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11: 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. …

12: 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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On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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External Links:

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

Referenced by:

§2.7(ii).

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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Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:

FullText, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

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13: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

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14: Software Index

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	Open Source													With Book						Commercial
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10.77(iii) Bessel Functions–Real Order and Argument	✓	✓	✓	✓	✓	✓		✓	✓		✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	NMS
10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions	✓	✓	✓									a	✓	✓			✓	✓	✓	✓	✓	✓	✓
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10.77(vi) Bessel Functions–Imaginary Order and Real Argument	✓	✓										✓	✓								✓	✓	✓
10.77(viii) Bessel Functions–Complex Order and Argument	✓	✓										✓	✓								✓	✓	✓
…

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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15: Philip J. Davis

… ►At CalTech, John Todd dedicated himself to the training of new researchers in numerical analysis, and Olga Taussky, who had been a full-time NBS consultant influential in establishing the field of matrix theory, became the first woman at CalTech to attain the academic rank of full professor. … ►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. …

16: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

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17: 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 $\alpha$). …

18: 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). …

19: 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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20: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

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