Summary

Four distinct solutions exist for the potential distribution around two equal circular parallel conducting cylinders, charged to the same potential. Their equivalence is demonstrated, and the resulting analytical identities are discussed. The identities relate the Jacobi elliptic function $sn$, the Jacobi theta functions $\theta _1 ,~\theta _2 $ and infinite series over trigonometric and hyperbolic functions.

中文翻译：

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两缸静电问题的四种解决方案以及它们的等效性

概要

对于在两个相等的圆形平行导电圆柱体周围的电势分布，存在四种不同的解决方案，这些圆柱体均充有相同的电势。证明了它们的等效性，并讨论了所得的分析恒等式。恒等式涉及Jacobi椭圆函数$ sn $，Jacobi theta函数$ \ theta _1，〜\ theta _2 $以及三角函数和双曲函数的无穷级数。