论文标题
HEUN功能的基本积分系列。使用黑洞扰动理论
Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
论文作者
论文摘要
HEUN微分方程是具有四个常规奇异性的最普遍的二阶二阶方程。迄今为止，仅涉及基本整合的HEUN函数的明确积分序列表示尚不清楚，并且在最近的评论中被认为是一个重要的开放问题。我们提供了Heun类的所有方程解决方案的明确积分表示：一般，汇合，双重共鸣，双重融合和三弦：仅涉及合理函数和指数积分的积分。所有系列都用具体的使用示例进行了说明。这些结果源于路径 - 湿度的技术，我们用来评估专门选择用于产生HEUN函数的可变矩阵的路径订购的指数。我们通过提供对旋转黑洞的度量扰动的Teukolsky径向方程的解决方案的首次表示，从而证明了整体序列的实用性，从黑洞地平线到空间无穷大。
Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide explicit integral representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly-confluent and triconfluent, with integrals involving only rational functions and exponential integrands. All the series are illustrated with concrete examples of use. These results stem from the technique of path-sums, which we use to evaluate the path-ordered exponential of a variable matrix chosen specifically to yield Heun functions. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.
