applied to asymptotic expansions

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1: 2.11 Remainder Terms; Stokes Phenomenon

… ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. ►A simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): … ► …

2: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

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§11.6(ii) Large $|\nu |$, Fixed $z$

… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …

3: 29.20 Methods of Computation

… ►The eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^m\left(k^2\right)$, and the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7. … ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►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. …

4: 33.21 Asymptotic Approximations for Large $|r|$

§33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

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When $r\to \pm \mathrm{\infty }$ with , Equations (33.16.10)–(33.16.13) are combined with

33.21.1

${\zeta }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(2r/\nu )}^{\nu },$

${\xi }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(2r/\nu )}^{-\nu }$ , $r\to \mathrm{\infty }$,

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Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E1

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §33.21(i), §33.21 and Ch.33

33.21.2

${\zeta }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(-2r/\nu )}^{-\nu },$

${\xi }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(-2r/\nu )}^{\nu },$ $r\to -\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E2

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §33.21(i), §33.21 and Ch.33

Corresponding approximations for $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ as $r\to \mathrm{\infty }$ can be obtained via (33.16.17), and as $r\to -\mathrm{\infty }$ via (33.16.18).

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§33.21(ii) Asymptotic Expansions

►For asymptotic expansions of $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$ with $ϵ$ and $\mathrm{\ell }$ fixed, see Curtis (1964a, §6).

5: 2.3 Integrals of a Real Variable

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As $t\to a+$ the asymptotic expansions (2.3.14) apply, and each is infinitely differentiable. Again $\lambda$, $\mu$, and $p_0$ are positive.

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6: 8.21 Generalized Sine and Cosine Integrals

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§8.21(vi) Series Expansions

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Power-Series Expansions

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Spherical-Bessel-Function Expansions

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§8.21(viii) Asymptotic Expansions

… ►For the corresponding expansions for $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ apply (8.21.20) and (8.21.21). …

7: 13.9 Zeros

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13.9.16 $$a=-n-\frac{2}{\pi }\sqrt{zn}-\frac{2z}{{\pi }^2}+\frac{1}{2}b+\frac{1}{4}+\frac{z^2\left(\frac{1}{3}-4{\pi }^{-2}\right)+z-{(b-1)}^2+\frac{1}{4}}{4\pi \sqrt{zn}}+O\left(\frac{1}{n}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii), DLMF Update; Version 1.0.13, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16)

Permalink:

http://dlmf.nist.gov/13.9.E16

Encodings:

pMML, png, TeX

Errata (effective with 1.0.13):

In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.

Reported 2016-09-12 by Adri Olde Daalhuis

Errata (effective with 1.0.12):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right hand side, $\sim$ was replaced by $=$.

Reported 2016-07-11 by Rudi Weikard

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

…

8: 18.15 Asymptotic Approximations

… ►For more powerful asymptotic expansions as $n\to \mathrm{\infty }$ in terms of elementary functions that apply uniformly when , $-1+\delta \le t\le 1-\delta$, or , where $t=x/\sqrt{2n+1}$ and $\delta$ is again an arbitrary small positive constant, see §§12.10(i)–12.10(iv) and 12.10(vi). And for asymptotic expansions as $n\to \mathrm{\infty }$ in terms of Airy functions that apply uniformly when or , see §§12.10(vii) and 12.10(viii). …

9: 35.10 Methods of Computation

… ►For small values of $\Vert 𝐓\Vert$ the zonal polynomial expansion given by (35.8.1) can be summed numerically. For large $\Vert 𝐓\Vert$ the asymptotic approximations referred to in §35.7(iv) are available. ►Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

10: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …as $\mu$ ($=\sqrt{-2a}$) $\to \mathrm{\infty }$, $s$ fixed. …