Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 075, 9 pages      arXiv:2402.14537      https://doi.org/10.3842/SIGMA.2024.075
Contribution to the Special Issue on Asymptotics and Applications of Special Functions in Memory of Richard Paris

McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions

Amparo Gil a, Javier Segura b and Nico M. Temme c
a) Departamento de Matemática Aplicada y CC, de la Computación, ETSI Caminos, Universidad de Cantabria, 39005 Santander, Spain
b) Departamento de Matemáticas, Estadistica y Computación, Universidad de Cantabria, 39005 Santander, Spain
c) Valkenierstraat 25, 1825BD Alkmaar, The Netherlands

Received February 23, 2024, in final form August 07, 2024; Published online August 10, 2024

Abstract
We derive asymptotic expansions of the large zeros of the Coulomb wave functions and for those of their derivatives. The new expansions have the same form as the McMahon expansions of the zeros of the Bessel functions and reduce to them when a parameter is equal to zero. Numerical tests are provided to demonstrate the accuracy of the expansions.

Key words: Coulomb wave functions; McMahon-type zeros; asymptotic expansions.

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