elementary symmetric functions

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11: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(i) Definition

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Jacobi Form

… ►Let $f:𝛀\to ℂ$ (a) be orthogonally invariant, so that $f(𝐓)$ is a symmetric function of $t_1,\mathrm{\dots },t_m$, the eigenvalues of the matrix argument $𝐓\in 𝛀$; (b) be analytic in $t_1,\mathrm{\dots },t_m$ in a neighborhood of $𝐓=𝟎$; (c) satisfy $f(𝟎)=1$. … ►These approximations are in terms of elementary functions. …

12: Bibliography L

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Evaluating elementary functions with guaranteed precision. Programming and Computer Software 27 (2), pp. 101–110.

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External Links:

ISSN 0361-7688, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(ii).

Encodings:

BibTeX

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Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

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External Links:

ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

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Asymptotic expansions of symmetric standard elliptic integrals. SIAM J. Math. Anal. 31 (4), pp. 754–775.

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External Links:

ISSN 1095-7154, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§19.27(vi), §2.6(ii).

Encodings:

BibTeX

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Elementary solutions of certain Painlevé equations. Differ. Uravn. 1 (3), pp. 731–735 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(3), pp. 561–564

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.10(iii), §32.10(iv), §32.9(i).

Encodings:

BibTeX

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Highly accurate tables for elementary functions. BIT 35 (3), pp. 352–360.

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External Links:

ISSN 0006-3835, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i), §4.46.

Encodings:

BibTeX

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13: Bibliography I

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Theta Functions. Springer-Verlag, New York.

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Notes:

Die Grundlehren der mathematischen Wissenschaften, Band 194

External Links:

ISBN 3540056998, MathReview (H. Klingen), ZentralBlatt

Referenced by:

§21.5(ii), §21.8, Ch.21.

Encodings:

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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IMSL Nuerical Libraries..

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Notes:

The IMSL Libraries are large general purpose numerical software libraries with broad coverage of elementary and special functions. Implementations are in single and double precision.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

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Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.

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External Links:

ZentralBlatt

Referenced by:

4th item, 2nd item.

Encodings:

BibTeX

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The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.

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External Links:

MathReview (H. Bateman), ZentralBlatt

Referenced by:

§29.1, §29.15(ii), 1st item, §29.7(i).

Encodings:

BibTeX

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14: Bibliography F

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Analysis on Symmetric Cones. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford-New York.

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External Links:

ISBN 0-19-853477-9, MathReview (Hong Ming Ding), ZentralBlatt

Referenced by:

§35.1, §35.4(i), §35.7(i), §35.7(iii), §35.8(i), §35.8(iv), §35.8(v), §35.8(v), Ch.35, Notations J, Notations K.

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

Encodings:

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

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Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

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External Links:

Document, ZentralBlatt

Referenced by:

§20.7(ii), §20.7(ii).

Encodings:

BibTeX

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15: 2.6 Distributional Methods

… ►The fact that expansion (2.6.6) misses all the terms in the second series in (2.6.7) raises the question: what went wrong with our process of reaching (2.6.6)? In the following subsections, we use some elementary facts of distribution theory (§1.16) to study the proper use of divergent integrals. … ►To each function in this equation, we shall assign a tempered distribution (i. …, a continuous linear functional) on the space $𝒯$ of rapidly decreasing functions on $ℝ$. … ►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►Also, …

16: 18.38 Mathematical Applications

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Zhedanov Algebra

►A symmetric Laurent polynomial is a function of the form …Define operators $K_0$ and $K_1$ acting on symmetric Laurent polynomials by $K_0=L$ ($L$ given by (18.28.6_2)) and $(K_{1}f)(z)=(z+z^{-1})f(z)$. …and $e_1,e_2,e_3,e_4$ are the elementary symmetric polynomials in $a,b,c,d$ given by $a+b+c+d$, $ab+ac+\mathrm{\cdots }+cd$, $abc+abd+acd+bcd$, $abcd$, respectively. … ►The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_n(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d|q)$, where $R_n(z)$ is given in (18.28.1_5). …

17: Bibliography M

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Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

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On the interval computation of elementary functions. C. R. Acad. Bulgare Sci. 34 (3), pp. 319–322.

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External Links:

ISSN 0366-8681, MathReview (David F. Griffiths), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

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The computation of elementary functions in radix $2^p$ . Computing 53 (3-4), pp. 219–232.

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Notes:

International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Vienna, 1993)

External Links:

ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

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Elementary Functions: Algorithms and Implementation. Birkhäuser Boston Inc., Boston, MA.

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Notes:

A 2nd edition was published in 2006.

External Links:

ISBN 0-8176-3990-X, MathReview (Warren E. Ferguson, Jr.), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

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18: 3.1 Arithmetics and Error Measures

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Rounding

… ►Symmetric rounding or rounding to nearest of $x$ gives $x_{-}$ or $x_{+}$, whichever is nearer to $x$, with maximum relative error equal to the machine precision $\frac{1}{2}{ϵ}_M=2^{-p}$. … ►The elementary arithmetical operations on intervals are defined as follows: … ►

3.1.10 $${ϵ}_{\mathrm{𝑟𝑝}}=\left|\ln \left(x^{\ast }/x\right)\right|,$$

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Defines:

${ϵ}_{\mathrm{𝑟𝑝}}$: relative precision (locally)

Symbols:

$\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/3.1.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(v), §3.1 and Ch.3

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19: 32.2 Differential Equations

… ►in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. … ►

§32.2(v) Symmetric Forms

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20: 18.39 Applications in the Physical Sciences

… ►defines the potential for a symmetric restoring force $-kx$ for displacements from $x=0$. … ►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►

§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

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a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

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c) Spherical Radial Coulomb Wave Functions

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