# symmetric

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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
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). …
##### 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 $\prod_{j=1}^{n}(1+tz_{j})=\sum_{s=0}^{n}t^{s}E_{s}(\mathbf{z}),$
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 $\mathbf{Z}$,
35.2.2 $f(\mathbf{X})=\dfrac{1}{(2\pi\mathrm{i})^{m(m+1)/2}}\int\operatorname{etr}% \left(\mathbf{Z}\mathbf{X}\right)g(\mathbf{Z})\,\mathrm{d}{\mathbf{Z}},$
##### 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 $pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),$ $pq=yz$.
##### 7: 19.18 Derivatives and Differential Equations
###### §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. … space of all real symmetric matrices. real symmetric matrices. … space of positive-definite real symmetric matrices. … complex symmetric matrix. …
##### 10: 35.11 Tables
Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.