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31: 32.14 Combinatorics

… ►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). … ►See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

32: 31.11 Expansions in Series of Hypergeometric Functions

… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. … ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. … ►In this case the accessory parameter $q$ is a root of the continued-fraction equation … ►

§31.11(v) Doubly-Infinite Series

…

33: 18.39 Applications in the Physical Sciences

… ►Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced. … ►By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are … ►There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi). … ►Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below. … ►thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure $dr$. …

34: 28.15 Expansions for Small $q$

… ►

§28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

►Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$: ►

28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

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Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

… ►

28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(ii), §28.15 and Ch.28

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35: Bibliography M

… ►

On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

Encodings:

BibTeX

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

Encodings:

BibTeX

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

… ►

Table of the ratio: Area to bounding ordinate, for any portion of normal curve. Biometrika 18, pp. 395–400.

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External Links:

FullText, ZentralBlatt

Referenced by:

§7.8.

Encodings:

BibTeX

…

36: Bibliography L

… ►

Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.

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Referenced by:

§19.14(ii).

Encodings:

BibTeX

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Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

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External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

Encodings:

BibTeX

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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:

ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

Encodings:

BibTeX

… ►

Algorithm 567: Extended-range arithmetic and normalized Legendre polynomials [A1], [C1]. ACM Trans. Math. Software 7 (1), pp. 141–146.

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Notes:

Double-precision Fortran, minumum accuracy 13S

External Links:

ISSN 0098-3500, Document, GAMS (module 567; package TOMS)

Referenced by:

6th item.

Encodings:

BibTeX

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The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

…

37: 3.2 Linear Algebra

… ►With $𝐲={[y_1,y_2,\mathrm{\dots },y_n]}^{\mathrm{T}}$ the process of solution can then be regarded as first solving the equation $𝐋𝐲=𝐛$ for $𝐲$ (forward elimination), followed by the solution of $𝐔𝐱=𝐲$ for $𝐱$ (back substitution). … ►A normalized eigenvector has Euclidean norm 1; compare (3.2.13) with $p=2$. … ►where $𝐱$ and $𝐲$ are the normalized right and left eigenvectors of $𝐀$ corresponding to the eigenvalue $\lambda$. … ►Define the Lanczos vectors ${𝐯}_j$ and coefficients ${\alpha }_j$ and ${\beta }_j$ by ${𝐯}_0=\mathrm{𝟎}$, a normalized vector ${𝐯}_1$ (perhaps chosen randomly), ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$, and for $j=1,2,\mathrm{\dots },n-1$ by the recursive scheme …

38: 22.18 Mathematical Applications

… ►

§22.18(i) Lengths and Parametrization of Plane Curves

… ►

§22.18(iii) Uniformization and Other Parametrizations

… ► … ►The special case $y^2=(1-x^2)(1-k^2{x}^2)$ is in Jacobian normal form. For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …

39: Bibliography H

… ►

Algorithm 442: Normal deviate. Comm. ACM 16 (1), pp. 51–52.

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Notes:

Includes Algol program, minimum accuracy 24D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(ii).

Encodings:

BibTeX

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Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.

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Notes:

Double-precision Fortran, maximum accuracy 12D.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

… ►

Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.

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External Links:

MathReview (Henrik Pedersen), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

BibTeX

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Estimates of the stability intervals for Hill’s equation. Proc. Amer. Math. Soc. 14 (6), pp. 930–932.

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External Links:

FullText, MathReview (P. Ungar), ZentralBlatt

Referenced by:

§28.29(iii).

Encodings:

BibTeX

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Differential Equations: A Modern Approach. Holt, Rinehart and Winston, New York.

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Notes:

Reprinted by Dover in 1975.

External Links:

MathReview (Z. Opial), ZentralBlatt

Referenced by:

§29.3(vi).

Encodings:

BibTeX

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40: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

… ►

Kernel Functions

… ►Here ${\tilde{\kappa }}_m$ is a normalization constant and $C$ is the contour of Example 1. … ►leads to the kernel equation … ►