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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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  • [1] M. Abramowitz & I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
  • [2] C. W. Clenshaw, "The numerical solution of linear differential equations in Chebyshev series," Proc. Cambridge Philos. Soc., v. 53, 1957, pp. 134-149. MR 0082196 (18:516a)
  • [3] C. W. Clenshaw, Chebyshev Series for Mathematical Functions, NPL Mathematical Tables, Vol. 5, HMSO London, 1962.
  • [4] J. P. Coleman, "A Fortran subroutine for the Bessel function $ {J_n}(x)$ of order 0 to 10," Comput. Phys. Comm., v. 21, 1980, pp. 109-118.
  • [5] P. J. Davis, Interpolation and Approximation, Blaisdell, Waltham, Mass., 1963. MR 0157156 (28:393)
  • [6] D. Elliott, "A Chebyshev series method for the numerical solution of Fredholm integral equations," Comput. J., v. 6, 1963, pp. 102-111. MR 0155452 (27:5386)
  • [7] G. H. Elliott, The Construction of Chebyshev Approximations in the Complex Plane, Ph.D. Thesis, University of London, 1978.
  • [8] C. Geiger & G. Opfer, "Complex Chebyshev polynomials in circular sectors," J. Approx. Theory, v. 24, 1978, pp. 93-118. MR 511466 (80m:41004)
  • [9] Y. L. Luke, The Special Functions and Their Approximations, Vol. II, Academic Press, New York and London, 1969. MR 0249668 (40:2909)
  • [10] Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York and London, 1975. MR 0501762 (58:19039)
  • [11] Y. L. Luke, Algorithms for the Computation of Mathematical Functions, Academic Press, New York and London, 1977. MR 0494840 (58:13624)
  • [12] G. F. Miller, "On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation," SIAM J. Numer. Anal, v. 3, 1966, pp. 390-409. MR 0203312 (34:3165)
  • [13] G. Opfer, "An algorithm for the construction of best approximations based on Kolmogorov's criterion," J. Approx. Theory, v. 23, 1978, pp. 299-317. MR 509560 (80i:65019)
  • [14] T. J. Rivlin, The Chebyshev Polynomials, Wiley, New York and London, 1974. MR 0450850 (56:9142)
  • [15] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1944. MR 0010746 (6:64a)

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Additional Information

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

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