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

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.Comment: 29 pages; accepted in Computer Physics Communication

Conserving ground-dwelling beetles in an endangered woodland community: Multi-scale habitat effects on assemblage diversity

Patterns of biodiversity are influenced by habitat features at multiple spatial scales, yet few studies have used a multi-scale approach to examine ground-dwelling beetle diversity patterns. We trapped and quantified ground-dwelling beetle assemblages at two spatial scales: (1) microhabitat elements, represented by open ground, ground under trees and ground next to logs and (2) macrohabitat, represented by three vegetation types in a box-gum grassy woodland in south-eastern Australia. Species richness and evenness was highest at samples from under trees and lowest at samples in the open. At the macrohabitat scale, species richness and evenness did not differ among vegetation types. Assemblage composition was significantly different between trees, logs and open elements. Assemblage composition was different only between vegetation types with contrasting high and low shrub cover. Estimation of true species richness indicated assemblages at logs may have a higher number of species compared to trees and open elements, and implied greater spatial heterogeneity in assemblages at logs. Significant spatial autocorrelation in beetle assemblages was detected for logs at up to 400 m, but not for ground under trees or in the open. In agreement with previous studies, a mix of vegetation types at the macrohabitat scale is important for beetle conservation. Assemblage composition, however, appears to be more closely linked with habitat elements at the microhabitat scale, where logs support a high diversity of beetle species. This strongly supports the idea that restoring logs to box-gum grassy woodlands would be useful for increasing beetle species richness and assemblage heterogeneity. Crown Copyright © 2009