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21: 9.14 Incomplete Airy Functions

… ►For information, including asymptotic approximations, computation, and applications, see Levey and Felsen (1969), Constantinides and Marhefka (1993), and Michaeli (1996).

22: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

23: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►In particular, asymptotic formulas in terms of elementary functions are given when $z=x$ is real and fixed. … ►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 two asymptotic expansions of $M_n(nx;\beta ,c)$ as $n\to \mathrm{\infty }$, with $\beta$ and $c$ fixed, see Jin and Wong (1998) and Wang and Wong (2011). … ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

24: 33.18 Limiting Forms for Large $\mathrm{\ell }$

… ►

$f(ϵ,\mathrm{\ell };r)\sim {\displaystyle \frac{{(2r)}^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!}},$

►

$h(ϵ,\mathrm{\ell };r)\sim {\displaystyle \frac{(2\mathrm{\ell })!}{\pi {(2r)}^{\mathrm{\ell }}}}.$

25: Richard B. Paris

… ►Paris published numerous papers in asymptotics, special functions, and MHD instability theory. His books are Asymptotics of High Order Differential Equations (with A. … Wood), published by Longman Scientific and Technical in 1986, and Asymptotics and Mellin-Barnes Integrals (with D. …

26: 10.30 Limiting Forms

… ►

10.30.1 $$I_{\nu }\left(z\right)\sim {(\frac{1}{2}z)}^{\nu }/\mathrm{\Gamma }\left(\nu +1\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.7

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E1

Encodings:

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See also:

Annotations for §10.30(i), §10.30 and Ch.10

►

10.30.2 $$K_{\nu }\left(z\right)\sim \frac{1}{2}\mathrm{\Gamma }\left(\nu \right){(\frac{1}{2}z)}^{-\nu },$$ $\mathrm{\Re }\nu >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.9

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E2

Encodings:

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See also:

Annotations for §10.30(i), §10.30 and Ch.10

►

10.30.3 $$K_0\left(z\right)\sim -\ln z.$$

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Symbols:

$\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.8

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E3

Encodings:

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See also:

Annotations for §10.30(i), §10.30 and Ch.10

… ►

10.30.4 $$I_{\nu }\left(z\right)\sim {\mathrm{e}}^z/\sqrt{2\pi z},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.30.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(ii), §10.30 and Ch.10

►

10.30.5 $$I_{\nu }\left(z\right)\sim {\mathrm{e}}^{\pm (\nu +\frac{1}{2})\pi \mathrm{i}}{\mathrm{e}}^{-z}/\sqrt{2\pi z},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.30.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(ii), §10.30 and Ch.10

…

27: 10.67 Asymptotic Expansions for Large Argument

§10.67 Asymptotic Expansions for Large Argument

►

§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 $\nu =0$

…

28: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

… ►

13.19.1 $$M_{\kappa ,\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}x}{x}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{x}^{-s},$$ $\mu -\kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

… ►

13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

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See also:

Annotations for §13.19 and Ch.13

… ►For an asymptotic expansion of $W_{\kappa ,\mu }\left(z\right)$ as $z\to \mathrm{\infty }$ that is valid in the sector $|\mathrm{ph}z|\le \pi -\delta$ and where the real parameters $\kappa$, $\mu$ are subject to the growth conditions $\kappa =o\left(z\right)$, $\mu =o\left(\sqrt{z}\right)$, see Wong (1973a).

29: Bibliography O

… ►

Asymptotic Enumeration Methods. In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.), pp. 1063–1229.

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External Links:

ISBN 0-444-82351-4, Pdf, MathReview (Konrad Engel), ZentralBlatt

Referenced by:

Ch.2.

Encodings:

BibTeX

… ►

On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

… ►

A paradox in asymptotics. SIAM J. Math. Anal. 1 (4), pp. 533–534.

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External Links:

Document, MathReview (L. Hoischen), ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

BibTeX

… ►

Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.

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External Links:

ISSN 0036-1445, Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

… ►

Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii), §2.7(ii).

Encodings:

BibTeX

…

30: 6.13 Zeros

… ►Values of $c_1$ and $c_2$ to 30D are given by MacLeod (1996b). … ►

6.13.2 $$c_k,s_k\sim \alpha +\frac{1}{\alpha }-\frac{16}{3}\frac{1}{{\alpha }^3}+\frac{1673}{15}\frac{1}{{\alpha }^5}-\frac{\mathrm{5\hspace{0.33em}07746}}{105}\frac{1}{{\alpha }^7}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $c_k$: zeros of $\mathrm{Ci}\left(x\right)$ and $s_k$: zeros of $\mathrm{Si}\left(x\right)$

Referenced by:

§6.13, §6.18(iii)

Permalink:

http://dlmf.nist.gov/6.13.E2

Encodings:

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See also:

Annotations for §6.13 and Ch.6

…