ds电子游戏平台【打开网址∶3399yule.com】ds博彩平台,ds电子游艺网站【官网地址∶3399yule.com】

Did you mean ds电子游戏平台【打开网址∶33929yule.com】ds博彩平台,ds电子游艺网站【官网地址∶33929yule.com】 ?

(0.003 seconds)

1—10 of 17 matching pages

1: 22.13 Derivatives and Differential Equations

… ►

Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{sn}z)$ $\;=\;$	$\operatorname{cn}z\operatorname{dn}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{dc}z)$ =	${k^{\prime}}^{2}\operatorname{sc}z\operatorname{nc}z$
…
$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{cd}z)$ $\;=\;$	$-{k^{\prime}}^{2}\operatorname{sd}z\operatorname{nd}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{ns}z)$ =	$-\operatorname{ds}z\operatorname{cs}z$
$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{sd}z)$ $\;=\;$	$\operatorname{cd}z\operatorname{nd}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{ds}z)$ =	$-\operatorname{cs}z\operatorname{ns}z$
$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{nd}z)$ $\;=\;$	$k^{2}\operatorname{sd}z\operatorname{cd}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{cs}z)$ =	$-\operatorname{ns}z\operatorname{ds}z$

► … ►

22.13.11

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ds}\left(z,k\right)\right)^{% 2}=\left({\operatorname{ds}}^{2}\left(z,k\right)-{k^{\prime}}^{2}\right)\left(% k^{2}+{\operatorname{ds}}^{2}\left(z,k\right)\right),

ⓘ

Symbols:: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.23

\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ds}\left(z,k\right)=(k% ^{2}-{k^{\prime}}^{2})\operatorname{ds}\left(z,k\right)+2{\operatorname{ds}}^{% 3}\left(z,k\right),

ⓘ

Symbols:: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(iii), §22.13 and Ch.22

…

2: 22.6 Elementary Identities

… ►

22.6.2

1+{\operatorname{cs}}^{2}\left(z,k\right)=k^{2}+{\operatorname{ds}}^{2}\left(z% ,k\right)={\operatorname{ns}}^{2}\left(z,k\right),

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.9.4
Permalink:: http://dlmf.nist.gov/22.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(i), §22.6 and Ch.22

… ►

22.6.14

\operatorname{ns}\left(2z,k\right)=\frac{{\operatorname{ns}}^{4}\left(z,k% \right)-k^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k% \right)\operatorname{ns}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.15

\operatorname{ds}\left(2z,k\right)=\frac{k^{2}{k^{\prime}}^{2}+{\operatorname{% ds}}^{4}\left(z,k\right)}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}% \left(z,k\right)\operatorname{ns}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.16

\operatorname{cs}\left(2z,k\right)=\frac{{\operatorname{cs}}^{4}\left(z,k% \right)-{k^{\prime}}^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}% \left(z,k\right)\operatorname{ns}\left(z,k\right)}.

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

… ►

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.

► ►►►

$\operatorname{sn}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sc}\left(z,k^{\prime}\right)$	$\operatorname{dc}\left(iz,k\right)$ $\;=\;$	$\operatorname{dn}\left(z,k^{\prime}\right)$
…
$\operatorname{sd}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sd}\left(z,k^{\prime}\right)$	$\operatorname{ds}\left(iz,k\right)$ $\;=\;$	$-i\operatorname{ds}\left(z,k^{\prime}\right)$
…

► …

3: 22.4 Periods, Poles, and Zeros

… ►

Table 22.4.1: Periods and poles of Jacobian elliptic functions.

► ►►►

Periods	$z$ -Poles
Periods	…
$4K$ , $2K+2iK^{\prime}$	$\operatorname{cn}$	$\operatorname{sd}$	$\operatorname{nc}$	$\operatorname{ds}$
…

► … ►

Table 22.4.2: Periods and zeros of Jacobian elliptic functions.

► ►►►

Periods	$z$ -Zeros
Periods	…
$4K$ , $2K+2iK^{\prime}$	$\operatorname{sd}$	$\operatorname{cn}$	$\operatorname{ds}$	$\operatorname{nc}$
…

► … ►

Table 22.4.3: Half- or quarter-period shifts of variable for the Jacobian elliptic functions.

► ►►►►►

	$u$
…
$\operatorname{cn}u$	$-k^{\prime}\operatorname{sd}z$	$-ik^{\prime}k^{-1}\operatorname{nc}z$	$-ik^{-1}\operatorname{ds}z$	$-\operatorname{cn}z$	$\operatorname{cn}z$	$-\operatorname{cn}z$
…
$\operatorname{sd}u$	${k^{\prime}}^{-1}\operatorname{cn}z$	$-i(kk^{\prime})^{-1}\operatorname{ds}z$	$ik^{-1}\operatorname{nc}z$	$-\operatorname{sd}z$	$\operatorname{sd}z$	$-\operatorname{sd}z$
…
$\operatorname{ds}u$	$k^{\prime}\operatorname{nc}z$	$ikk^{\prime}\operatorname{sd}z$	$-ik\operatorname{cn}z$	$-\operatorname{ds}z$	$\operatorname{ds}z$	$-\operatorname{ds}z$
…

►

4: 22.8 Addition Theorems

… ►

22.8.10

\operatorname{ns}(u+v)=\frac{\operatorname{ns}u\operatorname{ds}v\operatorname% {cs}v-\operatorname{ns}v\operatorname{ds}u\operatorname{cs}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u},

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

►

22.8.11

\operatorname{ds}(u+v)=\frac{\operatorname{ds}u\operatorname{cs}v\operatorname% {ns}v-\operatorname{ds}v\operatorname{cs}u\operatorname{ns}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u},

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

►

22.8.12

\operatorname{cs}(u+v)=\frac{\operatorname{cs}u\operatorname{ds}v\operatorname% {ns}v-\operatorname{cs}v\operatorname{ds}u\operatorname{ns}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u}.

