Stieltjes electrostatic interpretation

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21: 9.10 Integrals

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§9.10(vii) Stieltjes Transforms

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22: 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. …

23: 15.17 Mathematical Applications

… ►For special values of $\alpha$ and $\beta$ there are many group-theoretic interpretations. …

24: Bibliography W

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

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BibTeX

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Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions. Proc. Roy. Soc. London Ser. A 458, pp. 625–644.

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ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

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BibTeX

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25: 18.1 Notation

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Stieltjes–Wigert: $S_n(x;q)$.

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26: Bibliography J

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

BibTeX

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27: Bibliography N

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

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BibTeX

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28: Bibliography V

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

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BibTeX

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29: 15.2 Definitions and Analytical Properties

… ►This formula is also valid when $c=-m-\mathrm{\ell }$, $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, provided that we use the interpretation …(Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does $𝐅(a,b;c;z)$, which is analytic at $c=0,-1,-2,\mathrm{\dots }$.) ►For comparison of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. … ►In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F(-m,b;-m;z)$ is equal to the first $m+1$ terms of the Maclaurin series for ${(1-z)}^{-b}$.

30: 22.19 Physical Applications

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§22.19(v) Other Applications

►Numerous other physical or engineering applications involving Jacobian elliptic functions, and their inverses, to problems of classical dynamics, electrostatics, and hydrodynamics appear in Bowman (1953, Chapters VII and VIII) and Lawden (1989, Chapter 5). …