Hankel围线外的Laplace方程与超几何函数的新恒等式
The Laplace equation for the exterior of the Hankel contour and novel identities for hypergeometric functions
摘要：
利用保角映射，可以将拉普拉斯方程的某些边值问题的解表示为涉及给定边界数据的单个积分。我们表明，这种明确的公式可以用来获得新的身份，特殊的功能。推导这种类型的恒等式的一个方便的工具是所谓的\emph{global relation}，它最近出现在广泛的边值问题。作为一个具体的应用，我们分析了Neumann边值问题的拉普拉斯方程在外部的所谓的汉克尔轮廓，这是轮廓，出现在定义的伽玛和黎曼zeta函数。通过利用这个问题的显式解，我们推导出了大量的新的身份涉及超几何函数。
英文摘要：
By employing conformal mappings, it is possible to express the solution of certain boundary value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identities for special functions. A convenient tool for deriving this type of identities is the so-called \emph{global relation}, which has appeared recently in a wide range of boundary value problems. As a concrete application, we analyze the Neumann boundary value problem for the Laplace equation in the exterior of the so-called Hankel contour, which is the contour that appears in the definition of both the gamma and the Riemann zeta functions. By utilizing the explicit solution of this problem, we derive a plethora of novel identities involving the hypergeometric function.
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