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21: 19.34 Mutual Inductance of Coaxial Circles

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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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22: 19.20 Special Cases

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19.20.3 $$R_F(x,a,y)=R_{-\frac{1}{4}}(\frac{3}{4},\frac{1}{2};a^2,xy),$$ $a=\frac{1}{2}(x+y)$.

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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 and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.20(i)

Permalink:

http://dlmf.nist.gov/19.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(i), §19.20 and Ch.19

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19.20.23 $$R_D(x,y,a)=R_{-\frac{3}{4}}(\frac{5}{4},\frac{1}{2};a^2,xy),$$ $a=\frac{1}{2}x+\frac{1}{2}y$.

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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 and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.20.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

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19.20.25 $$R_{-c}(𝐛;𝐳)=\prod _{j=1}^n{z}_j^{-b_j},$$

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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 and $n$: nonnegative integer

Referenced by:

§19.18(ii), §19.21(ii)

Permalink:

http://dlmf.nist.gov/19.20.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(v), §19.20 and Ch.19

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19.20.26 $$R_{-a}(𝐛;𝐳)=\prod _{j=1}^n{z}_j^{-b_j}{R}_{-a^{\prime }}(𝐛;{𝒛}^{\mathbf{-}\mathrm{𝟏}}),$$ $a+a^{\prime }=c$, ${𝒛}^{\mathbf{-}\mathrm{𝟏}}=(z_1^{-1},\mathrm{\dots },z_n^{-1})$.

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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 and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/19.20.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(v), §19.20 and Ch.19

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23: Bibliography P

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Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.

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External Links:

ISSN 0090-5364, FullText, MathReview (Takeaki Kariya), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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24: 19.25 Relations to Other Functions

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19.25.3 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{1}{2}\pi R_{-\frac{1}{2}}(\frac{1}{2},-\frac{1}{2},1;k_{}^{\prime }{}_{}{}^2,1,1-{\alpha }^2),$$

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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, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.8 $$E(\varphi ,k)=R_{-\frac{1}{2}}(\frac{1}{2},-\frac{1}{2},\frac{3}{2};c-1,c-k^2,c),$$

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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, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.15 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=R_{-\frac{1}{2}}(\frac{1}{2},\frac{1}{2},-\frac{1}{2},1;c-1,c-k^2,c,c-{\alpha }^2).$$

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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, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.42 $${}_2{}^{}F_1^{}(a,b;c;z)=R_{-a}(b,c-b;1-z,1),$$

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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 and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Referenced by:

§19.20(i), §19.20(iv)

Permalink:

http://dlmf.nist.gov/19.25.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vii), §19.25 and Ch.19

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19.25.43 $$R_{-a}(b_1,b_2;z_1,z_2)=z_2^{-a}{}_2{}^{}F_1^{}(a,b_1;b_1+b_2;1-(z_1/z_2)).$$

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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 and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.25.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vii), §19.25 and Ch.19

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25: Bibliography F

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Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96049-X, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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26: Bibliography W

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The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

Encodings:

BibTeX

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27: 19.24 Inequalities

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28: 19.28 Integrals of Elliptic Integrals

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19.28.4 $${\int }_0^1{t}^{\sigma -1}{(1-t)}^{c-1}{R}_{-a}(b_1,b_2;t,1)dt=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(\sigma \right)\mathrm{\Gamma }\left(\sigma +b_2-a\right)}{\mathrm{\Gamma }\left(\sigma +c-a\right)\mathrm{\Gamma }\left(\sigma +b_2\right)},$$ $c=b_1+b_2>0$, $\mathrm{\Re }\sigma >\max (0,a-b_2)$.

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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, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sigma$: parameter

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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29: Bibliography C

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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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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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30: Bibliography M

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Aspects of Multivariate Statistical Theory. John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-09442-0, MathReview (Nariaki Sugiura), ZentralBlatt

Referenced by:

§35.11, §35.2, §35.4(i), §35.4(ii), §35.6(i), §35.6(ii), §35.7(i), §35.7(ii), §35.7(iii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), §35.9, Ch.35, Profile Robb J. Muirhead.

Encodings:

BibTeX

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