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

…

5: 22.14 Integrals

… ►

22.14.10

\int\operatorname{ns}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ds}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.10
Permalink:: http://dlmf.nist.gov/22.14.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.11

\int\operatorname{ds}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.11
Permalink:: http://dlmf.nist.gov/22.14.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.12

\int\operatorname{cs}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{ds}\left(x,k\right)\right).

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.12
Permalink:: http://dlmf.nist.gov/22.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

…

6: 22.5 Special Values

… ►

Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.

► ►►►

	$z$
…
$\operatorname{ds}z$	$\infty,1$	$k^{\prime},0$	$0,ikk^{\prime}$	$-ik,0$	$\infty,-1$	$\infty,1$	$\infty,-1$
…

► … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as

k\to 0

► ►►►

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\sin z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$\cos z$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$\sec z$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\csc z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\cos z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sin z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\sec z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\csc z$
…

► ►

Table 22.5.4: Limiting forms of Jacobian elliptic functions as

k\to 1

► ►►►

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\tanh z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\coth z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$
…

► …

7: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions

\operatorname{sn}\left(z,k\right)

\operatorname{cn}\left(z,k\right)

\operatorname{dn}\left(z,k\right)

; the nine subsidiary Jacobian elliptic functions

\operatorname{cd}\left(z,k\right)

\operatorname{sd}\left(z,k\right)

\operatorname{nd}\left(z,k\right)

\operatorname{dc}\left(z,k\right)

\operatorname{nc}\left(z,k\right)

\operatorname{sc}\left(z,k\right)

\operatorname{ns}\left(z,k\right)

\operatorname{ds}\left(z,k\right)

\operatorname{cs}\left(z,k\right)

; the amplitude function

\operatorname{am}\left(x,k\right)

; Jacobi’s epsilon and zeta functions

\mathcal{E}\left(x,k\right)

and

\mathrm{Z}\left(x|k\right)

. …

8: 22.15 Inverse Functions

… ►are denoted respectively by … ►

22.15.22

\operatorname{arcds}\left(x,k\right)=\int_{x}^{\infty}\frac{\,\mathrm{d}t}{% \sqrt{(t^{2}+k^{2})(t^{2}-{k^{\prime}}^{2})}},

k^{\prime}\leq x<\infty

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcds}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 17.4.50
Permalink:: http://dlmf.nist.gov/22.15.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §22.15(ii), §22.15 and Ch.22

…

9: 22.16 Related Functions

… ►

22.16.24

\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({\operatorname{ns}}^{2}\left(t,% k\right)-t^{-2}\right)\,\mathrm{d}t+x^{-1}+x-\operatorname{cn}\left(x,k\right)% \operatorname{ds}\left(x,k\right),

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus
A&S Ref:: 16.26.10
Referenced by:: §22.16(ii)
Permalink:: http://dlmf.nist.gov/22.16.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

►

22.16.25

\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({\operatorname{ds}}^{2}\left(t,% k\right)-t^{-2}\right)\,\mathrm{d}t+x^{-1}+{k^{\prime}}^{2}x-\operatorname{cn}% \left(x,k\right)\operatorname{ds}\left(x,k\right),

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.26.11
Permalink:: http://dlmf.nist.gov/22.16.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

►

22.16.26

\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({\operatorname{cs}}^{2}\left(t,% k\right)-t^{-2}\right)\,\mathrm{d}t+x^{-1}-\operatorname{cn}\left(x,k\right)% \operatorname{ds}\left(x,k\right).

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus
A&S Ref:: 16.26.12
Referenced by:: §22.16(ii)
Permalink:: http://dlmf.nist.gov/22.16.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

…

10: 22.3 Graphics

… ►Line graphs of the functions

\operatorname{sn}\left(x,k\right)

\operatorname{cn}\left(x,k\right)

\operatorname{dn}\left(x,k\right)

\operatorname{cd}\left(x,k\right)

\operatorname{sd}\left(x,k\right)

\operatorname{nd}\left(x,k\right)

\operatorname{dc}\left(x,k\right)

\operatorname{nc}\left(x,k\right)

\operatorname{sc}\left(x,k\right)

\operatorname{ns}\left(x,k\right)

\operatorname{ds}\left(x,k\right)

, and

\operatorname{cs}\left(x,k\right)

for representative values of real

x

and real

k

illustrating the near trigonometric (

k=0

), and near hyperbolic (

k=1

) limits. …