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Mathematics of Computation

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Chebyshev expansions for the Bessel function $ J\sb{n}(z)$ in the complex plane


Authors: J. P. Coleman and A. J. Monaghan
Journal: Math. Comp. 40 (1983), 343-366
MSC: Primary 65A05; Secondary 30E10, 33A40, 65D20
DOI: https://doi.org/10.1090/S0025-5718-1983-0679451-4
MathSciNet review: 679451
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Abstract: Polynomial-based approximations for $ {J_0}(z)$ and $ {J_1}(z)$ are presented. The first quadrant of the complex plane is divided into six sectors, and separate approximations are given for $ \vert z\vert \leqslant 8$ and for $ \vert z\vert \geqslant 8$ on each sector. Each approximation is based on a Chebyshev expansion in which the argument of the Chebyshev polynomials is real on the central ray of the sector. The errors involved in extrapolation off the central ray are discussed. The approximation obtained for $ \vert z\vert \geqslant 8$ can also be used to evaluate the Bessel functions $ {Y_0}(z)$ and $ {Y_1}(z)$ and the Hankel functions of the first and second kinds.


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DOI: https://doi.org/10.1090/S0025-5718-1983-0679451-4
Article copyright: © Copyright 1983 American Mathematical Society