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11: 19.31 Probability Distributions

§19.31 Probability Distributions

► $R_G(x,y,z)$ and $R_F(x,y,z)$ occur as the expectation values, relative to a normal probability distribution in ${ℝ}^2$ or ${ℝ}^3$, of the square root or reciprocal square root of a quadratic form. … ►

19.31.2 $${\int }_{{ℝ}^n}{({𝐱}^{\mathrm{T}}𝐀𝐱)}^{\mu }\exp \left(-{𝐱}^{\mathrm{T}}𝐁𝐱\right)d{x}_1\mathrm{\cdots }d{x}_n=\frac{{\pi }^{n/2}\mathrm{\Gamma }\left(\mu +\frac{1}{2}n\right)}{\sqrt{det𝐁}\mathrm{\Gamma }\left(\frac{1}{2}n\right)}{R}_{\mu }(\frac{1}{2},\mathrm{\dots },\frac{1}{2};{\lambda }_1,\mathrm{\dots },{\lambda }_n),$$ $\mu >-\frac{1}{2}n$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $ℝ$: real line, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $n$: nonnegative integer and ${\lambda }_n$: eigenvalues

Referenced by:

§19.31

Permalink:

http://dlmf.nist.gov/19.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.31 and Ch.19

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12: 25.10 Zeros

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§25.10(i) Distribution

… ►In the region , called the critical strip, $\zeta \left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the critical line $\mathrm{\Re }s=\frac{1}{2}$. … ►

25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

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See also:

Annotations for §25.10(i), §25.10 and Ch.25

►is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. …

13: 9.12 Scorer Functions

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9.12.6 $$\mathrm{Gi}\left(0\right)=\frac{1}{2}\mathrm{Hi}\left(0\right)=\frac{1}{3}\mathrm{Bi}\left(0\right)=1/\left(3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)\right)=\mathrm{0.20497\hspace{0.33em}55424}\mathrm{\dots },$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function) and $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Source:

Olver (1997b, p. 431)

Permalink:

http://dlmf.nist.gov/9.12.E6

Encodings:

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See also:

Annotations for §9.12(iii), §9.12 and Ch.9

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9.12.24 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{2{\pi }^2\mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)\mathrm{\Gamma }\left(-t\right){(3^{1/3}{\mathrm{e}}^{\pi \mathrm{i}}z)}^{t}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Keywords:

Mellin transform

Source:

Exton (1983, (2.1), p. 100)

Permalink:

http://dlmf.nist.gov/9.12.E24

Encodings:

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See also:

Annotations for §9.12(vii), §9.12(vii), §9.12 and Ch.9

►where the integration contour separates the poles of $\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$. … ►where $\gamma$ is Euler’s constant (§5.2(ii)). … ►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …

14: Bibliography J

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Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.11, §35.4(i), §35.4(ii), §35.8(i), §35.8(ii), §35.8(iv), §35.9, Ch.35.

Encodings:

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

Encodings:

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Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.

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External Links:

ISBN 978-3-319-72453-9, ZentralBlatt

Referenced by:

§4.13.

Encodings:

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Continuous Univariate Distributions. 2nd edition, Vol. I, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-58495-9, MathReview (D. N. Shanbhag), ZentralBlatt

Referenced by:

§7.20(iii), §8.23.

Encodings:

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Continuous Univariate Distributions. 2nd edition, Vol. II, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-58494-0, MathReview (D. N. Shanbhag), ZentralBlatt

Referenced by:

§8.23.

Encodings:

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15: 10.21 Zeros

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§10.21(i) Distribution

… ►Some information on the distribution of ${\rho }_{\nu }(t)$ and ${\sigma }_{\nu }(t)$ for real values of $\nu$ and $t$ is given in Muldoon and Spigler (1984). … ►

§10.21(ix) Complex Zeros

… ►For describing the distribution of complex zeros by methods based on the Liouville–Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013). … ► …

16: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

In previous versions of the DLMF, in §8.18(ii), the notation $\tilde{\mathrm{\Gamma }}(z)$ was used for the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$. Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).

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It is now mentioned that (25.2.5), defines the Stieltjes constants ${\gamma }_n$. Consequently, ${\gamma }_n$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

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Several changes have been made to

1. make consistent use of the Fourier transform notations $ℱ\left(f\right)$, $ℱ\left(\varphi \right)$ and $ℱ\left(u\right)$ where $f$ is a function of one real variable, $\varphi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. introduce the partial differential operator $𝐃$ in (1.16.30);

3. clarify the definition (1.16.32) of the partial differential operator $P(𝐃)$; and

4. clarify the use of $P(𝐃)$ and $P(𝐱)$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

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An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

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17: Bibliography L

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Monotonicity results and inequalities for the gamma and error functions. J. Comput. Appl. Math. 23 (1), pp. 25–33.

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External Links:

ISSN 0377-0427, Document, MathReview (M. E. Cohen), ZentralBlatt

Referenced by:

§7.8.

Encodings:

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Further inequalities for the gamma function. Math. Comp. 42 (166), pp. 597–600.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

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Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.

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Notes:

Single-precision Fortran, maximum accuracy 8S. For Remarks see same journal 13(1970), pp. 124 and 449.

External Links:

ISSN 0001-0782, Document, Remark 1 and Remark 2

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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Sur la distribution des nombres premiers. Comptes Rendus de l’Academie des Sciences, Paris 158, pp. 1869–1872 (French).

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External Links:

ZentralBlatt

Referenced by:

§27.12.

Encodings:

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Inequalities for ultraspherical polynomials and the gamma function. J. Approx. Theory 40 (2), pp. 115–120.

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External Links:

ISSN 0021-9045, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§18.14(i).

Encodings:

BibTeX

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18: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

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12.11.2 $${\tau }_s=\left(2s+\frac{1}{2}-a\right)\pi +\mathrm{i}\ln \left({\pi }^{-\frac{1}{2}}{2}^{-a-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)\right),$$

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Defines:

${\tau }_s$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $s$: nonnegative integer and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.11.E2

Encodings:

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See also:

Annotations for §12.11(ii), §12.11 and Ch.12

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19: 25.16 Mathematical Applications

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§25.16(i) Distribution of Primes

►In studying the distribution of primes $p\le x$, Chebyshev (1851) introduced a function $\psi \left(x\right)$ (not to be confused with the digamma function used elsewhere in this chapter), given by … ►

25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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20: Bibliography D

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The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.

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External Links:

ISSN 0031-8914, Document, Link

Referenced by:

§18.40(ii).

Encodings:

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Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§8.12, §8.25(iii).

Encodings:

BibTeX

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Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 13 (3), pp. 318–319.

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Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 654; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

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Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39

External Links:

ISSN 0001-0782, Document

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function. Methods Appl. Anal. 5 (3), pp. 223–247.

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External Links:

ISSN 1073-2772, Pdf, MathReview (John G. Stalker), ZentralBlatt

Referenced by:

§8.12.

Encodings:

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