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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Chebyshev expansions for the Bessel function $J_{n}(z)$ in the complex plane
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by J. P. Coleman and A. J. Monaghan PDF
Math. Comp. 40 (1983), 343-366 Request permission


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 $|z| \leqslant 8$ and for $|z| \geqslant 8$ 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 $|z| \geqslant 8$ 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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Math. Comp. 40 (1983), 343-366
  • MSC: Primary 65A05; Secondary 30E10, 33A40, 65D20
  • DOI:
  • MathSciNet review: 679451