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21: 27.16 Cryptography

… ►For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. …

22: 1.4 Calculus of One Variable

… ►

Square-Integrable Functions

►A function $f(x)$ is square-integrable if … ►

1.4.33 $${𝒱}_{a,b}\left(f\right)=sup\sum _{j=1}^n\left|f(x_j)-f(x_{j-1})\right|,$$

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Defines:

$𝒱\left(f\right)$: total variation and ${𝒱}_{a,b}\left(f\right)$: total variation

Symbols:

$sup$: least upper bound (supremum), $j$: integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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23: Mathematical Introduction

… ► ►►►

$ℂ$	complex plane (excluding infinity).
…
$sup$	least upper bound (supremum).
…

► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
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$lim\; inf$	least limit point.
…

…

24: 28.6 Expansions for Small $q$

… ►It is conjectured that for large $n$, the radii increase in proportion to the square of the eigenvalue number $n$; see Meixner et al. (1980, §2.4). … ►

28.6.20 $$\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{{\rho }_n^{(j)}}{n^2}\ge k{k}^{\prime }{(K\left(k\right))}^2=\mathrm{2.04183\hspace{0.33em}4}\mathrm{\dots },$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $lim\; inf$: least limit point, $n$: integer, $j$: integer, ${\rho }_n^{(j)}$: radii of convergence and $k$: root

Referenced by:

§28.6(i), §28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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25: 1.1 Special Notation

… ► ►►►►►►►

$x,y$	real variables.
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$L^2(X,d\alpha )$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $d\alpha$.
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${𝐀}^{-1}$	inverse of the square matrix $𝐀$
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$det(𝐀)$	determinant of the square matrix $𝐀$
$\mathrm{tr}(𝐀)$	trace of the square matrix $𝐀$
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${𝐀}^{\ast }$	adjoint of the square matrix $𝐀$
…

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26: 8.23 Statistical Applications

§8.23 Statistical Applications

… ►Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). …

27: Bibliography L

… ►

The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:

ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

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

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

… ►

An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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28: 18.39 Applications in the Physical Sciences

… ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. … ►where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►with an infinite set of orthonormal $L^2$ eigenfunctions … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. … ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

29: 20.7 Identities

… ►

§20.7(i) Sums of Squares

… ►See Lawden (1989, pp. 19–20). … ►In the following equations ${\tau }^{\prime }=-1/\tau$, and all square roots assume their principal values. … ►

20.7.34 $$\frac{{\theta }_1(z,q^2){\theta }_3(z,q^2)}{{\theta }_1(z,\mathrm{i}q)}=\frac{{\theta }_2(z,q^2){\theta }_4(z,q^2)}{{\theta }_2(z,\mathrm{i}q)}={\mathrm{i}}^{-1/4}\sqrt{\frac{{\theta }_2(0,q^2){\theta }_4(0,q^2)}{2}}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{i}$: imaginary unit, $z$: complex and $q$: nome

Referenced by:

§20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/20.7.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. See Walker (2012).

See also:

Annotations for §20.7(ix), §20.7 and Ch.20

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30: 19.29 Reduction of General Elliptic Integrals

… ►These theorems reduce integrals over a real interval $(y,x)$ of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over $(0,\mathrm{\infty })$ containing the square root of a cubic polynomial (compare §19.16(i)). … ►The only cases that are integrals of the third kind are those in which at least one $m_j$ with $j>h$ is a negative integer and those in which $h=4$ and ${\sum }_{j=1}^n{m}_j$ is a positive integer. … ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as $R_G(a_1{b}_2,a_2{b}_1,0)$ multiplied either by $-2/(b_1{b}_2)$ or by $-2/(a_1{a}_2)$ in the cases of (19.29.20) or (19.29.21), respectively. …