essentially self-adjoint operator

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11: 32.2 Differential Equations

… ►An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. … ►

32.2.13 $$z(1-z)𝐼\left({\int }_{\mathrm{\infty }}^w\frac{dt}{\sqrt{t(t-1)(t-z)}}\right)=\sqrt{w(w-1)(w-z)}\left(\alpha +\frac{\beta z}{w^2}+\frac{\gamma (z-1)}{{(w-1)}^2}+(\delta -\frac{1}{2})\frac{z(z-1)}{{(w-z)}^2}\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: real, $\alpha$: arbitrary constant, $𝐼$: operator, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(iv), §32.2 and Ch.32

►where ►

32.2.14 $$𝐼=z(1-z)\frac{d^2}{{dz}^2}+(1-2z)\frac{d}{dz}-\frac{1}{4}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: real and $𝐼$: operator

Permalink:

http://dlmf.nist.gov/32.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(iv), §32.2 and Ch.32

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12: 34.12 Physical Applications

… ►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …

13: Mathematical Introduction

… ►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … ► ►►►►

$ℂ$	complex plane (excluding infinity).
…
$\mathrm{\Delta }$ (or ${\mathrm{\Delta }}_x$)	forward difference operator: $\mathrm{\Delta }f(x)=f(x+1)-f(x)$.
$\nabla$ (or ${\nabla }_x$)	backward difference operator: $\nabla f(x)=f(x)-f(x-1)$. (See also del operator in the Notations section.)
…

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14: Bibliography K

… ►

Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187, pp. 139–170 (Russian).

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Notes:

English translation: J. Math. Sci. 73(1995), no. 4, pp. 500–517

External Links:

ISSN 0373-2703, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt

Referenced by:

§32.12(ii).

Encodings:

BibTeX

… ►

Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

§18.9(iii).

Encodings:

BibTeX

… ►

Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

… ►

Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§18.36(i).

Encodings:

BibTeX

15: DLMF Project News

error generating summary

16: 1.10 Functions of a Complex Variable

… ►Lastly, if $a_n\ne 0$ for infinitely many negative $n$, then $z_0$ is an isolated essential singularity. … ► … ►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … ►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $ℂ$ with at most one exception. …

17: Notices

… ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. …

18: 25.17 Physical Applications

… ►This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. …

19: Errata

… ►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … ►The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

There has been disagreement about the identification of the Chebyshev polynomials of the third and fourth kinds, denoted $V_n\left(x\right)$ and $W_n\left(x\right)$, in published references. Originally, DLMF used the definitions given in (Andrews et al., 1999, Remark 2.5.3). However, those definitions were the reverse of those used by Mason and Handscomb (2003), Gautschi (2004) following Mason (1993) and Gautschi (1992), as was noted in several warnings added in Version 1.0.10 (August 7, 2015) of the DLMF. Since the latter definitions are more widely established, the DLMF is now adopting the definitions of Mason and Handscomb (2003). Essentially, what we previously denoted $V_n(x)$ is now written as $W_n(x)$, and vice-versa.

This notational interchange necessitated changes in Tables 18.3.1, 18.5.1, and 18.6.1, and in Equations (18.5.3), (18.5.4), (18.7.5), (18.7.6), (18.7.17), (18.7.18), (18.9.11), and (18.9.12).

… ►

A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

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20: 18.42 Software

… ►Also included is a website (CAOP) operated by a university department. …