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21: 26.13 Permutations: Cycle Notation

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26.13.2

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Referenced by:

§26.13, §26.13, §26.13

Permalink:

http://dlmf.nist.gov/26.13.E2

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pMML, png, TeX

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Annotations for §26.13 and Ch.26

… ►Cycles of length one are fixed points. … ►An element of ${𝔖}_n$ with $a_1$ fixed points, $a_2$ cycles of length $2,\mathrm{\dots },a_n$ cycles of length $n$, where $n=a_1+2a_2+\mathrm{\cdots }+n{a}_n$, is said to have cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$. … ►A derangement is a permutation with no fixed points. The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: …

22: 2.1 Definitions and Elementary Properties

… ►Let $𝐗$ be a point set with a limit point $c$. As $x\to c$ in $𝐗$ … ►If $c$ is a finite limit point of $𝐗$, then … ►Similarly for finite limit point $c$ in place of $\mathrm{\infty }$. … ►where $c$ is a finite, or infinite, limit point of $𝐗$. …

23: 13.27 Mathematical Applications

… ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …

24: Mark J. Ablowitz

… ►ODEs which do not have moveable branch point singularities. …

25: 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). … ►Let $P(t)=P(x(t),y(t))$ be any point on the projected spiral. …

26: 21.7 Riemann Surfaces

… ►Consider the set of points in ${ℂ}^2$ that satisfy the equation …Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. …This compact curve may have singular points, that is, points at which the gradient of $\tilde{P}$ vanishes. … ►The zeros ${\lambda }_j$, $j=1,2,\mathrm{\dots },2g+1$ of $Q(\lambda )$ specify the finite branch points $P_j$, that is, points at which ${\mu }_j=0$, on the Riemann surface. Denote the set of all branch points by $B=\{P_1,P_2,\mathrm{\dots },P_{2g+1},P_{\mathrm{\infty }}\}$. …

27: 10.20 Uniform Asymptotic Expansions for Large Order

… ►Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2. … ►The points $P_1,P_2$ where these curves intersect the imaginary axis are $\pm \mathrm{i}c$, where … ►

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Figure 10.20.1: $z$-plane. $P_1$ and $P_2$ are the points $\pm \mathrm{i}c$. … Magnify

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Figure 10.20.2: $\zeta$-plane. $E_1$ and $E_2$ are the points ${\mathrm{e}}^{\mp \pi \mathrm{i}/3}{(3\pi /2)}^{2/3}.$ Magnify

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28: 10.41 Asymptotic Expansions for Large Order

… ►Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the $z$-plane and the $\eta$-plane. …Thus $B$ is the point $z=c$, where $c$ is given by (10.20.18). …

29: Bibliography O

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Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.

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External Links:

Document, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Second-order linear differential equations with two turning points. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 137–174.

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External Links:

FullText, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§2.8(vi).

Encodings:

BibTeX

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Improved error bounds for second-order differential equations with two turning points. J. Res. Nat. Bur. Standards Sect. B 80B (4), pp. 437–440.

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External Links:

MathReview (Peter Deuflhard), ZentralBlatt

Referenced by:

§2.8(vi).

Encodings:

BibTeX

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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

Encodings:

BibTeX

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Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:

FullText, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

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30: 2.9 Difference Equations

… ►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …