Abstract In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the F 1 2 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the complex Higgins functions, the complex Christov functions, and their sine-like and cosine-like versions. Then, after studying the asymptotic behavior of the resulting expressions, we discuss the numerical difficulties in their implementation, and develop a method using arbitrary-precision arithmetic that computes them accurately. We also explain how to create the differentiation matrices associated to the complex Higgins functions and to the complex Christov functions, and study their condition numbers. In this regard, we show how arbitrary-precision arithmetic is the natural tool to deal with ill-conditioned systems. Finally, we simulate numerically the fractional nonlinear Schrodinger equation using the developed tools.

中文翻译：

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使用正交族对 R 上的分数拉普拉斯算子进行数值逼近

摘要 在本文中，我们使用众所周知的复变量技术，根据 F 1 2 高斯超几何函数、复希金斯函数的一维分数拉普拉斯算子、复克里斯托夫函数及其类正弦函数明确计算和类似余弦的版本。然后，在研究了结果表达式的渐近行为之后，我们讨论了它们实现中的数值困难，并开发了一种使用任意精度算术准确计算它们的方法。我们还解释了如何创建与复 Higgins 函数和复 Christov 函数相关的微分矩阵，并研究它们的条件数。在这方面，我们展示了任意精度算术是如何处理病态系统的自然工具。最后，