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21: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For $T$ to be actually self adjoint it is necessary to also show that $𝒟(T^{\ast })=𝒟(T)$, as it is often the case that $T$ and $T^{\ast }$ have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator $\frac{d}{dx}$. …

22: Bibliography G

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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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23: Bibliography K

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Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.

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Notes:

This system provides a uniform interface in C++, named e_float, to perform arithmetic operations and to calculate mathematical functions that are implemented in several different MP packages. It is a free, open-source, multiple precision package that allows the computation of many special functions. It supports calculations with 30….300 decimal digits and is interoperable with Microsoft’s CLR, Python and Mathematica.

External Links:

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

Referenced by:

Erratum (V1.0.9) for References, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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24: Bibliography R

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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

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External Links:

ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)

Referenced by:

§1.18(x).

Encodings:

BibTeX

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:

ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

Referenced by:

§29.20(ii).

Encodings:

BibTeX

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On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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25: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
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$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
$G\times H$	Cartesian product of groups $G$ and $H$, that is, the set of all pairs of elements $(g,h)$ with group operation $(g_1,h_1)+(g_2,h_2)=(g_1+g_2,h_1+h_2)$.

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26: 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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27: 2.1 Definitions and Elementary Properties

… ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. …

28: 1.15 Summability Methods

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1.15.47 $${𝐼}^{\alpha }f(x)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^x{(x-t)}^{\alpha -1}f(t)dt.$$

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Defines:

${𝐼}^{\alpha }$: fractional integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.15(vi), §1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)

Permalink:

http://dlmf.nist.gov/1.15.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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1.15.48 $${𝐼}^{\alpha }{𝐼}^{\beta }={𝐼}^{\alpha +\beta },$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\beta >0$.

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Symbols:

$\mathrm{\Re }$: real part and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vi)

Permalink:

http://dlmf.nist.gov/1.15.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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1.15.50 $${𝐼}^{\alpha }f(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{k!}{\mathrm{\Gamma }\left(k+\alpha +1\right)}{a}_k{x}^{k+\alpha }.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)

Permalink:

http://dlmf.nist.gov/1.15.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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1.15.51 $${𝐷}^{\alpha }f(x)=\frac{d^n}{{dx}^n}{𝐼}^{n-\alpha }f(x),$$

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Defines:

${𝐷}^{\alpha }$: fractional derivative (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E51

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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1.15.52 $${𝐷}^k{𝐼}^{\alpha }={𝐷}^n{𝐼}^{\alpha +n-k},$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$k$: integer, $n$: nonnegative integer, ${𝐼}^{\alpha }$: fractional integral and ${𝐷}^{\alpha }$: fractional derivative

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E52

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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29: 1.16 Distributions

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1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

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Defines:

$𝐃$: vector differential operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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30: Guide to Searching the DLMF

… ►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: … ►

and and or

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adj, prec/n, and near/n, where n is any positive natural number.

… ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. … ►DLMF search recognizes just the essential font differences, that is, the font style differences deemed important for the DLMF contents: …