Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons. A simple accurate approximation of J₀(x) is found on the interval [0, 2].

中文翻译：

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多边形上的一些 Neumann-Bessel 级数和拉普拉斯算子

具有贝塞尔函数和三角函数的诺依曼级数的几个和被评估为三角函数的有限和。它们源于正多边形中拉普拉斯算子本征态的诺依曼展开式的推广。在区间 [0, 2] 上找到了J ₀ ( x )的简单精确近似值。