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Chebyshev series approximations for the Bessel function ${Y}_{n}\left(z\right)$ of complex argument. (English) Zbl 0918.65016
Bessel functions of the first kind ${J}_{n}\left(z\right)$ and the second kind ${Y}_{n}\left(z\right)$ of integer order play an important role in mathematical physics and engineering sciences. Numerical methods for efficiently computing these functions are therefore of interest to computational physicists and engineers. The authors employ the truncated Chebychev series to approximate the Bessel function of the second kind ${Y}_{n}\left(z\right)$ for $|z|\le 8$. Detailed manipulations and discussions for ${Y}_{0}\left(z\right)$ and ${Y}_{1}\left(z\right)$ are given. Results of numerical experiments are presented to demonstrate the computed accuracy by using the Chevychev series approximation. The computed accuracy is comparable with that computed by the tau-method approximations, especially when $argz$ is small. Advantages and disadvantages of the Chebychev series approximation compared with the tau-method approximation are discussed.
##### MSC:
 65D20 Computation of special functions, construction of tables 65E05 Numerical methods in complex analysis 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$