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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Self-Adjoint and Symmetric Operators

… ►One then needs a self-adjoint extension of a symmetric operator to carry out its spectral theory in a mathematically rigorous manner. ►An essential feature of such symmetric operators is that their eigenvalues $\lambda$ are real, and eigenfunctions … ►

Self-adjoint extensions of a symmetric Operator

… ► We have a direct sum of linear spaces: $𝒟(T^{\ast })=𝒟(T^{\ast \ast })+N_{\mathrm{i}}+N_{-\mathrm{i}}$. …

2: 35.2 Laplace Transform

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35.2.3

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Defines:

$\ast$: convolution operator (locally)

Symbols:

$\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, : less-than for matrices, $f(𝐗)$: complex-valued function of matrix argument and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

3: 18.38 Mathematical Applications

… ►Define operators $K_0$ and $K_1$ acting on symmetric Laurent polynomials by $K_0=L$ ($L$ given by (18.28.6_2)) and $(K_{1}f)(z)=(z+z^{-1})f(z)$. … ►The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_n(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d|q)$, where $R_n(z)$ is given in (18.28.1_5). …

4: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►Real symmetric ($𝐀={𝐀}^{\mathrm{T}}$) and Hermitian ($𝐀={𝐀}^{\mathrm{H}}$) matrices are self-adjoint operators on ${𝐄}_n$. …

5: 18.39 Applications in the Physical Sciences

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§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

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6: 21.7 Riemann Surfaces

… ►Define the operation ►

21.7.12 $$T_1\ominus T_2=(T_1\cup T_2)\setminus (T_1\cap T_2).$$

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Symbols:

$\cap$: intersection, $\setminus$: set subtraction, $\cup$: union and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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21.7.14 $$𝜼(T_1\ominus T_2)=𝜼(T_1)+𝜼(T_2),$$

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Symbols:

$T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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21.7.15 $$4{𝜼}^1(T)\cdot {𝜼}^2(T)=\frac{1}{2}\left(|T\ominus U|-g-1\right)(mod2),$$

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Symbols:

$g$: positive integer, $U$: subset of $B$ and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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7: 2.6 Distributional Methods

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2.6.32 $${\int }_0^{\mathrm{\infty }}\frac{f(t)}{{(t+z)}^{\rho }}dt,$$ $\rho >0$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $f(t)$: locally integrable function

Permalink:

http://dlmf.nist.gov/2.6.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …

8: 32.2 Differential Equations

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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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§32.2(v) Symmetric Forms

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9: Bibliography G

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Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

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Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

Encodings:

BibTeX

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The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.

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External Links:

ISSN 0002-9939, Link, MathReview (Raj Wilson), ZentralBlatt

Referenced by:

§18.28(xi).

Encodings:

BibTeX

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WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.

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External Links:

FullText, MathReview (Ioannis K. Purnaras), ZentralBlatt

Referenced by:

§18.15(v), §18.15(v).

Encodings:

BibTeX

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10: 3.1 Arithmetics and Error Measures

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Rounding

… ►Symmetric rounding or rounding to nearest of $x$ gives $x_{-}$ or $x_{+}$, whichever is nearer to $x$, with maximum relative error equal to the machine precision $\frac{1}{2}{ϵ}_M=2^{-p}$. … ►The elementary arithmetical operations on intervals are defined as follows: …