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1—10 of 19 matching pages

1: 36.5 Stokes Sets

§36.5 Stokes Sets

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§36.5(i) Definitions

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§36.5(ii) Cuspoids

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Elliptic Umbilic Stokes Set (Codimension three)

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§36.5(iv) Visualizations

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2: 7.20 Mathematical Applications

… ►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). …

3: 8.22 Mathematical Applications

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8.22.1 $$F_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }{z}^{1-p}{E}_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }\mathrm{\Gamma }(1-p,z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $F_p\left(z\right)$: terminant function, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(i), §8.22 and Ch.8

►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. …

4: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b) when $\rho >0$, and Wong and Zhao (1999a) when . … ►This reference includes exponentially-improved asymptotic expansions for $E_{a,b}\left(z\right)$ when $|z|\to \mathrm{\infty }$, together with a smooth interpretation of Stokes phenomena. …

5: 2.11 Remainder Terms; Stokes Phenomenon

§2.11 Remainder Terms; Stokes Phenomenon

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§2.11(iv) Stokes Phenomenon

… ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. Where should the change-over take place? Can it be accomplished smoothly? … ►For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004). …

6: Bibliography P

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Stokes phenomenon demystified. Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.

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External Links:

ISSN 0905-5628, MathReview Entry, ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.

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External Links:

ISSN 0962-8444, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.

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External Links:

ISSN 0377-0427, Document, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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The Stokes phenomenon associated with the Hurwitz zeta function $\zeta (s,a)$ . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.

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External Links:

ISSN 1364-5021, Document, MathReview (Marc Prevost), ZentralBlatt

Referenced by:

(25.11.43), §25.11(xii).

Encodings:

BibTeX

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7: Bibliography W

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Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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Smoothing of Stokes’s discontinuity for the generalized Bessel function. Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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The Stokes set of the cusp diffraction catastrophe. J. Phys. A 13 (9), pp. 2913–2928.

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External Links:

ISSN 0305-4470, Document, MathReview (I. Stewart), ZentralBlatt

Referenced by:

§36.5(ii).

Encodings:

BibTeX

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8: Bibliography I

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Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16 (1), pp. 363–386.

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External Links:

ISSN 0951-7715, Document, MathReview (Youmin Lu), ZentralBlatt

Referenced by:

§32.12(ii).

Encodings:

BibTeX

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9: Bibliography O

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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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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.

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External Links:

ISSN 0377-0427, Document, MathReview (Jorge Mozo Fernández), ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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10: 7.12 Asymptotic Expansions

… ►For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by $z^2$; compare (7.11.2). ►For re-expansions of the remainder terms leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv) and use (7.11.3). …