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1: 19.16 Definitions

… ►

§19.16(i) Symmetric Integrals

… ►A fourth integral that is symmetric in only two variables is defined by … ►which is homogeneous and of degree

-a

in the

z

’s, and unchanged when the same permutation is applied to both sets of subscripts

1,\dots,n

. Thus

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

is symmetric in the variables

z_{j}

and

z_{\ell}

if the parameters

b_{j}

and

b_{\ell}

are equal. … …

2: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ► ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

3: 19.35 Other Applications

… ► … ►

4: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►Define the elementary symmetric function

E_{s}(\mathbf{z})

by ►

19.19.4

\prod_{j=1}^{n}(1+tz_{j})=\sum_{s=0}^{n}t^{s}E_{s}(\mathbf{z}),

ⓘ

Defines:: $E_{s}(\mathbf{z})$ : symmetric function (locally)
Symbols:: $n$ : nonnegative integer
Referenced by:: §19.36(i)
Permalink:: http://dlmf.nist.gov/19.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.19 and Ch.19

… ►The number of terms in

T_{N}

can be greatly reduced by using variables

\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)

with

A

chosen to make

E_{1}(\mathbf{Z})=0

. … ►

5: 35.2 Laplace Transform

… ►For any complex symmetric matrix

\mathbf{Z}

, ►

35.2.1

g(\mathbf{Z})=\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{Z}% \mathbf{X}\right)f(\mathbf{X})\,\mathrm{d}{\mathbf{X}},

ⓘ

Defines:: $g$ : Laplace transformation of $f$ (locally)
Symbols:: $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $\mathbf{Z}$ : complex symmetric $m\times m$ matrix and $f(\mathbf{X})$ : complex-valued function of matrix argument
Keywords:: Laplace transform
Referenced by:: §35.2
Permalink:: http://dlmf.nist.gov/35.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.2, §35.2 and Ch.35

… ►

35.2.2

f(\mathbf{X})=\dfrac{1}{(2\pi\mathrm{i})^{m(m+1)/2}}\int\operatorname{etr}% \left(\mathbf{Z}\mathbf{X}\right)g(\mathbf{Z})\,\mathrm{d}{\mathbf{Z}},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\,\mathrm{d}$ : differential of matrix, $\mathbf{Z}$ : complex symmetric $m\times m$ matrix, $f(\mathbf{X})$ : complex-valued function of matrix argument, $m$ : positive integer and $g$ : Laplace transformation of $f$
Permalink:: http://dlmf.nist.gov/35.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.2, §35.2 and Ch.35

… ►

35.2.3

f_{1}*f_{2}(\mathbf{T})=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}f_% {1}(\mathbf{T}-\mathbf{X})f_{2}(\mathbf{X})\,\mathrm{d}{\mathbf{X}}.

ⓘ

Defines:: $*$ : convolution operator (locally)
Symbols:: $\int$ : integral, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\,\mathrm{d}$ : differential of matrix, $<$ : less-than for matrices, $f(\mathbf{X})$ : complex-valued function of matrix argument and $\boldsymbol{{0}}$ : $m\times m$ 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

6: 19.21 Connection Formulas

§19.21 Connection Formulas

… ►

R_{D}\left(x,y,z\right)

is symmetric only in

x

and

y

, but either (nonzero)

x

or (nonzero)

y

can be moved to the third position by using …Because

R_{G}

is completely symmetric,

x, y, z

can be permuted on the right-hand side of (19.21.10) so that

(x-z)(y-z)\leq 0

if the variables are real, thereby avoiding cancellations when

R_{G}

is calculated from

R_{F}

and

R_{D}

(see §19.36(i)). … ►

§19.21(iii) Change of Parameter of $R_{J}$

… ►

19.21.15

pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),

pq=yz

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.21.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

7: 19.18 Derivatives and Differential Equations

… ►

§19.18(i) Derivatives

… ►

§19.18(ii) Differential Equations

… ►and two similar equations obtained by permuting

x, y, z

in (19.18.10). … ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). … ►Similarly, the function

u=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+iy,x-iy\right)

satisfies an equation of axially symmetric potential theory: …

8: 19.36 Methods of Computation

§19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …where the elementary symmetric functions

E_{s}

are defined by (19.19.4). … ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). …

9: 35.1 Special Notation

… ► ►►►►►►

$a, b$	complex variables.
…
$\boldsymbol{\mathcal{S}}$	space of all real symmetric matrices.
$\mathbf{S},\mathbf{T},\mathbf{X}$	real symmetric matrices.
…
${\boldsymbol{\Omega}}$	space of positive-definite real symmetric matrices.
…
$\mathbf{Z}$	complex symmetric matrix.
…

…

10: 35.11 Tables

… ►Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.