twofold integrals we present a method which allows to derive straightforwardly and systematically: (a) different sets of poles which correspond to different convergent double series representations of a given integral, (b) the regions of convergence of all these series (without an a priori full knowledge of their general term), and (c) the general term of each series (this may be performed, if necessary, once the relevant domain of convergence has been found). This systematic procedure is illustrated with some integrals which appear, among others, in the calculation of the two-loop hexagon Wilson loop in N = 4 SYM theory. Mellin-Barnes integrals of higher dimension are also considered.
New proofs for the two Barnes lemmas and an additional lemma
Mellin-Barnes（MB）表示已成为计算量子场论微扰计算中出现的Feynman圈积分的广泛使用的工具。一些MB积分可以解析地解决封闭形式的帮助下，两个巴恩斯引理已被称为数学已经一个世纪。这些引理的原始证明通过取无穷级数的残基并通过超几何函数将其求和来解决积分。本文提出了新的，优雅的证明巴恩斯引理，只依赖于著名的MB表示的基本身份，避免任何系列求和。它们对于向量子场论的学生介绍和证明巴恩斯引理特别有用，而不需要超几何函数的知识。本文还引入并证明了涉及相位因子exp（+-i pi z）的MB积分\int dz的附加引理。
Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1
本文研究了Usyukina-Davydychev（UD）函数的Mellin-Barnes（MB）变换的性质。在我们以前的论文[核物理B 870（2013）243]中，[Nuclear Physics B 876（2013）322]我们表明，Usyukina-Davydychev（UD）函数的多重Mellin-Barnes（MB）变换可以简化为双重MB变换，并且更高的Mellin-Barnes（MB）变换可以简化为更高的MB变换。阶UD函数通过将其应用于稍微修改的第一UD函数而获得。该结果在$d=4$维中是有效的，并且其在$d=4-2\varepsilon$维中的类似物也存在[Theoretical and数学物理177（2013）1515]。在[Nuclear Physics B 870（2013）243]中，通过比较具有不同循环阶的图之间的图解关系的左手侧和右手侧，隐式地获得了解析正则化UD函数的递归关系链。反过来，这些图表的关系得到了由于方法的循环减少三角梯形图提出了在1983年由Belokurov和Usyukina。在这里，我们重现这些经常性的关系，通过巴恩斯引理计算明确的轮廓积分所产生的左手边的图解关系。在这样的方式中，我们明确地计算一族的多重轮廓积分的某些比率的欧拉伽玛函数。我们做了一个猜想，类似的结果是有效的轮廓积分更广泛的家庭，其中包括MB变换的UD函数的光滑函数。
