asymptotic approximations to zeros

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11: 2.2 Transcendental Equations

§2.2 Transcendental Equations

… ►Higher approximations are obtainable by successive resubstitutions. … ►An important case is the reversion of asymptotic expansions for zeros of special functions. …Applications to real and complex zeros of Airy functions are given in Fabijonas and Olver (1999). For other examples see de Bruijn (1961, Chapter 2).

12: 29.20 Methods of Computation

… ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to \mathrm{\infty }$. … ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … ►

§29.20(iii) Zeros

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13: 18.16 Zeros

… ►Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of $L_n^{(\pm \frac{1}{2})}\left(x\right)$ lead immediately to results for the zeros of $H_n\left(x\right)$. …

14: 6.18 Methods of Computation

… ►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

15: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►Asymptotic approximations are also provided for the zeros of $K_n(x;p,N)$ in various cases depending on the values of $p$ and $\mu$. … ►For asymptotic approximations for the zeros of $M_n(nx;\beta ,c)$ in terms of zeros of $\mathrm{Ai}\left(x\right)$ (§9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … ►For asymptotic approximations to $P_n^{(\lambda )}(x;\varphi )$ as $|x+i\lambda |\to \mathrm{\infty }$, with $n$ fixed, see Temme and López (2001). …Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

16: 8.13 Zeros

… ►For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to -\mathrm{\infty }$ see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012). … ►As $x$ increases the positive zeros coalesce to form a double zero at ($a_n^{\ast },x_n^{\ast }$). The values of the first six double zeros are given to 5D in Table 8.13.1. …Approximations to $a_n^{\ast }$, $x_n^{\ast }$ for large $n$ can be found in Kölbig (1970). …

17: 14.16 Zeros

… ►For uniform asymptotic approximations for the zeros of ${𝖯}_n^{-m}\left(x\right)$ in the interval when $n\to \mathrm{\infty }$ with $m$ $(\ge 0)$ fixed, see Olver (1997b, p. 469). …

18: 9.17 Methods of Computation

… ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

19: 10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

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§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►Lastly, there are two conjugate sets, with $n$ zeros in each set, that are asymptotically close to the boundary of $𝐊$ as $n\to \mathrm{\infty }$. … ►are simple and the asymptotic expansion of the $m$th positive zero as $m\to \mathrm{\infty }$ is given by … ►This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …

20: 2.7 Differential Equations

… ►The first of these references includes extensions to complex variables and reversions for zeros. …