Stokes line

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1: 2.11 Remainder Terms; Stokes Phenomenon

… ►The choice of ${\alpha }^{\prime }$ and ${\beta }^{\prime }$ is facilitated by a knowledge of the relevant Stokes lines; see §2.11(iv) below. … ►Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ($\theta =\pi$ in the present example). … ►The relevant Stokes lines are $\mathrm{ph}z=\pm \pi$ for $w_1(z)$, and $\mathrm{ph}z=0,2\pi$ for $w_2(z)$. … ►For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.

2: 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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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:

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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. … ►The function $\mathrm{\Gamma }(a,z)$, with $|\mathrm{ph}a|\le \frac{1}{2}\pi$ and $\mathrm{ph}z=\frac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)) on the critical line $\mathrm{\Re }s=\frac{1}{2}$. …

4: Bibliography D

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.

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External Links:

ISSN 0962-8444, FullText, MathReview (Alexander Tovbis), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

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