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31: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►Analogous to (18.39.7) the 3D Schrödinger operator is …where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►Here tridiagonal representations of simple Schrödinger operators play a similar role. The radial operator (18.39.28) …

32: Preface

… ► Stegun, editors); and to disseminate essentially the same information from a public website operated by NIST. … ►The Web pages contain many active links, for example, to the definitions of symbols within the DLMF, and to external sources of reviews, full texts of articles, and items of mathematical software. …

33: Bibliography L

… ►

Exact computation of the $3$-$j$ and $6$-$j$ symbols. Comput. Phys. Comm. 61 (3), pp. 350–360.

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Notes:

A Fortran program is included.

External Links:

Document, ZentralBlatt

Referenced by:

§34.15.

Encodings:

BibTeX

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Exact computation of the $9$-$j$ symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

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Notes:

Double-precision Fortran

External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

7th item.

Encodings:

BibTeX

… ►

Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (2), pp. 425–452.

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Notes:

For a Maple implementation of the algorithm given in this paper for Jack and zonal polynomials see Zeilberger (website).

External Links:

ISSN 0010-3616, Link, MathReview (Kazuhiro Hikami), ZentralBlatt

Referenced by:

§35.12.

Encodings:

BibTeX

… ►

Introduction to spectral theory: selfadjoint ordinary differential operators. Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I..

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Notes:

Translated from the Russian by Amiel Feinstein

External Links:

MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

… ►

Simplified recursive algorithm for Wigner $3j$ and $6j$ symbols. Phys. Rev. E 57 (6), pp. 7274–7277.

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External Links:

Document

Referenced by:

§34.13, §34.3(iii).

Encodings:

BibTeX

…

34: 3.11 Approximation Techniques

… ►Here the single prime on the summation symbol means that the first term is to be halved. … ► ${\delta }_{k,\mathrm{\ell }}$ being Kronecker’s symbol and the bar denoting complex conjugate. … ►In consequence of this structure the number of operations can be reduced to $nm=n{\log }_{2}n$ operations. …

35: 18.3 Definitions

… ►

As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations $A(x)p_n^{\prime \prime }(x)+B(x)p_n^{\prime }(x)+{\lambda }_n{p}_n(x)=0$, in Table 18.8.1.

…

36: 18.27 $q$-Hahn Class

… ►The $q$-Hahn class OP’s comprise systems of OP’s $\{p_n(x)\}$, $n=0,1,\mathrm{\dots },N$, or $n=0,1,2,\mathrm{\dots }$, that are eigenfunctions of a second order $q$-difference operator. …In the $q$-Hahn class OP’s the role of the operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative ${𝒟}_q$, as defined in (17.2.41). … ►

18.27.9 $$v_x=\frac{{(a^{-1}x,c^{-1}x;q)}_{\mathrm{\infty }}}{{(x,b{c}^{-1}x;q)}_{\mathrm{\infty }}},$$ , , ,

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_x$

Permalink:

http://dlmf.nist.gov/18.27.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.12 $$v_x=\frac{{(qx/c,-qx/d;q)}_{\mathrm{\infty }}}{{(q^{\alpha +1}x/c,-q^{\beta +1}x/d;q)}_{\mathrm{\infty }}},$$ $\alpha ,\beta >-1$, $c,d>0$.

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_x$

Permalink:

http://dlmf.nist.gov/18.27.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

… ►

18.27.14_1 $$h_n=\frac{{\left(aq\right)}^n}{1-ab{q}^{2n+1}}\frac{{(q,bq;q)}_n}{{(aq;q)}_n}\frac{{(ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq;q)}_{\mathrm{\infty }}}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$

Source:

Koekoek et al. (2010, (14.12.2))

Referenced by:

§18.27(iv), (18.28.27), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E14_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iv), §18.27 and Ch.18

…

37: 1.5 Calculus of Two or More Variables

… ►

1.5.3 $$\frac{\partial f}{\partial x}=D_{x}f=f_x=\underset{h\to 0}{lim}\frac{f(x+h,y)-f(x,y)}{h},$$

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Defines:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $D_x$: differential operator (locally)

Permalink:

http://dlmf.nist.gov/1.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

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1.5.4 $$\frac{\partial f}{\partial y}=D_{y}f=f_y=\underset{h\to 0}{lim}\frac{f(x,y+h)-f(x,y)}{h}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $D_x$: differential operator

Permalink:

http://dlmf.nist.gov/1.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

… ►For mathematicians the symbols $\theta$ and $\varphi$ now are usually interchanged. …

38: Bibliography S

… ►

An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.

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Notes:

This is a reprint, with corrections, of the first edition [University of Pennsylvania Press, Philadelphia, Pa., 1924].

External Links:

MathReview, ZentralBlatt

Referenced by:

§24.1, §24.6.

Encodings:

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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

External Links:

ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

Encodings:

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Arbitrary $3n-j$ symbols for $\mathrm{SU}(2)$ . Comput. Phys. Comm. 1 (3), pp. 207–215.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

9th item.

Encodings:

BibTeX

… ►

The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.

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External Links:

Document, MathReview Entry

Referenced by:

§1.18(viii).

Encodings:

BibTeX

… ►

Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

Encodings:

BibTeX

…