TY - JOUR
T1 - Evaluation of Abramowitz functions in the right half of the complex plane
AU - Gimbutas, Zydrunas
AU - Jiang, Shidong
AU - Luo, Li Shi
N1 - Funding Information:
S. Jiang was supported by the National Science Foundation under grant DMS-1720405 , and by the Flatiron Institute, a division of the Simons Foundation . L.-S. Luo was supported by the National Science Foundation under grant DMS-1720408 . The authors would like to thank Vladimir Rokhlin at Yale University for sharing his unpublished pioneer work on the evaluation of Hankel functions in the complex plane and Manas Rachh at the Flatiron Institute, Simons Foundation for helpful discussions. Certain commercial software products and equipment are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the software products and equipment identified are necessarily the best available for the purpose. Appendix A
Funding Information:
S. Jiang was supported by the National Science Foundation under grant DMS-1720405, and by the Flatiron Institute, a division of the Simons Foundation. L.-S. Luo was supported by the National Science Foundation under grant DMS-1720408. The authors would like to thank Vladimir Rokhlin at Yale University for sharing his unpublished pioneer work on the evaluation of Hankel functions in the complex plane and Manas Rachh at the Flatiron Institute, Simons Foundation for helpful discussions. Certain commercial software products and equipment are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the software products and equipment identified are necessarily the best available for the purpose.
Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2020/3/15
Y1 - 2020/3/15
N2 - 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.
AB - 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.
KW - Abramowitz functions
KW - Laurent series
KW - Least squares method
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U2 - 10.1016/j.jcp.2019.109169
DO - 10.1016/j.jcp.2019.109169
M3 - Article
AN - SCOPUS:85078010856
SN - 0021-9991
VL - 405
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 109169
ER -