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

Abstract The fast computation of the Gauss hypergeometric function _2 F_1with all its parameters complex is a difficult task. Although the _2 F_1function 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, in the vicinity of z = e^(± i frac(π, 3)), are inaccessible using _2 F_1power series li... Title of program: hyp_2F1, PTG_wf Catalogue Id: AEAE_v1_0 Nature of problem The Gauss hypergeometric function 2 F 1 , with all its parameters complex, is uniquely calculated in the frame of transformation theory with power series summations, thus providing a very fast algorithm. The evaluation of the wave functions of the analytical Pöschl-Teller-Ginocchio potential is treated as a physical application. Versions of this program held in the CPC repository in Mendeley Data AEAE_v1_0; hyp_2F1, PTG_wf; 10.1016/j.cpc.2007.11.007 This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2019)摘要 所有参数均为复数的高斯超几何函数（Gauss hypergeometric function）₂F₁的快速计算是一项极具挑战性的工作。尽管₂F₁函数具备诸多涉及幂级数展开（power series expansions）的解析性质，且此类展开式的实现看似直观简便，但由于极大量项之间的抵消效应会引发数值不稳定问题，导致其实际应用受到阻碍。此外，复平面（complex plane）内z=e^(±iπ/3)附近的极小区域，无法通过₂F₁的幂级数展开方法完成计算（原文此处未完整显示）。 程序名称：hyp_2F1、PTG_wf 目录编号：AEAE_v1_0 问题本质 所有参数均为复数的高斯超几何函数₂F₁，可通过基于变换理论（transformation theory）结合幂级数求和的方式实现唯一求解，由此得到一种极具效率的快速算法。本程序将解析形式的珀施尔-泰勒-吉诺基奥势（Pöschl-Teller-Ginocchio potential）的波函数（wave functions）求值作为物理应用场景。 Mendeley数据中CPC（Computer Physics Communications）库收录的本程序版本：AEAE_v1_0；hyp_2F1、PTG_wf；DOI:10.1016/j.cpc.2007.11.007 本程序源自贝尔法斯特女王大学馆藏的CPC程序库（1969-2019年）。