DLMF: Untitled Document

… ►

See accompanying text — Figure 22.3.4: $k=0.999999$ , $-3K\leq x\leq 3K$ , $K=7.9474\dots$ . Magnify

… ►

… ►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. … ►In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase. …

… ►With

q

as in (22.2.1) and

\zeta=\pi x/(2K)

, … ►In Equations (22.16.21)–(22.16.23),

-K<x<K.

… ►In Equations (22.16.24)–(22.16.26),

-2K<x<2K

. … ►With

E\left(k\right)

and

K\left(k\right)

as in §19.2(ii) and

x\in\mathbb{R}

, …(Sometimes in the literature

\mathrm{Z}\left(x|k\right)

is denoted by

\mathrm{Z}(\operatorname{am}\left(x,k\right),k^{2})

.) …

… ►

J. Faraut and A. Korányi (1994) Analysis on Symmetric Cones. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford-New York.

ⓘ

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.
Encodings:: BibTeX

… ►

J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

ⓘ

External Links:: FullText, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.10.
Encodings:: BibTeX

… ►

N. Fleury and A. Turbiner (1994) Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt
Referenced by:: §13.3(ii), §15.5(i), §16.3(i).
Encodings:: BibTeX

… ►

T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:: MathReview (W. J. Trjitzinsky), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

►

L. Fox and I. B. Parker (1968) Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London.

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Notes:: Reprinted in 1972 with corrections.
External Links:: MathReview (G. N. Lance), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

…

… ►

35.7.2

P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

;

\gamma,\delta,\nu\in\mathbb{C}

;

\Re\left(\gamma\right)>-1

.

ⓘ

Defines:: $P^{(\NVar{\gamma},\NVar{\delta})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Jacobi function of matrix argument
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\mathbb{C}$ : complex plane, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

… ►

35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►Let

f:{\boldsymbol{\Omega}}\to\mathbb{C}

(a) be orthogonally invariant, so that

f(\mathbf{T})

is a symmetric function of

t_{1},\dots,t_{m}

, the eigenvalues of the matrix argument

\mathbf{T}\in{\boldsymbol{\Omega}}

; (b) be analytic in

t_{1},\dots,t_{m}

in a neighborhood of

\mathbf{T}=\boldsymbol{{0}}

; (c) satisfy

f(\boldsymbol{{0}})=1

. … ►Systems of partial differential equations for the

{{}_{0}F_{1}}

(defined in §35.8) and

{{}_{1}F_{1}}

functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). … ►These approximations are in terms of elementary functions. …

… ► 1924 in Croydon, U. …degrees in mathematics from the University of London in 1945, 1948, and 1961, respectively. … ►In 1992 he retired, and was appointed Professor Emeritus. … ►In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olver’s 65th birthday, with Proceedings published by Marcel Dekker in 1990. … ►Most notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). …

… ►The results have been published in book form as the NIST Handbook of Mathematical Functions, by Cambridge University Press, and disseminated in the free electronic Digital Library of Mathematical Functions. …Details of the early history of the DLMF Project are given in the Preface and on pp. ix–xi in the NIST Handbook of Mathematical Functions. … ►After the death in April 2013 of Frank W. … ►They were selected as recognized leaders in the research communities interested in the mathematics and applications of special functions and orthogonal polynomials; in the presentation of mathematics reference information online and in handbooks; and in the presentation of mathematics on the web. …

… ►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument

z

and the modulus

k

is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►By application of the transformations given in §§22.7(i) and 22.7(ii),

k

or

k^{\prime}

can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. … ►From the first two terms in (22.10.6) we find

\operatorname{dn}\left(0.19,\tfrac{1}{19}\right)=0.999951

. Then by using (22.7.4) we have

\operatorname{dn}\left(0.2,\sqrt{0.19}\right)=0.996253

. … ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

… ►

►

… ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. … ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …

… ►

►

… ►

… ►If we set

\ifrac{{\mathrm{d}}^{2}u}{{\mathrm{d}x}^{2}}=0

in (32.3.2) and solve for

u

, then …

… ►They are denoted by

a_{k}

,

a^{\prime}_{k}

,

b_{k}

,

b^{\prime}_{k}

, respectively, arranged in ascending order of absolute value for

k=1,2,\ldots.

… ►They lie in the sectors

\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi

and

-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi

, and are denoted by

\beta_{k}

,

\beta^{\prime}_{k}

, respectively, in the former sector, and by

\overline{\beta_{k}}

,

\overline{\beta^{\prime}_{k}}

, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for

k=1,2,\ldots.

See §9.3(ii) for visualizations. ►For the distribution in

\mathbb{C}

of the zeros of

\operatorname{Ai}'\left(z\right)-\sigma\operatorname{Ai}\left(z\right)

, where

\sigma

is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). … ►For error bounds for the asymptotic expansions of

a_{k}

,

b_{k}

,

a^{\prime}_{k}

, and

b^{\prime}_{k}

see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). … ►Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of

\operatorname{Bi}

and

\operatorname{Bi}'

in the upper half plane. …

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11—20 of 913 matching pages

11: 22.3 Graphics

12: 22.16 Related Functions

13: Bibliography F

14: 35.7 Gaussian Hypergeometric Function of Matrix Argument

15: Frank W. J. Olver

16: About the Project

17: 22.20 Methods of Computation

18: 20.3 Graphics

19: 32.3 Graphics

20: 9.9 Zeros