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11: 9.16 Physical Applications

… ►The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. …The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). … ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. … ►An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).

12: Bibliography D

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Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.

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External Links:

ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

Referenced by:

§30.9(i).

Encodings:

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

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Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.

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External Links:

ISSN 1073-2772, Pdf, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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External Links:

ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

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Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.22(ii).

Encodings:

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13: 14.32 Methods of Computation

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Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

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14: 28.8 Asymptotic Expansions for Large $q$

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§28.8(iv) Uniform Approximations

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Barrett’s Expansions

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Dunster’s Approximations

►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►

15: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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§28.26(ii) Uniform Approximations

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16: 22.18 Mathematical Applications

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§22.18(iii) Uniformization and Other Parametrizations

… ►This circumvents the cumbersome branch structure of the multivalued functions $x(y)$ or $y(x)$, and constitutes the process of uniformization; see Siegel (1988, Chapter II). …

17: Bibliography T

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Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii).

Encodings:

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Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

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External Links:

ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

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Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

Ch.2.

Encodings:

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Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters. Methods Appl. Anal. 3 (3), pp. 335–344.

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External Links:

ISSN 1073-2772, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§8.12, §8.2(i), §8.2(ii), §8.6(ii).

Encodings:

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18: 10.41 Asymptotic Expansions for Large Order

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§10.41(ii) Uniform Expansions for Real Variable

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10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

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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, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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§10.41(iii) Uniform Expansions for Complex Variable

… ► … ►Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. …

19: 36.15 Methods of Computation

… ►Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. …

20: 33.12 Asymptotic Expansions for Large $\eta$

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§33.12(ii) Uniform Expansions

… ►The first set is in terms of Airy functions and the expansions are uniform for fixed $\mathrm{\ell }$ and , where $\delta$ is an arbitrary small positive constant. … ►The second set is in terms of Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$, and they are uniform for fixed $\mathrm{\ell }$ and $0\le z\le 1-\delta$, where $\delta$ again denotes an arbitrary small positive constant. … ►