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Evaluation of Abramowitz functions in the right half of the complex plane
Zydrunas Gimbutas1, Shidong Jiang2, Li-Shi Luo3,4
1Information Technology Laboratory, National Institute of Standards and Technology, Boulder, CO 80305, USA.
Abstract:
A numerical scheme is developed for the evaluation of Abramowitz functions Jn in the right half of the complex plane. For n = - 1, … , 2, the scheme utilizes series expansions for ∣z∣ < 1, asymptotic expansions for ∣z∣ > R with R determined by the required precision, and least squares Laurent polynomial approximations on each sub-region in the intermediate region 1 ≤ ∣z∣ ≤ R. For n > 2, Jn is evaluated via a forward recurrence relation. The scheme achieves nearly machine precision for n = -1, … , 2 at a cost that is competitive as compared with software packages for the evaluation of other special functions in the complex domain.
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