right-hand rule for cross products

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1: 24.19 Methods of Computation

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24.19.1 $$N_{2n}=\frac{2(2n)!}{{(2\pi )}^{2n}}\left(\prod _{p-1|2n}p\right)\left(\prod _p\frac{p^{2n}}{p^{2n}-1}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $p$: prime

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.19(i), §24.19 and Ch.24

►

$D_{2n}={\displaystyle \prod _{p-1|2n}}p,$

… ►If ${\tilde{N}}_{2n}$ denotes the right-hand side of (24.19.1) but with the second product taken only for $p\le \lfloor {(\pi \mathrm{e})}^{-1}2n\rfloor +1$, then $N_{2n}=\lceil {\tilde{N}}_{2n}\rceil$ for $n\ge 2$. …

2: 16.5 Integral Representations and Integrals

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16.5.1 $$\left(\prod _{k=1}^p\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{1}{2\pi \mathrm{i}}{\int }_L\left(\prod _{k=1}^p\mathrm{\Gamma }\left(a_k+s\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k+s\right)\right)\mathrm{\Gamma }\left(-s\right){(-z)}^{s}ds,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§13.6(vi), §16.11(i), §16.5, §16.5

Permalink:

http://dlmf.nist.gov/16.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

… ►In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when . In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector ; compare §16.2(iii). Lastly, when $p>q+1$ the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side. In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta$ is an arbitrary small positive constant. …

3: 19.21 Connection Formulas

… ►The case $z=1$ shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is $\pi /4$. … ►

19.21.7 $$(x-y)R_D(y,z,x)+(z-y)R_D(x,y,z)=3R_F(x,y,z)-3y^{1/2}{x}^{-1/2}{z}^{-1/2},$$

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Symbols:

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

Referenced by:

§19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E7

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 §19.21(ii), §19.21 and Ch.19

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19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

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Symbols:

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

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

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 §19.21(ii), §19.21 and Ch.19

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19.21.10 $$2R_G(x,y,z)=z{R}_F(x,y,z)-\frac{1}{3}(x-z)(y-z)R_D(x,y,z)+x^{1/2}{y}^{1/2}{z}^{-1/2},$$ $z\ne 0$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)

Permalink:

http://dlmf.nist.gov/19.21.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

An incorrect factor of 3 was removed in the last term on the right-hand side.

Suggested 2022-06-27 by Abdulhafeez Abdulsalam

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 §19.21(ii), §19.21 and Ch.19

►Because $R_G$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\le 0$ if the variables are real, thereby avoiding cancellations when $R_G$ is calculated from $R_F$ and $R_D$ (see §19.36(i)). …

4: 19.25 Relations to Other Functions

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19.25.7 $$E(\varphi ,k)=2R_G(c-1,c-k^2,c)-(c-1)R_F(c-1,c-k^2,c)-\sqrt{c-1}\sqrt{c-k^2}/\sqrt{c},$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $E(\varphi ,k)$: 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), §19.25(i), §19.25(iii), Figure 19.3.4, Figure 19.3.4, Figure 19.3.4, §19.36(ii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E7

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 §19.25(i), §19.25 and Ch.19

… ►All terms on the right-hand sides are nonnegative when $k^2\le 0$, $0\le k^2\le 1$, or $1\le k^2\le c$, respectively. … ►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). … ►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which , for some $j$. … ►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which , for some $j$. …

5: 1.6 Vectors and Vector-Valued Functions

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Dot Product (or Scalar Product)

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Cross Product (or Vector Product)

… ►where $𝐧$ is the unit vector normal to $𝐚$ and $𝐛$ whose direction is determined by the right-hand rule; see Figure 1.6.1. ►

►►

Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify

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6: 11.9 Lommel Functions

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11.9.4 $$a_k(\mu ,\nu )=\prod _{m=1}^k\left({(\mu +2m-1)}^2-{\nu }^2\right)=4^k{\left(\frac{\mu -\nu +1}{2}\right)}_k{\left(\frac{\mu +\nu +1}{2}\right)}_k,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Referenced by:

§11.9(iii), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/11.9.E4

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

An alternative Pochhammer symbol representation was added.

See also:

Annotations for §11.9(i), §11.9 and Ch.11

… ►the right-hand side being replaced by its limiting form when $\mu \pm \nu$ is an odd negative integer. … ►If either of $\mu \pm \nu$ equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents $S_{\mu ,\nu }\left(z\right)$ exactly. …

7: 4.24 Inverse Trigonometric Functions: Further Properties

… ►The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. …

8: 25.19 Tables

… ►

Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots }$, 20D (p. 811); ${\mathrm{Li}}_2\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ }(1^{\circ }){30}^{\circ }(2^{\circ }){90}^{\circ }(5^{\circ }){180}^{\circ }$, $f(\theta )+\theta \ln \theta$, $\theta =0(1^{\circ }){15}^{\circ }$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

…

9: 32.15 Orthogonal Polynomials

… ►For this result and applications see Fokas et al. (1991): in this reference, on the right-hand side of Eq. …

10: 16.11 Asymptotic Expansions

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16.11.2 $$H_{p,q}(z)=\sum _{m=1}^p\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\mathrm{\Gamma }\left(a_m+k\right)\left(\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne m\end{array}}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}-a_m-k\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-a_m-k\right)\right)z^{-a_m-k}.$$

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Defines:

$H_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.11(ii), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

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16.11.5 $$e_{k,m}=\sum _{j=1}^{q+1}{\left(1-\nu -\kappa b_j+m\right)}_{\kappa +k-m}\left(\prod _{\mathrm{\ell }=1}^p(a_{\mathrm{\ell }}-b_j)/\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne j\end{array}}^{q+1}(b_{\mathrm{\ell }}-b_j)\right),$$

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Defines:

$e_{k,m}$ (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $p$: nonnegative integer, $q$: nonnegative integer, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $\kappa$ and $\nu$

Referenced by:

§16.11, Erratum (V1.1.9) for Section 16.11(i)

Permalink:

http://dlmf.nist.gov/16.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

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16.11.6 $$\left(\prod _{\mathrm{\ell }=1}^{q+1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=H_{q+1,q}(-z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$;

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $H_{p,q}(z)$: formal infinite series

Permalink:

http://dlmf.nist.gov/16.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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16.11.7 $$\left(\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_q{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_q}{b_1,\mathrm{\dots },b_q};z)\sim H_{q,q}(z{\mathrm{e}}^{\mp \pi \mathrm{i}})+E_{q,q}(z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series and $H_{p,q}(z)$: formal infinite series

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►(Either sign may be used when $\mathrm{ph}z=0$ since the first term on the right-hand side becomes exponentially small compared with the second term.) …