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11: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

Encodings:

BibTeX

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Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §2.11(v), §9.7(v).

Encodings:

BibTeX

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

BibTeX

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Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v), §32.12(i).

Encodings:

BibTeX

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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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12: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. …The order of the approximating Bessel functions, or modified Bessel functions, is $1/(\lambda +2)$, except in the case when $g(z)$ has a double pole at $z_0$. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …

13: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

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10.57.1 $${𝗃}_n^{\prime }\left((n+\frac{1}{2})z\right)=\frac{{\pi }^{\frac{1}{2}}}{{((2n+1)z)}^{\frac{1}{2}}}{J}_{n+\frac{1}{2}}^{\prime }\left((n+\frac{1}{2})z\right)-\frac{{\pi }^{\frac{1}{2}}}{{((2n+1)z)}^{\frac{3}{2}}}{J}_{n+\frac{1}{2}}\left((n+\frac{1}{2})z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.57

Permalink:

http://dlmf.nist.gov/10.57.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.57 and Ch.10

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14: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

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§10.41(i) Asymptotic Forms

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§10.41(ii) Uniform Expansions for Real Variable

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15: 10.26 Graphics

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§10.26(i) Real Order and Variable

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§10.26(ii) Real Order, Complex Variable

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§10.26(iii) Imaginary Order, Real Variable

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Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.8: ${\tilde{I}}_1\left(x\right)$, ${\tilde{K}}_1\left(x\right)$, $0.01\le x\le 3$. Magnify

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16: 14.6 Integer Order

§14.6 Integer Order

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§14.6(i) Nonnegative Integer Orders

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§14.6(ii) Negative Integer Orders

… ►For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). …

17: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

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10.69.2 $${\mathrm{ber}}_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{{(2\pi \nu \xi )}^{1/2}}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.37

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

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10.69.3 $${\ker }_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }\left(\nu x\right)\sim {\mathrm{e}}^{-\nu \xi }{\left(\frac{\pi }{2\nu \xi }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.38

Permalink:

http://dlmf.nist.gov/10.69.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

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18: Bibliography Q

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A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

Encodings:

BibTeX

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A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.

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External Links:

ISSN 0168-9673, Document, MathReview (Dmitrii Petrovich Novikov), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

BibTeX

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.

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External Links:

Document, Link

Referenced by:

§18.38(iii).

Encodings:

BibTeX

19: 2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols

►Let $𝐗$ be a point set with a limit point $c$. As $x\to c$ in $𝐗$ … ►

§2.1(ii) Integration and Differentiation

►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. …

20: 14.1 Special Notation

§14.1 Special Notation

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$x$, $y$, $\tau$	real variables.
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$m$, $n$	unless stated otherwise, nonnegative integers, used for order and degree, respectively.
$\mu$, $\nu$	general order and degree, respectively.
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$𝐅(a,b;c;z)$	Olver’s scaled hypergeometric function: $F(a,b;c;z)/\mathrm{\Gamma }\left(c\right)$.

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