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1: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982).

2: 30.16 Methods of Computation

… ►If $|{\gamma }^2|$ is large we can use the asymptotic expansions in §30.9. … ►If $|{\gamma }^2|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$. …

3: 10.74 Methods of Computation

… ►If $x$ or $|z|$ is large compared with ${|\nu |}^2$, then the asymptotic expansions of §§10.17(i)–10.17(iv) are available. … ►And since there are no error terms they could, in theory, be used for all values of $z$; however, there may be severe cancellation when $|z|$ is not large compared with $n^2$. …

4: 4.13 Lambert $W$-Function

… ►where ${\ln }_k(z)=\ln \left(z\right)+2\pi \mathrm{i}k$. … ►

4.13.5_3 $${\left(1+W_0\left(z\right)\right)}^2=1-2\sum _{n=1}^{\mathrm{\infty }}\frac{n^{n-2}}{n!}{\left(-z\right)}^n,$$ .

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

Notes:

See https://vixra.org/abs/2308.0054.

Referenced by:

§4.13, Erratum (V1.1.11) for Additions

Permalink:

http://dlmf.nist.gov/4.13.E5_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.11):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

… ►As $\left|z\right|\to \mathrm{\infty }$ …where ${\xi }_k=\ln \left(z\right)+2\pi \mathrm{i}k$. For large enough $\left|z\right|$ the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. …

5: 15.12 Asymptotic Approximations

… ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$, …

6: 11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

… ►If the series on the right-hand side of (11.6.1) is truncated after $m(\ge 0)$ terms, then the remainder term $R_m(z)$ is $O\left(z^{\nu -2m-1}\right)$. If $\nu$ is real, $z$ is positive, and $m+\frac{1}{2}-\nu \ge 0$, then $R_m(z)$ is of the same sign and numerically less than the first neglected term. … ►

§11.6(ii) Large $|\nu |$, Fixed $z$

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§11.6(iii) Large $|\nu |$, Fixed $z/\nu$

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7: 10.40 Asymptotic Expansions for Large Argument

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10.40.2 $$K_{\nu }\left(z\right)\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.2

Referenced by:

§10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.41(iv), §10.45

Permalink:

http://dlmf.nist.gov/10.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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8: 2.11 Remainder Terms; Stokes Phenomenon

… ►For large $|z|$, with $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$ (), the Whittaker function of the second kind has the asymptotic expansion (§13.19) …

9: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for . … ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector . … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …

10: 10.41 Asymptotic Expansions for Large Order

… ►To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $z{\mathrm{e}}^{\pm \pi \mathrm{i}}$ is to replace $\eta$, ${(1+z^2)}^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm \mathrm{i}{(1+z^2)}^{\frac{1}{4}}$, and $-p$, respectively. …