# 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(ii) Large $|\nu|$, Fixed $z$
More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions2.1(v)). …
##### 3: 29.20 Methods of Computation
The eigenvalues $a^{m}_{\nu}\left(k^{2}\right)$, $b^{m}_{\nu}\left(k^{2}\right)$, and the Lamé functions $\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)$, 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 $n\to\infty$. … 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(i) Limiting Forms
• (b)

When $r\to\pm\infty$ with $\epsilon<0$, Equations (33.16.10)–(33.16.13) are combined with

33.21.1
$\zeta_{\ell}(\nu,r)\sim e^{-r/\nu}(2r/\nu)^{\nu},$
$\xi_{\ell}(\nu,r)\sim e^{r/\nu}(2r/\nu)^{-\nu}$ , $r\to\infty$,
33.21.2
$\zeta_{\ell}(-\nu,r)\sim e^{r/\nu}(-2r/\nu)^{-\nu},$
$\xi_{\ell}(-\nu,r)\sim e^{-r/\nu}(-2r/\nu)^{\nu},$ $r\to-\infty$.

Corresponding approximations for $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$ as $r\to\infty$ can be obtained via (33.16.17), and as $r\to-\infty$ via (33.16.18).

• ###### §33.21(ii) AsymptoticExpansions
For asymptotic expansions of $f\left(\epsilon,\ell;r\right)$ and $h\left(\epsilon,\ell;r\right)$ as $r\to\pm\infty$ with $\epsilon$ and $\ell$ fixed, see Curtis (1964a, §6).
##### 5: 2.3 Integrals of a Real Variable
• (b)

As $t\to a+$ the asymptotic expansions (2.3.14) apply, and each is infinitely differentiable. Again $\lambda$, $\mu$, and $p_{0}$ are positive.

• ##### 6: 8.21 Generalized Sine and Cosine Integrals
###### §8.21(viii) AsymptoticExpansions
For the corresponding expansions for $\mathrm{si}\left(a,z\right)$ and $\mathrm{ci}\left(a,z\right)$ apply (8.21.20) and (8.21.21). …
##### 7: 13.9 Zeros
13.9.16 $a=-n-\frac{2}{\pi}\sqrt{zn}-\frac{2z}{\pi^{2}}+\tfrac{1}{2}b+\tfrac{1}{4}+% \frac{z^{2}\left(\frac{1}{3}-4\pi^{-2}\right)+z-(b-1)^{2}+\frac{1}{4}}{4\pi% \sqrt{zn}}+O\left(\frac{1}{n}\right),$
##### 8: 18.15 Asymptotic Approximations
For more powerful asymptotic expansions as $n\to\infty$ in terms of elementary functions that apply uniformly when $1+\delta\leq t<\infty$, $-1+\delta\leq t\leq 1-\delta$, or $-\infty, where $t=\ifrac{x}{\sqrt{2n+1}}$ 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 $n\to\infty$ in terms of Airy functions that apply uniformly when $-1+\delta\leq t<\infty$ or $-\infty, see §§12.10(vii) and 12.10(viii). …
##### 9: 12.11 Zeros
###### §12.11(ii) AsymptoticExpansions 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) AsymptoticExpansions for Large Parameter
For large negative values of $a$ the real zeros of $U\left(a,x\right)$, $U'\left(a,x\right)$, $V\left(a,x\right)$, and $V'\left(a,x\right)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …as $\mu$ ($=\sqrt{-2a}$) $\to\infty$, $s$ fixed. …
##### 10: 18.16 Zeros
For an asymptotic expansion of $x_{n,m}$ as $n\to\infty$ that applies uniformly for $m=1,2,\dots,\left\lfloor\tfrac{1}{2}n\right\rfloor$, see Olver (1959, §14(i)). …