applied to asymptotic expansions

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11: 18.16 Zeros

… ►For an asymptotic expansion of $x_{n,m}$ as $n\to \mathrm{\infty }$ that applies uniformly for $m=1,2,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$, see Olver (1959, §14(i)). …

12: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►where $\delta$ denotes an arbitrary small positive constant. … ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►This reference also contains explicit formulas for $b_k(\lambda )$ in terms of Stirling numbers and for the case $\lambda >1$ an asymptotic expansion for $b_k(\lambda )$ as $k\to \mathrm{\infty }$. … ►

13: 28.34 Methods of Computation

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

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Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

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Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

14: 10.41 Asymptotic Expansions for Large Order

… ►We first prove that for the expansions (10.20.6) for the Hankel functions $H_{\nu }^{(1)}\left(\nu z\right)$ and $H_{\nu }^{(2)}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to \pm \mathrm{i}\mathrm{\infty }$, respectively. …

15: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. … ►The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when $\mu \to \mathrm{\infty }$ uniformly with respect to $t\in [1+\delta ,\mathrm{\infty })$. … … ►

Modified Expansions

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16: 6.12 Asymptotic Expansions

§6.12 Asymptotic Expansions

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§6.12(i) Exponential and Logarithmic Integrals

… ►For the function $\chi$ see §9.7(i). … ►

§6.12(ii) Sine and Cosine Integrals

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17: 10.74 Methods of Computation

… ►For large positive real values of $\nu$ the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used. … ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. … ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). … ►Similar considerations apply to the spherical Bessel functions and Kelvin functions. … ►Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of $2D$ graphs in $ℝ$ (e. …

18: 5.21 Methods of Computation

… ►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). … ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …

19: 6.18 Methods of Computation

… ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. … ►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). … ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. … ►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. …

20: 10.67 Asymptotic Expansions for Large Argument

§10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). … ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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