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1: 3.2 Linear Algebra

… ►The $p$-norm of a matrix $𝐀=[a_{jk}]$ is … ►Then we have the a posteriori error bound … ►If $𝐀$ is an $n\times n$ matrix, then a real or complex number $\lambda$ is called an eigenvalue of $𝐀$, and a nonzero vector $𝐱$ a corresponding (right) eigenvector, if … ►

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

… ►has the same eigenvalues as $𝐀$. …

2: 26.16 Multiset Permutations

… ►Let $S=\{1^{a_1},2^{a_2},\mathrm{\dots },n^{a_n}\}$ be the multiset that has $a_j$ copies of $j$, $1\le j\le n$. ${𝔖}_S$ denotes the set of permutations of $S$ for all distinct orderings of the $a_1+a_2+\mathrm{\cdots }+a_n$ integers. The number of elements in ${𝔖}_S$ is the multinomial coefficient (§26.4) $\left(\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right)$. … ►The $q$-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by …and again with $S=\{1^{a_1},2^{a_2},\mathrm{\dots },n^{a_n}\}$ we have …

3: 4.30 Elementary Properties

… ►

Table 4.30.1: Hyperbolic functions: interrelations. …

► ►►►►►►►

	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
$\sinh \theta$	$a$	${(a^2-1)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(1-a^2)}^{1/2}$	${(a^2-1)}^{-1/2}$
$\cosh \theta$	${(1+a^2)}^{1/2}$	$a$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a^{-1}$	$a{(a^2-1)}^{-1/2}$
$\tanh \theta$	$a{(1+a^2)}^{-1/2}$	$a^{-1}{(a^2-1)}^{1/2}$	$a$	${(1+a^2)}^{-1/2}$	${(1-a^2)}^{1/2}$	$a^{-1}$
$\mathrm{csch}\theta$	$a^{-1}$	${(a^2-1)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	$a{(1-a^2)}^{-1/2}$	${(a^2-1)}^{1/2}$
$\mathrm{sech}\theta$	${(1+a^2)}^{-1/2}$	$a^{-1}$	${(1-a^2)}^{1/2}$	$a{(1+a^2)}^{-1/2}$	$a$	$a^{-1}{(a^2-1)}^{1/2}$
…

►

4: 1.1 Special Notation

… ► ►►►►►►

$x,y$	real variables.
…
$\varphi$	a testing function.
…
$𝐀$	or $[a_{i,j}]$ or $[a_{ij}]$ matrix with elements $a_{i,j}$ or $a_{ij}$.
…
$det(𝐀)$	determinant of the square matrix $𝐀$
$\mathrm{tr}(𝐀)$	trace of the square matrix $𝐀$
…

►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

5: Alexander A. Its

Profile
Alexander A. Its

… ►Alexander A. Its (b. … ► Belokolos, A. … Fokas, A.  A. …

6: Sidebar 9.SB2: Interference Patterns in Caustics

… ►A thin beam of light refracted by an irregularity in bathroom-window glass produces this image on a distant screen. The bright sharp-edged triangle is a caustic, that is a line of focused light. …

7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …

8: 8.13 Zeros

… ►

one negative zero $x_{-}(a)$ and no positive zeros when ;

►

one negative zero $x_{-}(a)$ and one positive zero $x_{+}(a)$ when .

… ►For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to -\mathrm{\infty }$ see Tricomi (1950b), with corrections by Kölbig (1972b). … ►

two zeros in each of the intervals when ;

… ►When $x>x_n^{\ast }$ a pair of conjugate trajectories emanate from the point $a=a_n^{\ast }$ in the complex $a$-plane. …

9: Annie A. M. Cuyt

Profile
Annie A. M. Cuyt

… ►Annie A. M. Cuyt (b. …As a consequence her expertise spans a wide range of activities from pure abstract mathematics to computer arithmetic and different engineering applications. …In 2013 she was elected a life-time member of the Flemish Royal Society of the Sciences and Arts. ►In November 2015, Cuyt was named a Senior Associate Editor of the DLMF.

10: 12.4 Power-Series Expansions

… ►

12.4.1 $$U(a,z)=U(a,0)u_1(a,z)+U^{\prime }(a,0)u_2(a,z),$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.2 $$V(a,z)=V(a,0)u_1(a,z)+V^{\prime }(a,0)u_2(a,z),$$

ⓘ

Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►where the initial values are given by (12.2.6)–(12.2.9), and $u_1(a,z)$ and $u_2(a,z)$ are the even and odd solutions of (12.2.2) given by ►

12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

…