symmetric forms

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1: 32.2 Differential Equations

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§32.2(v) Symmetric Forms

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2: Mathematical Introduction

… ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …

3: 22.15 Inverse Functions

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4: 18.38 Mathematical Applications

… ►A symmetric Laurent polynomial is a function of the form …

5: Bille C. Carlson

… ►If some of the parameters are equal, then the

R

-function is symmetric in the corresponding variables. This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

6: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►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}

. …The corresponding eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

can be chosen such that they form a complete orthonormal basis in

\mathbf{E}_{n}

. ►Let the columns of matrix

\mathbf{S}

be these eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

, then

{\mathbf{S}}^{-1}={\mathbf{S}}^{{\rm H}}

, and the similarity transformation (1.2.73) is now of the form

{\mathbf{S}}^{{\rm H}}\mathbf{A}\mathbf{S}=\lambda_{i}\delta_{i,j}

. For Hermitian matrices

\mathbf{S}

is unitary, and for real symmetric matrices

\mathbf{S}

is an orthogonal transformation. …

7: 19.31 Probability Distributions

§19.31 Probability Distributions

►

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

and

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

occur as the expectation values, relative to a normal probability distribution in

{\mathbb{R}}^{2}

{\mathbb{R}}^{3}

, of the square root or reciprocal square root of a quadratic form. …

8: 19.18 Derivatives and Differential Equations

… ►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)). …

9: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►A fourth integral that is symmetric in only two variables is defined by … ►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …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

. … …

10: 19.25 Relations to Other Functions

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§19.25(i) Legendre’s Integrals as Symmetric Integrals

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§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

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§19.25(vii) Hypergeometric Function

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