power function

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31—40 of 137 matching pages

31: Bibliography G

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P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

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External Links:: ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

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32: 5.19 Mathematical Applications

… ►Many special functions

f(z)

can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of

z

, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …

33: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …

34: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. See Faierman (1992) for power series and asymptotic expansions of a solution of (12.15.1).

35: Bibliography B

… ►

D. K. Bhaumik and S. K. Sarkar (2002) On the power function of the likelihood ratio test for MANOVA. J. Multivariate Anal. 82 (2), pp. 416–421.

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External Links:: ISSN 0047-259X, Document, MathReview (Yasuko Chikuse), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

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36: 29.3 Definitions and Basic Properties

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§29.3(vii) Power Series

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37: 22.14 Integrals

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§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions

… ►The indefinite integral of the 3rd power of a Jacobian function can be expressed as an elementary function of Jacobian functions and a product of Jacobian functions. The indefinite integral of a 4th power can be expressed as a complete elliptic integral, a polynomial in Jacobian functions, and the integration variable. … ►For indefinite integrals of squares and products of even powers of Jacobian functions in terms of symmetric elliptic integrals, see Carlson (2006b). …

38: 19.28 Integrals of Elliptic Integrals

… ►Also,

\mathrm{B}

again denotes the beta function (§5.12). … ►

19.28.1

\int_{0}^{1}t^{\sigma-1}R_{F}\left(0,t,1\right)\,\mathrm{d}t=\tfrac{1}{2}\left% (\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sigma$ : parameter
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

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19.28.2

\int_{0}^{1}t^{\sigma-1}R_{G}\left(0,t,1\right)\,\mathrm{d}t=\frac{\sigma}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},

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Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sigma$ : parameter
Permalink:: http://dlmf.nist.gov/19.28.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

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19.28.4

\int_{0}^{1}t^{\sigma-1}(1-t)^{c-1}R_{-a}\left(b_{1},b_{2};t,1\right)\,\mathrm% {d}t=\frac{\Gamma\left(c\right)\Gamma\left(\sigma\right)\Gamma\left(\sigma+b_{% 2}-a\right)}{\Gamma\left(\sigma+c-a\right)\Gamma\left(\sigma+b_{2}\right)},

c=b_{1}+b_{2}>0

\Re\sigma>\max(0,a-b_{2})

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : 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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19.28.9

\int_{0}^{\pi/2}R_{F}\left({\sin}^{2}\theta{\cos}^{2}\left(x+y\right),{\sin}^{% 2}\theta{\cos}^{2}\left(x-y\right),1\right)\,\mathrm{d}\theta=R_{F}\left(0,{% \cos}^{2}x,1\right)R_{F}\left(0,{\cos}^{2}y,1\right),

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

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39: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). …

40: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters

\mu

and

\nu

the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. …