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arXiv 0708.0116math-phmath.MP
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全参数复高斯超几何函数的快速计算及其在Poschl-Teller-Ginocchio势波函数中的应用

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Poschl-Teller-Ginocchio potential wave functions

N. Michel, M. V. Stoitsov

备注：
29 pages; accepted in Computer Physics Communications
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摘要：

高斯超几何函数2F 1的快速计算是一个困难的任务。虽然2F 1函数验证了涉及幂级数展开的许多分析性质，其实施显然是立即的，但它们的使用受到非常大的项之间的抵消引起的不稳定性的阻碍。此外，仅使用2F 1幂级数公式无法访问复平面的小区域，因此在纯分析的基础上不可能进行2F 1评估。为了解决这些问题，本文提出了一种推广的R.C. Forrey的转换理论得到了发展。后者已成功地处理了2F 1函数与实参数。与实际情况的变换理论一样，2F 1解析公式中出现的大的抵消项被严格地处理，但通过一种新的方法，直接适用于复平面。泰勒级数展开被用来进入幂级数解析公式有效域之外的复杂区域。然而，当|a|、|b|、|c|为中等或大时，所提出的算法通常变得不稳定。作为一个物理应用，考虑了解析Poschl-Teller-Ginocchio势的波函数的计算。

英文摘要：

The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forrey's transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.

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