right-hand rule for cross products

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21: 1.5 Calculus of Two or More Variables

… ►

Chain Rule

… ►where

f

and its partial derivatives on the right-hand side are evaluated at

(a,b)

, and

R_{n}/(\lambda^{2}+\mu^{2})^{n/2}\to 0

(\lambda,\mu)\to(0,0)

. … ►where the right-hand side is interpreted as the repeated integral …

22: 3.7 Ordinary Differential Equations

… ►If, for example,

\beta_{0}=\beta_{1}=0

, then on moving the contributions of

w(z_{0})

and

w(z_{P})

to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of

\mathbf{A}_{P}

that lie below the main diagonal and its two adjacent diagonals. … ►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. … ►For

w^{\prime}=f(z,w)

the standard fourth-order rule reads … ►For

w^{\prime\prime}=f(z,w,w^{\prime})

the standard fourth-order rule reads …

23: 19.20 Special Cases

… ►

19.20.19

R_{D}\left(x,y,z\right)\sim 3x^{-1/2}y^{-1/2}z^{-1/2},

z/\sqrt{xy}\to 0

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $\sim$ : asymptotic equality
Referenced by:: §19.20(iv), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.20.E19
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

… ►

19.20.25

R_{-c}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}},

ⓘ

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 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

►

19.20.26

R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}}R_{-a^{% \prime}}\left(\mathbf{b};\boldsymbol{{z^{-1}}}\right),

a+a^{\prime}=c

\boldsymbol{{z^{-1}}}=(z_{1}^{-1},\dots,z_{n}^{-1})

ⓘ

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 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

…

24: 18.15 Asymptotic Approximations

… ►Also, when

\tfrac{1}{6}\pi<\theta<\tfrac{5}{6}\pi

, the right-hand side of (18.15.12) with

M=\infty

converges; paradoxically, however, the sum is

2P_{n}\left(\cos\theta\right)

and not

P_{n}\left(\cos\theta\right)

as stated erroneously in Szegő (1975, §8.4(3)). … ►

18.15.22

L^{(\alpha)}_{n}\left(\nu x\right)=(-1)^{n}\frac{{\mathrm{e}}^{\frac{1}{2}\nu x% }}{2^{\alpha-\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\*\left(\frac{\zeta% }{x-1}\right)^{\frac{1}{4}}\left(\frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}% }\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{m=0}^{M-1}\frac{E_{m}(\zeta)}{\nu^{2m}}% +\frac{\operatorname{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}% }}\sum_{m=0}^{M-1}\frac{F_{m}(\zeta)}{\nu^{2m}}+\operatorname{envAi}\left(\nu^% {\frac{2}{3}}\zeta\right)O\left(\frac{1}{\nu^{2M-\frac{2}{3}}}\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\sim$ : Poincaré asymptotic expansion, $\operatorname{envAi}\left(\NVar{x}\right)$ : envelope of Airy function $\operatorname{Ai}\left(\NVar{x}\right)$ , $\mathrm{e}$ : base of natural logarithm, $m$ : nonnegative integer, $n$ : nonnegative integer, $\nu$ , $\zeta$ , $E_{m}$ : coefficient, $F_{m}$ : coefficient and $x$ : real variable
Referenced by:: Erratum (V1.0.11) for Equation (18.15.22)
Permalink:: http://dlmf.nist.gov/18.15.E22
Encodings:: pMML, png, TeX
Errata (effective with 1.0.11):: Originally this equation was expressed in terms of the asymptotic symbol $\sim$ . As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$ .
See also:: Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

…

25: 16.8 Differential Equations

… ►

16.8.8

{{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=% \sum_{j=1}^{q+1}\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\frac{\Gamma\left(a_{k}-a_{j}\right)}{\Gamma\left(% a_{k}\right)}}{\prod\limits_{k=1}^{q}\frac{\Gamma\left(b_{k}-a_{j}\right)}{% \Gamma\left(b_{k}\right)}}}\right)\widetilde{w}_{j}(z),

|\operatorname{ph}\left(-z\right)|\leq\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $w$ : solution
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

… ►

16.8.9

\left({\textstyle\ifrac{\prod\limits_{k=1}^{q+1}\Gamma\left(a_{k}\right)}{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{q+1}F_{q}}\left({a% _{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=\sum_{j=1}^{q+1}\left(z_{0% }-z\right)^{-a_{j}}\sum_{n=0}^{\infty}\frac{\Gamma\left(a_{j}+n\right)}{n!}\*% \left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\Gamma\left(a_{k}-a_{j}-n\right)}{\prod\limits_{k=% 1}^{q}\Gamma\left(b_{k}-a_{j}-n\right)}}\right)\*{{}_{q+1}F_{q}}\left({a_{1}-a% _{j}-n,\dots,a_{q+1}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}% \right)\left(z-z_{0}\right)^{-n}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

►(Note that the generalized hypergeometric functions on the right-hand side are polynomials in

z_{0}

.) …

26: 33.14 Definitions and Basic Properties

… ►

33.14.11

A(\epsilon,\ell)=\prod_{k=0}^{\ell}(1+\epsilon k^{2}).

ⓘ

Defines:: $A(\epsilon,\ell)$ : function (locally)
Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer and $\epsilon$ : real parameter
Referenced by:: §33.14(iv), §33.16(ii), §33.16(iii), §33.19, §33.20(iii)
Permalink:: http://dlmf.nist.gov/33.14.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §33.14(iv), §33.14 and Ch.33

… ►where the right-hand side is the Dirac delta (§1.17). …

27: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

… ►When

|\zeta-1|<1

the series on the right-hand side converges in the half-plane

\Re z<\frac{1}{2}

. …

28: 2.6 Distributional Methods

… ►But the right-hand side is meaningful for all values of

\alpha

and

\beta

, other than nonpositive integers. We may therefore define the integral on the left-hand side of (2.6.4) by the value on the right-hand side, except when

\alpha,\beta=0,-1,-2,\dots

. … ►On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes … ►We now derive an asymptotic expansion of

I^{\mu}f(x)

for large positive values of

x

. ►In terms of the convolution product …

29: 3.4 Differentiation

… ►

3.4.3

hR^{\prime}_{n,t}=\frac{h^{n+1}}{(n+1)!}\left(f^{(n+1)}(\xi_{0})\frac{\mathrm{% d}}{\mathrm{d}t}\prod_{k=n_{0}}^{n_{1}}(t-k)+f^{(n+2)}(\xi_{1})\prod_{k=n_{0}}% ^{n_{1}}(t-k)\right),

ⓘ

Defines:: $R^{\prime}_{n,t}(x)$ : remainder (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/3.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §3.4(i), §3.4 and Ch.3

… ►The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. …

30: 10.23 Sums

… ►For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). … ►(Note that when

x=1

the left-hand side is 1 and the right-hand side is 0.) …