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1: 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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See also:

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: …

2: 3.8 Nonlinear Equations

… ►and the solutions are called fixed points of $\varphi$. … ►

§3.8(vii) Systems of Nonlinear Equations

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§3.8(viii) Fixed-Point Iterations: Fractals

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3: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

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Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1), pp. 177–232 (French).

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§22.19(iv).

Encodings:

BibTeX

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4: 36.4 Bifurcation Sets

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§36.4(i) Formulas

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Critical Points for Cuspoids

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Critical Points for Umbilics

… ►This is the codimension-one surface in $𝐱$ space where critical points coalesce, satisfying (36.4.1) and … ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid …

5: 12.11 Zeros

… ►For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. …as $\mu$ ($=\sqrt{-2a}$) $\to \mathrm{\infty }$, $s$ fixed. … ►

12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

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See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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6: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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External Links:

ISSN 1095-7170, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(v).

Encodings:

BibTeX

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§28.7.

Encodings:

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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7: Bibliography G

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Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33 (1), pp. 1–55.

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§32.10(ii), §32.10(iv), §32.7(ii).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys 23 (1), pp. 5–48.

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External Links:

ISSN 0360-0300, Document

Referenced by:

§3.1(i).

Encodings:

BibTeX

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Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..

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Notes:

Published for the National Science Foundation by the Israel Program for Scientific Translations, 1960.

External Links:

MathReview, ZentralBlatt

Referenced by:

§22.19(iv).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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8: 25.12 Polylogarithms

… ►In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. … ►

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

… ►For each fixed complex $s$ the series defines an analytic function of $z$ for . …