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On the paper “A continued fraction approximation of the modified Bessel function \(I_ 1(t)''\) by P. R. Parthasarathy and N. Balakrishnan. (English) Zbl 0733.65010

In the paper mentioned in the title [ibid. 3, No.4, 13-15 (1990; Zbl 0709.65013)] the authors use the continued fraction \[ f(x)=\frac{1/2}{s+1}-\frac{1/4}{s+1}-\frac{1/4}{s+1}-... \] and its convergents to approximate \((1)\quad F_ 1(x)=e^{-x}I_ 1(x)/x,\) where \(I_ 1\) is the modified Bessel function of order 1. Here we point out that (i) the approximation found is simply a Gauss-Chebyshev quadrature approximation of an integral representation for the function (1); (ii) the author’s statement concerning the accuracy of their approximation on the interval \(0\leq x\leq 20\) is inaccurate and misleading; (iii) the approximation is “from below”, but can be supplemented by an approximation “from above” using Gauss-Radau quadrature; (iv) similar approximations can be had for Bessel functions \(I_{\nu}\) of arbitrary order \(\nu >-1/2\).

MSC:

65D20 Computation of special functions and constants, construction of tables
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Citations:

Zbl 0709.65013
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References:

[1] Parthasarathy, P. R.; Balakrishnan, N., A continued fraction approximation of the modified Bessel function \(I_1(t)\), Appl. Math. Letters, 4, 25-27 (1991) · Zbl 0734.41020
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