Iterated integrals in Feynman integral computations

费曼积分的计算对于现代粒子对撞机的理论预测是至关重要的。Feynman积分可以用多重多对数或经典多对数的相关推广来表示，已有强大的计算方法。这个项目的一个重点是通过使用一类代表多重多对数的迭代积分来改进和进一步自动化其中的两种技术，这种积分在这种计算中具有特别好的实际应用性质。借助于具体的算法及其对这类非常一般的函数的符号计算的实现，我们尝试通过直接参数积分和超几何函数的展开来改进Feynman积分的导数。项目的第二个焦点是Feynman积分的子集，根据本领域的现状，它不能用多重多对数来表示。到目前为止，依赖于使用迭代积分的计算方法还不能应用于这些情况。最近的进展表明，椭圆多对数是计算此类Feynman积分的一类合适的函数。在这个项目中，我们探索了椭圆多对数在Feynman积分计算中的进一步应用，并争取建立一个椭圆多对数。用重积分表示椭圆多对数的可能性可能导致对这类有问题的Feynman积分的第一个系统方法的发展。

The computation of Feynman integrals is crucial for theoretical predictions at modern particle colliders. Powerful computational methods exist for Feynman integrals which can be expressed in terms of multiple polylogarithms or related generalizations of classical polylogarithms. One focus of this project is the improvement and further automatization of two of these techniques by use of a certain class of iterated integrals which represents the multiple polylogarithms and which has particularly good properties for the practical use in such computations. With the help of concrete algorithms and their implementation for the symbolical computation with this very general class of functions, we attempt the improvement of the derivation of Feynman integrals via direct parametric integration and via the expansion of hypergeometric functions.A second focus of the project is a subset of Feynman integrals, which according to the present state of the art cannot be expressed in terms of multiple polylogarithms. Computational methods which rely on the use of iterated integrals cannot be applied to these cases so far. Recent progress suggests that elliptic polylogarithms are an appropriate class of functions for the computation of such Feynman integrals. In this project we explore the further use and strive for an establishment of elliptic polylogarithms in Feynman integral computations. The possibility of expressing elliptic polylogarithms in terms of iterated integrals may lead to the development of a first systematic method for this problematic class of Feynman integrals.