斗地主免费领现金是真的吗【杏彩体育qee9.com】DCS

SearchAdvancedHelp

(0.001 seconds)

11—17 of 17 matching pages

11: 22.16 Related Functions

… ►

22.16.21 $$ℰ(x,k)=-{\int }_0^x{\mathrm{dc}}^2(t,k)dt+x+\mathrm{sn}(x,k)\mathrm{dc}(x,k),$$

ⓘ

Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real and $k$: modulus

A&S Ref:

16.26.7

Referenced by:

§22.16(ii)

Permalink:

http://dlmf.nist.gov/22.16.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

►

22.16.22 $$ℰ(x,k)=-k_{}^{\prime }{}_{}{}^2{\int }_0^x{\mathrm{nc}}^2(t,k)dt+k_{}^{\prime }{}_{}{}^{2}x+\mathrm{sn}(x,k)\mathrm{dc}(x,k),$$

ⓘ

Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.26.8

Permalink:

http://dlmf.nist.gov/22.16.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

►

22.16.23 $$ℰ(x,k)=-k_{}^{\prime }{}_{}{}^2{\int }_0^x{\mathrm{sc}}^2(t,k)dt+\mathrm{sn}(x,k)\mathrm{dc}(x,k).$$

ⓘ

Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{sc}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.26.9

Referenced by:

§22.16(ii)

Permalink:

http://dlmf.nist.gov/22.16.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

…

12: 22.2 Definitions

… ►

22.2.8 $$\mathrm{cd}(z,k)=\frac{{\theta }_3(0,q)}{{\theta }_2(0,q)}\frac{{\theta }_2(\zeta ,q)}{{\theta }_3(\zeta ,q)}=\frac{1}{\mathrm{dc}(z,k)},$$

ⓘ

Defines:

$\mathrm{cd}(z,k)$: Jacobian elliptic function and $\mathrm{dc}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/22.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

…

13: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

$${\mathrm{Dc}}_0(n,m,z)={\mathrm{Mc}}_n^{(3)}\left(z\right){\mathrm{Mc}}_m^{(4)}\left(z\right)-{\mathrm{Mc}}_n^{(4)}\left(z\right){\mathrm{Mc}}_m^{(3)}\left(z\right),$$

►

$${\mathrm{Dc}}_1(n,m,z)=\mathrm{Mc}_n^{(3)}{}_{}{}^{\prime }\left(z\right){\mathrm{Mc}}_m^{(4)}\left(z\right)-\mathrm{Mc}_n^{(4)}{}_{}{}^{\prime }\left(z\right){\mathrm{Mc}}_m^{(3)}\left(z\right),$$

… ►

28.28.47 $$\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{ce}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(c)}{\mathrm{Dc}}_0(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

►

28.28.48 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{ce}}_n^{\prime }(t,h^2){\mathrm{ce}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(c)}{\mathrm{Dc}}_1(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

… ►

28.28.49 $${\widehat{\alpha }}_{n,m}^{(c)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{ce}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)dt={(-1)}^{p+1}\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{ce}}_n(0,h^2){\mathrm{ce}}_m(0,h^2)}{h{\mathrm{Dc}}_0(n,m,0)}.$$

ⓘ

Defines:

${\widehat{\alpha }}_{n,m}$ (locally)

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.28.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

…

14: 22.11 Fourier and Hyperbolic Series

… ►

22.11.10 $$\mathrm{dc}(z,k)-\frac{\pi }{2K}\sec \zeta =\frac{2\pi }{K}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{2n+1}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $\sec z$: secant function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.7

Permalink:

http://dlmf.nist.gov/22.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

…

15: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.8 $$2K\mathrm{dc}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t+\frac{1}{2}-m-n\tau }\right),$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

…

16: Errata

… ►

Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime }$). The correct entries are $-k^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}{K}^{\prime }$	$z+\mathrm{i}{K}^{\prime }$	$z+2K$	$z+2K+2\mathrm{i}{K}^{\prime }$	$z+2\mathrm{i}{K}^{\prime }$
$\mathrm{⋮}$
$\mathrm{cd}u$	$-\mathrm{sn}z$	$-k^{-1}\mathrm{ns}z$	$k^{-1}\mathrm{dc}z$	$-\mathrm{cd}z$	$-\mathrm{cd}z$	$\mathrm{cd}z$
$\mathrm{⋮}$
$\mathrm{dc}u$	$-\mathrm{ns}z$	$-k\mathrm{sn}z$	$k\mathrm{cd}z$	$-\mathrm{dc}z$	$-\mathrm{dc}z$	$\mathrm{dc}z$
$\mathrm{⋮}$

Reported 2014-02-28 by Svante Janson.

… ►

Originally the limiting form for $\mathrm{sc}(z,k)$ in the last line of this table was incorrect ($\cosh z$, instead of $\sinh z$).

$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
$\mathrm{cn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{sd}(z,k)$ $\to$	$\sinh z$	$\mathrm{nc}(z,k)$ $\to$	$\cosh z$	$\mathrm{ds}(z,k)$ $\to$	$\mathrm{csch}z$
$\mathrm{dn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{nd}(z,k)$ $\to$	$\cosh z$	$\mathrm{sc}(z,k)$ $\to$	$\sinh z$	$\mathrm{cs}(z,k)$ $\to$	$\mathrm{csch}z$

Reported 2010-11-23.

…

17: 19.25 Relations to Other Functions

… ►where we assume $0\le x^2\le 1$ if $x=\mathrm{sn}$, $\mathrm{cn}$, or $\mathrm{cd}$; $x^2\ge 1$ if $x=\mathrm{ns}$, $\mathrm{nc}$, or $\mathrm{dc}$; $x$ real if $x=\mathrm{cs}$ or $\mathrm{sc}$; $k^{\prime }\le x\le 1$ if $x=\mathrm{dn}$; $1\le x\le 1/k^{\prime }$ if $x=\mathrm{nd}$; $x^2\ge k_{}^{\prime }{}_{}{}^2$ if $x=\mathrm{ds}$; $0\le x^2\le 1/k_{}^{\prime }{}_{}{}^2$ if $x=\mathrm{sd}$. …