办假的怀俄明州结婚证【仿证 微kaa77788】】incorrectlis

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1: 10.37 Inequalities; Monotonicity

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2: 31.16 Mathematical Applications

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3: 22.19 Physical Applications

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right)),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $\theta (t)$: angular displacement and $\alpha$: initial displacement

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.2)

Permalink:

http://dlmf.nist.gov/22.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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22.19.3 $$\theta (t)=2\mathrm{am}(t\sqrt{E/2},\sqrt{2/E}),$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\theta (t)$: angular displacement and $E$: energy

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.3)

Permalink:

http://dlmf.nist.gov/22.19.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{am}$ was given incorrectly as $t$. The correct argument is $t\sqrt{E/2}$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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22.19.6 $$x(t)=a\mathrm{cn}(t\sqrt{1+2\eta },k).$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $k$: modulus, $x(t)$: coordinate, $t$: time, $a$: displacement and $\eta$: parameter

Referenced by:

Erratum (V1.0.9) for Equation (22.19.6), Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

Permalink:

http://dlmf.nist.gov/22.19.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.9):

Originally the term $\sqrt{1+2\eta }$ was given incorrectly as $\sqrt{1+\eta }$ in this equation and in the line above. Additionally, for improved clarity, the modulus $k$ has been defined in the line above. Previously $1/\sqrt{2+{\eta }^{-1}}$ was given explicitly in the equation.

Reported 2014-04-28 by Svante Janson

See also:

Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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4: 32.13 Reductions of Partial Differential Equations

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5: 10.19 Asymptotic Expansions for Large Order

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10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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6: Errata

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Previously the exponents of the associated Legendre differential equation (14.2.2) at infinity were given incorrectly by $\{-\nu -1,\nu \}$. These were replaced by $\{\nu +1,-\nu \}$.

Reported by Hans Volkmer on 2019-01-30

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Originally it was stated incorrectly that $\rho$ was fixed. This has been corrected to state that $\eta \rho$ is fixed.

Reported by Ian Thompson on 2018-05-17

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Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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Originally the Stirling number $s(10,6)$ was given incorrectly as 6327. The correct number is 63273.

$n$	$k$
	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
10	$0$	$-\mathrm{3\hspace{0.17em}62880}$	$\mathrm{10\hspace{0.17em}26576}$	$-\mathrm{11\hspace{0.17em}72700}$	$\mathrm{7\hspace{0.17em}23680}$	$-\mathrm{2\hspace{0.17em}69325}$	$63273$	$-9450$	$870$	$-45$	$1$

Reported 2013-11-25 by Svante Janson.

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7: 22.9 Cyclic Identities

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22.9.8 $$s_{1,3}^{(4)}{s}_{2,3}^{(4)}+s_{2,3}^{(4)}{s}_{3,3}^{(4)}+s_{3,3}^{(4)}{s}_{1,3}^{(4)}=\frac{{\kappa }^2-1}{k^2},$$

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Symbols:

$k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $s_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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22.9.9 $$c_{1,3}^{(4)}{c}_{2,3}^{(4)}+c_{2,3}^{(4)}{c}_{3,3}^{(4)}+c_{3,3}^{(4)}{c}_{1,3}^{(4)}=-\frac{\kappa (\kappa +2)}{{(1+\kappa )}^2},$$

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Symbols:

$c_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $c_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2).$$

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Symbols:

$d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

§22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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8: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

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9: 17.13 Integrals

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

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10: 21.3 Symmetry and Quasi-Periodicity

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21.3.4 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶+{𝐦}_1}{𝜷+{𝐦}_2}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}𝜶\cdot {𝐦}_2}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Referenced by:

Erratum (V1.0.5) for Equation (21.3.4)

Permalink:

http://dlmf.nist.gov/21.3.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the vector ${𝐦}_2$ on the right-hand-side was given incorrectly as ${𝐦}_1$.

Reported 2012-08-27 by Klaas Vantournhout

See also:

Annotations for §21.3(ii), §21.3 and Ch.21

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