symmetric case

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11: 19.36 Methods of Computation

… ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. …

12: 19.26 Addition Theorems

§19.26 Addition Theorems

… ►

§19.26(ii) Case $x=0$

… ►

§19.26(iii) Duplication Formulas

… ►

19.26.20

R_{D}\left(x,y,z\right)=2R_{D}\left(x+\lambda,y+\lambda,z+\lambda\right)+\frac% {3}{\sqrt{z}(z+\lambda)}.

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

►

19.26.21

2R_{G}\left(x,y,z\right)=4R_{G}\left(x+\lambda,y+\lambda,z+\lambda\right)-% \lambda R_{F}\left(x,y,z\right)-\sqrt{x}-\sqrt{y}-\sqrt{z}.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.36(i)
Permalink:: http://dlmf.nist.gov/19.26.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

…

13: 3.2 Linear Algebra

… ►Then

\left|\ell_{jk}\right|\leq 1

in all cases. … ►The Euclidean norm is the case

p=2

. … ►When

\mathbf{A}

is a symmetric matrix, the left and right eigenvectors coincide, yielding

\kappa(\lambda)=1

, and the calculation of its eigenvalues is a well-conditioned problem. ►

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

►Let

\mathbf{A}

be an

n\times n

symmetric matrix. …

14: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►In the case of a real matrix

{\mathbf{A}}^{*}=\mathbf{A}^{\mathrm{T}}

and in the complex case

{\mathbf{A}}^{*}={\mathbf{A}}^{{\rm H}}

. ►Real symmetric (

\mathbf{A}=\mathbf{A}^{\mathrm{T}}

) and Hermitian (

\mathbf{A}={\mathbf{A}}^{{\rm H}}

) matrices are self-adjoint operators on

\mathbf{E}_{n}

. … ►For Hermitian matrices

\mathbf{S}

is unitary, and for real symmetric matrices

\mathbf{S}

is an orthogonal transformation. …

15: 19.17 Graphics

§19.17 Graphics

►See Figures 19.17.1–19.17.8 for symmetric elliptic integrals with real arguments. … ►For

R_{F}

R_{G}

, and

R_{J}

, which are symmetric in

x, y, z

, we may further assume that

z

is the largest of

x, y, z

if the variables are real, then choose

z=1

, and consider only

0\leq x\leq 1

and

0\leq y\leq 1

. The cases

x=0

y=0

correspond to the complete integrals. The case

y=1

corresponds to elementary functions. …

16: 19.29 Reduction of General Elliptic Integrals

… ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the

8+8+12=28

formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking

x^{2}

as the variable of integration in 3. …

17: 19.22 Quadratic Transformations

… ►

Bartky’s Transformation

… ►

§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

… ►Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … ►

18: 19.7 Connection Formulas

… ►

§19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$

… ►If

k^{2}

and

\alpha^{2}

are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)). ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of

\Pi\left(\phi,\alpha^{2},k\right)

when

\alpha^{2}>{\csc}^{2}\phi

(see (19.6.5) for the complete case). …

19: 19.33 Triaxial Ellipsoids

… ►

19.33.1

S=3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right),

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $V$ : volume and $S$ : area
Referenced by:: §19.33(i)
Permalink:: http://dlmf.nist.gov/19.33.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(i), §19.33 and Ch.19

… ►

§19.33(ii) Potential of a Charged Conducting Ellipsoid

… ►

19.33.5

V(\lambda)=QR_{F}\left(a^{2}+\lambda,b^{2}+\lambda,c^{2}+\lambda\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $Q$ : charge and $V(\lambda)$ : potential
Permalink:: http://dlmf.nist.gov/19.33.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(ii), §19.33 and Ch.19

… ►

19.33.6

1/C=R_{F}\left(a^{2},b^{2},c^{2}\right).

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $C$ : electric capacity
Permalink:: http://dlmf.nist.gov/19.33.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(ii), §19.33 and Ch.19

►A conducting elliptic disk is included as the case

c=0

. …

20: 18.38 Mathematical Applications

… ►

Zhedanov Algebra

►A symmetric Laurent polynomial is a function of the form … ►Analogues of the original Dunkl operator (the rational case) were introduced by Heckman and Cherednik for the trigonometric case, and by Cherednik for the

q

-case. …In the

q

-case this algebraic structure is called the double affine Hecke algebra (DAHA), introduced by Cherednik. … ►In the one-variable case the Dunkl operator eigenvalue equation …