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31: 2.3 Integrals of a Real Variable

… ►Then … ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►When $p(t)$ is real and $x$ is a large positive parameter, the main contribution to the integral … ►When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in … ► $k$ ($\le \mathrm{\infty }$) and $\lambda$ are positive constants, $\alpha$ is a variable parameter in an interval ${\alpha }_1\le \alpha \le {\alpha }_2$ with ${\alpha }_1\le 0$ and , and $x$ is a large positive parameter. …

32: Bibliography B

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Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.

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External Links:

Document, ZentralBlatt

Referenced by:

§36.14(iii), §9.16.

Encodings:

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Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.

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Referenced by:

§36.14(i), §9.16.

Encodings:

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Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

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External Links:

ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

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Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.

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External Links:

MathReview (L. Fox)

Referenced by:

3rd item.

Encodings:

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Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

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33: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

Encodings:

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

Encodings:

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Coulomb Functions for Large Values of the Parameter $\eta$ . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.

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Notes:

Rep. CRT-526

External Links:

MathReview (A. Erdélyi)

Referenced by:

§33.7.

Encodings:

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34: 10.1 Special Notation

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$m,n$	integers. In §§10.47–10.71 $n$ is nonnegative.
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$\nu$	real or complex parameter (the order).
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… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

35: 2.1 Definitions and Elementary Properties

… ►If $\sum a_s{x}^{-s}$ converges for all sufficiently large $|x|$, then it is automatically the asymptotic expansion of its sum as $x\to \mathrm{\infty }$ in $ℂ$. … ►

§2.1(iv) Uniform Asymptotic Expansions

… ►The asymptotic property may also hold uniformly with respect to parameters. Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $𝐔$, and for each nonnegative integer $n$ … ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. …

36: 28.33 Physical Applications

… ►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. …with $W(x,y,t)={\mathrm{e}}^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with $k^2={\omega }^2\rho /\tau$. … ►

28.33.2 $$V_n(\xi ,\eta )=\left(c_n{\mathrm{M}}_n^{(1)}(\xi ,\sqrt{q})+d_n{\mathrm{M}}_n^{(2)}(\xi ,\sqrt{q})\right){\mathrm{me}}_n(\eta ,q),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $q=h^2$: parameter, $n$: integer, $V(\xi ,\eta )$: solution, $d_n$: constants, $\eta$: variable, $\xi$: variable and $c_n$: constants

Permalink:

http://dlmf.nist.gov/28.33.E2

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See also:

Annotations for §28.33(ii), §28.33 and Ch.28

… ►If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as $\cos \left(2\omega t\right)$. … ►In particular, the equation is stable for all sufficiently large values of $\omega$. …