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31: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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Case $m=2$

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35.7.5 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(c-a\right)>\frac{1}{2}(m-1)$, .

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $c$: complex variable, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.7(iv)

Permalink:

http://dlmf.nist.gov/35.7.E5

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Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►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 $f(𝟎)=1$. … ►

35.7.10 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha a,\alpha b}{\alpha c};𝐓)$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iv), §35.7 and Ch.35

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35.7.11 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈-{\alpha }^{-1}𝐓)$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E11

Encodings:

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See also:

Annotations for §35.7(iv), §35.7 and Ch.35

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32: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Self-Adjoint and Symmetric Operators

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§1.18(vii) Continuous Spectra: More General Cases

… ►A linear operator $T$ with dense domain is called symmetric if … ►

Self-adjoint extensions of a symmetric Operator

… ►By Weyl’s alternative $n_1$ equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for $n_2$. …

33: 3.1 Arithmetics and Error Measures

… ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{\min }=-126$, $E_{\max }=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{\min }=-1022$, $E_{\max }=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{\min }=-16382$, $E_{\max }=16383$) are as in Figure 3.1.1. … ►

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 $\frac{1}{2}{ϵ}_M=2^{-p}$. … ►Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases. …

34: 21.7 Riemann Surfaces

… ►In contrast, a $g$-dimensional Riemann theta function arising from a compact Riemann surface of genus $g$ ($>1$) depends on at most $3g-3$ complex parameters (one complex parameter for the case $g=1$). … ►Fay derives (21.7.10) as a special case of a more general class of addition theorems for Riemann theta functions on Riemann surfaces. … ►

21.7.12 $$T_1\ominus T_2=(T_1\cup T_2)\setminus (T_1\cap T_2).$$

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Symbols:

$\cap$: intersection, $\setminus$: set subtraction, $\cup$: union and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E12

Encodings:

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Annotations for §21.7(iii), §21.7 and Ch.21

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21.7.14 $$𝜼(T_1\ominus T_2)=𝜼(T_1)+𝜼(T_2),$$

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Symbols:

$T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E14

Encodings:

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See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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21.7.15 $$4{𝜼}^1(T)\cdot {𝜼}^2(T)=\frac{1}{2}\left(|T\ominus U|-g-1\right)(mod2),$$

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Symbols:

$g$: positive integer, $U$: subset of $B$ and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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35: 21.5 Modular Transformations

… ►($𝐁$ symmetric with integer elements and even diagonal elements.) …($𝐁$ symmetric with integer elements.) … ►For explicit results in the case $g=1$, see §20.7(viii).

36: 2.6 Distributional Methods

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2.6.6 $$S(x)\sim \frac{2\pi }{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)x^{-s-(1/3)},$$ $x\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter and $S(x)$: integral

Referenced by:

§2.6(i)

Permalink:

http://dlmf.nist.gov/2.6.E6

Encodings:

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See also:

Annotations for §2.6(i), §2.6 and Ch.2

… ►Similarly, in the case $\alpha =1$, we define …In either case, we define the distribution associated with $f_n(t)$ by … ►

2.6.32 $${\int }_0^{\mathrm{\infty }}\frac{f(t)}{{(t+z)}^{\rho }}dt,$$ $\rho >0$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $f(t)$: locally integrable function

Permalink:

http://dlmf.nist.gov/2.6.E32

Encodings:

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Annotations for §2.6(ii), §2.6 and Ch.2

… ►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). …

37: 3.5 Quadrature

… ►When $f\in C^{\mathrm{\infty }}$, the Romberg method affords a means of obtaining high accuracy in many cases with a relatively simple adaptive algorithm. … ►The Gauss nodes $x_k$ (the zeros of $p_n$) are the eigenvalues of the (symmetric tridiagonal) Jacobi matrix of order $n\times n$: … ►For these cases the integration path may need to be deformed; see §3.5(ix). … ►The integrand can be extended as a periodic $C^{\mathrm{\infty }}$ function on $ℝ$ with period $2\pi$ and as noted in §3.5(i), the trapezoidal rule is exceptionally efficient in this case. … ►A special case is the rule for Hilbert transforms (§1.14(v)): …

38: 19.3 Graphics

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . …The function tends to $+\mathrm{\infty }$ as ${\sin }^2\varphi \to \frac{1}{2}$, except in the last case below. … Magnify 3D Help

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Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The real part is symmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The real part is symmetric under reflection in the real axis. … Magnify 3D Help

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39: 1.2 Elementary Algebra

… ►Special cases are the Euclidean length or $l^2$ norm … ►a real symmetric matrix if …

40: 18.28 Askey–Wilson Class

… ►The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of $q$-Racah polynomials, and cases of these families obtained by specialization of parameters. …$y$) such that $P_n(z)=p_n(\frac{1}{2}(z+z^{-1}))$ in the Askey–Wilson case, and $P_n(y)=p_n(q^{-y}+c{q}^{y+1})$ in the $q$-Racah case, and both are eigenfunctions of a second order $q$-difference operator similar to (18.27.1). … ►The polynomials $p_n(x;a,b,c,d|q)$ are symmetric in the parameters $a,b,c,d$. … ►For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006). … ►These systems are the $q$-Racah polynomials and its limit cases. …