multivariate beta function

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11: 19.34 Mutual Inductance of Coaxial Circles

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19.34.1 $$\frac{c^{2}M}{2\pi }=ab{\int }_0^{2\pi }{(h^2+a^2+b^2-2ab\cos \theta )}^{-1/2}\cos \theta d\theta =2ab{\int }_{-1}^1\frac{tdt}{\sqrt{(1+t)(1-t)(a_3-2abt)}}=2abI({𝐞}_5),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $I(𝐦)$: integral, $M$: mutual inductance and $h$: distance

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

… ►Application of (19.29.4) and (19.29.7) with $\alpha =1$, $a_{\beta }+b_{\beta }t=1-t$, $\delta =3$, and $a_{\gamma }+b_{\gamma }t=1$ yields … ►

19.34.7 $$M=(2/c^2)(\pi a^2)(\pi b^2)R_{-\frac{3}{2}}(\frac{3}{2},\frac{3}{2};r_{+}^2,r_{-}^2).$$

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

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12: 19.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The first set of main functions treated in this chapter are Legendre’s complete integrals … ►The second set of main functions treated in this chapter is … ► $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). ►A third set of functions, introduced by Bulirsch (1965b, a, 1969a), is …

13: Bibliography C

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Calcolo delle funzioni speciali $\mathrm{\Gamma }(x)$, $\log \mathrm{\Gamma }(x)$, $\beta (x,y)$, $\mathrm{erf}(x)$, $\mathrm{erfc}(x)$ alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).

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Notes:

Translated title: Computation of the special functions $\mathrm{\Gamma }(x),\log \mathrm{\Gamma }(x),\beta (x,y),\mathrm{erf}(x),\mathrm{erfc}(x)$ to a high degree of precision

External Links:

MathReview, ZentralBlatt

Referenced by:

§5.24(ii).

Encodings:

BibTeX

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Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.

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Notes:

Single-precision Fortran.

External Links:

FullText, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Multivariate Splines. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:

With an appendix by Harvey Diamond

External Links:

ISBN 0-89871-226-2, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.11(vi).

Encodings:

BibTeX

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Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha }{1}\cdot \frac{\beta }{\gamma }x+\frac{\alpha \cdot \alpha +1}{1\cdot 2}\cdot \frac{\beta \cdot \beta +1}{\gamma \cdot \gamma +1}{x}^2+$ etc. ein Quadrat von der Form $z=1+\frac{{\alpha }^{\prime }}{1}\cdot \frac{{\beta }^{\prime }}{{\gamma }^{\prime }}\cdot \frac{{\delta }^{\prime }}{{ϵ}^{\prime }}x+\frac{{\alpha }^{\prime }\cdot {\alpha }^{\prime }+1}{1\cdot 2}\cdot \frac{{\beta }^{\prime }\cdot {\beta }^{\prime }+1}{{\gamma }^{\prime }\cdot {\gamma }^{\prime }+1}\cdot \frac{{\delta }^{\prime }\cdot {\delta }^{\prime }+1}{{ϵ}^{\prime }\cdot {ϵ}^{\prime }+1}{x}^2+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.

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External Links:

Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.4(i), §35.4(ii).

Encodings:

BibTeX

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14: 19.29 Reduction of General Elliptic Integrals

… ►and $\alpha ,\beta ,\gamma ,\delta$ is any permutation of the numbers $1,2,3,4$, then …where …where … ►where $\alpha ,\beta ,\gamma$ is any permutation of the numbers $1,2,3$, and … ►It depends primarily on multivariate recurrence relations that replace one integral by two or more. …