right-hand rule for cross products

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11: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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12: 3.8 Nonlinear Equations

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§3.8(ii) Newton’s Rule

… ► … ►Newton’s rule is given by … ►Another iterative method is Halley’s rule: … ►For moderate or large values of $n$ it is not uncommon for the magnitude of the right-hand side of (3.8.14) to be very large compared with unity, signifying that the computation of zeros of polynomials is often an ill-posed problem. …

13: 6.7 Integral Representations

… ►The path of integration does not cross the negative real axis or pass through the origin. …The first integrals on the right-hand sides apply when ; the second ones when $\mathrm{\Re }z\ge 0$ and (in the case of (6.7.14)) $z\ne 0$. …

14: 8.2 Definitions and Basic Properties

… ►However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ take their principal values; compare §4.2(i). … ►(8.2.9) also holds when $a$ is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. …

15: 10.67 Asymptotic Expansions for Large Argument

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). … ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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16: 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 $\mathbf{A}$ 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 …

17: 18.15 Asymptotic Approximations

… ►Also, when , the right-hand side of (18.15.12) with $M=\mathrm{\infty }$ 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 $$L_n^{(\alpha )}\left(\nu x\right)={(-1)}^n\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha -\frac{1}{2}}{x}^{\frac{1}{2}\alpha +\frac{1}{4}}}{\left(\frac{\zeta }{x-1}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{m=0}^{M-1}\frac{E_m(\zeta )}{{\nu }^{2m}}+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{m=0}^{M-1}\frac{F_m(\zeta )}{{\nu }^{2m}}+\mathrm{envAi}\left({\nu }^{\frac{2}{3}}\zeta \right)O\left(\frac{1}{{\nu }^{2M-\frac{2}{3}}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\sim$: Poincaré asymptotic expansion, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$ , $\mathrm{e}$: 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

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18: 21.6 Products

§21.6 Products

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21.6.3 $$\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{\mathbf{z}}_k\right|\mathbf{\Omega })=\frac{1}{{𝒟}^g}\sum _{\mathbf{A}\in 𝒦}\sum _{\mathbf{B}\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}tr\left[\frac{1}{2}{\mathbf{A}}^{\mathrm{T}}\mathbf{\Omega }\mathbf{A}+{\mathbf{A}}^{\mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\prod _{j=1}^h\theta \left({\mathbf{z}}_j+\mathbf{\Omega }{\mathbf{a}}_j+{\mathbf{b}}_j\right|\mathbf{\Omega }),$$

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Symbols:

$\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $tr\mathbf{A}$: trace of matrix, $g$: positive integer, $h$: positive integer, $\mathbf{\Omega }$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

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21.6.4 $$\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{\sum }_{k=1}^h{T}_{jk}{\mathbf{c}}_k}{{\sum }_{k=1}^h{T}_{jk}{\mathbf{d}}_k}\right]\left(\sum _{k=1}^h{T}_{jk}{\mathbf{z}}_k\right|\mathbf{\Omega })=\frac{1}{{𝒟}^g}\sum _{\mathbf{A}\in 𝒦}\sum _{\mathbf{B}\in 𝒦}{\mathrm{e}}^{-2\pi \mathrm{i}{\sum }_{j=1}^h{\mathbf{b}}_j\cdot {\mathbf{c}}_j}\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{\mathbf{a}}_j+{\mathbf{c}}_j}{{\mathbf{b}}_j+{\mathbf{d}}_j}\right]\left({\mathbf{z}}_j\right|\mathbf{\Omega }),$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $g$: positive integer, $h$: positive integer, $\mathbf{\Omega }$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

… ►Many identities involving products of theta functions can be established using these formulas. … ►

21.6.5

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Referenced by:

Erratum (V1.0.17) for Equation (21.6.5)

Permalink:

http://dlmf.nist.gov/21.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the prefactor $\frac{1}{2}$ on the right-hand side was missing.

Reported 2017-08-12 by Wolfgang Bauhardt

See also:

Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

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19: 16.8 Differential Equations

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16.8.8 $${}_{q+1}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}\left(\prod _{\begin{array}{c}\hfill k=1\hfill \\ \hfill k\ne j\hfill \end{array}}^{q+1}\frac{\mathrm{\Gamma }\left(a_k-a_j\right)}{\mathrm{\Gamma }\left(a_k\right)}/\prod _{k=1}^q\frac{\mathrm{\Gamma }\left(b_k-a_j\right)}{\mathrm{\Gamma }\left(b_k\right)}\right){\tilde{w}}_j(z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$.

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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 ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$
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 $w$: 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

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16.8.9 $$\left(\prod _{k=1}^{q+1}\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_{q+1}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}{\left(z_0-z\right)}^{-a_j}\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a_j+n\right)}{n!}\left(\prod _{\begin{array}{c}\hfill k=1\hfill \\ \hfill k\ne j\hfill \end{array}}^{q+1}\mathrm{\Gamma }\left(a_k-a_j-n\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k-a_j-n\right)\right){}_{q+1}F_q(\genfrac{}{}{0pt}{}{a_1-a_j-n,\mathrm{\dots },a_{q+1}-a_j-n}{b_1-a_j-n,\mathrm{\dots },b_q-a_j-n};z_0){\left(z-z_0\right)}^{-n}.$$

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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 ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$
generalized hypergeometric function, $!$: factorial (as in $n!$), $q$: 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:

§16.8(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 $z_0$.) …

20: 26.10 Integer Partitions: Other Restrictions

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26.10.2 $$\sum _{n=0}^{\mathrm{\infty }}p(𝒟,n)q^n=\prod _{j=1}^{\mathrm{\infty }}(1+q^j)=\prod _{j=1}^{\mathrm{\infty }}\frac{1}{1-q^{2j-1}}=1+\sum _{m=1}^{\mathrm{\infty }}\frac{q^{m(m+1)/2}}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^m)}=1+\sum _{m=1}^{\mathrm{\infty }}{q}^m(1+q)(1+q^2)\mathrm{\cdots }(1+q^{m-1}),$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. ►

26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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26.10.5 $$\sum _{n=0}^{\mathrm{\infty }}p(\in S,n)q^n=\prod _{j\in S}\frac{1}{1-q^j}.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $n$: nonnegative integer, $S$: set and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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