in terms of Bessel functions of fixed order
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§2.8(iv) Case III: Simple Pole… ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …
… ►The second set is in terms of Bessel functions of orders and , and they are uniform for fixed and , where again denotes an arbitrary small positive constant. …
… ►where , , , and are continuous real functions of and , with the branches of and chosen to satisfy (10.68.18) and (10.68.21) as . … ►Equations (10.68.8)–(10.68.14) also hold with the symbols , , , and replaced throughout by , , , and , respectively. In place of (10.68.7), … ►When is fixed, , and … ►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …
… ►The latter expansions are in terms of Bessel functions, and are uniform in complex -domains not containing neighborhoods of 1. … ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►
In Terms of Elementary Functions… ►
In Terms of Bessel Functions… ►
In Terms of Airy Functions…
§10.19 Asymptotic Expansions for Large Order►
§10.19(i) Asymptotic Forms… ►
§10.19(ii) Debye’s Expansions… ►
§10.19(iii) Transition Region… ►See also §10.20(i).
§10.40(ii) Error Bounds for Real Argument and Order… ►For the error term in (10.40.1) see §10.40(iii). ►
§10.40(iii) Error Bounds for Complex Argument and Order… ►For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).
§14.15(i) Large , Fixed… ►Here and are the modified Bessel functions (§10.25(ii)). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). ►
§14.15(iii) Large , Fixed… ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials as with fixed. …
8: 13.9 Zeros
… ►When and let , , be the positive zeros of arranged in increasing order of magnitude, and let be the th positive zero of the Bessel function (§10.21(i)). … ►For fixed the large -zeros of satisfy … ►For fixed and in the large -zeros of are given by … ►For fixed and in the function has only a finite number of -zeros. … ►For fixed and in the large -zeros of are given by …
§10.17 Asymptotic Expansions for Large Argument… ► … ►
§10.17(iii) Error Bounds for Real Argument and Order… ►If these expansions are terminated when , then the remainder term is bounded in absolute value by the first neglected term, provided that . … ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).