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Mathematical approximation of special functions. Ten papers on Chebyshev expansions. (English) Zbl 0860.33004
Commack, NY: Nova Science Publishers. vii, 188 p. \$ 47.00 (1992).
This book represents a collection of ten papers on approximation of some special functions by means of Chebyshev expansions. Bessel functions and related ones are considered. The book is aimed towards applied mathematicians and other scientists with some background in numerical analysis and special functions. It is divided into ten chapters. – The first three chapters are devoted to the expansion of generalized hypergeometric functions in Chebyshev polynomials (some applications are also given), polynomial approximants to the Gauss error integral ${\int }_{0}^{x}{e}^{-{u}^{2}}du$ and Chebyshev expansions to the Dawson integral ${e}^{-{x}^{2}/2}{\int }_{0}^{x}{e}^{{u}^{2}/2}du$, respectively. Chapters 4 and 5 include Chebyshev expansions of the Bessel functions ${J}_{\nu }\left(x\right)$, ${N}_{\nu }\left(x\right)$, and of ${I}_{\nu }\left(x\right)$, ${K}_{\nu }\left(x\right)$, respectively. In Chapter 6, a numerical calculation of the Chebyshev expansion coefficients for ${J}_{\nu }\left(x\right)$ and also for ${I}_{\nu }\left(x\right)$ is obtained by applying Miller’s recurrence algorithm (the convergence of Miller’s method is only proved for the case $0\le x\le a$). Tables of the coefficients (with 15 decimal digit accuracy) of Chebyshev expansions for the first ten zeros of the functions ${J}_{\nu }\left(x\right)$, ${J}_{\nu }^{\text{'}}\left(x\right)$ and ${\left[{x}^{1/2}{J}_{\nu }\left(x\right)\right]}^{\text{'}}$ $\left(0<\nu <\infty \right)$ and ${\left[{x}^{-1/2}{J}_{\nu }\left(x\right)\right]}^{\text{'}}$ $\left(\nu >\frac{1}{2}\right)$ are presented in Chapter 7. Chebyshev polynomial expansions of the Airy functions $\text{Ai}\left(x\right)$, $\text{Bi}\left(x\right)$, their first derivatives and integrals, of the Kelvin functions $\text{ber}\phantom{\rule{4.pt}{0ex}}x$, $\text{bei}\phantom{\rule{4.pt}{0ex}}x$, $\text{ker}\phantom{\rule{4.pt}{0ex}}x$ and $\text{kei}\phantom{\rule{4.pt}{0ex}}x$, and some integrals of Bessel functions (of the form ${\int }_{x}^{\infty }{J}_{0}\left(t\right)dt$, ${\int }_{0}^{x}{I}_{0}\left(t\right)dt$, ${\int }_{x}^{\infty }\left({N}_{0}\left(t\right)/t\right)dt,\cdots \right)$, are, respectively, performed in Chapters 8, 9 and 10. At the end of chapters 1, 2, 3, 6, 8, 9 and 10, coefficients in the corresponding Chebyshev expansions are computed and listed in tabulated forms.
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33C99 Hypergeometric functions 41A30 Approximation by other special function classes 41A50 Best approximation, Chebyshev systems
##### Keywords:
Bessel functions; Chebyshev expansions