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11: 19.1 Special Notation

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$l, m, n$	nonnegative integers.
…
$k$	real or complex modulus.
$k^{\prime}$	complementary real or complex modulus, $k^{2}+{k^{\prime}}^{2}=1$ .
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12: 19.7 Connection Formulas

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19.7.1

E\left(k\right){K^{\prime}}\left(k\right)+{E^{\prime}}\left(k\right)K\left(k% \right)-K\left(k\right){K^{\prime}}\left(k\right)=\tfrac{1}{2}\pi.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, ${E^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind and $k$ : real or complex modulus
Referenced by:: §19.21(i), §19.35(i), §19.7(i)
Permalink:: http://dlmf.nist.gov/19.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.7(i), §19.7(i), §19.7 and Ch.19

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19.7.8

\Pi\left(\phi,\alpha^{2},k\right)+\Pi\left(\phi,\omega^{2},k\right)=F\left(% \phi,k\right)+\sqrt{c}R_{C}\left((c-1)(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})% \right),

\alpha^{2}\omega^{2}=k^{2}

.

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Defines:: $\omega^{2}$ : change of variable (locally)
Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter and $c$ : change of variable
Referenced by:: §19.25(i), §19.6(i)
Permalink:: http://dlmf.nist.gov/19.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.9

(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(k^{2}-\omega^{2})\Pi\left% (\phi,\omega^{2},k\right)=k^{2}F\left(\phi,k\right)-\alpha^{2}\omega^{2}\sqrt{% c-1}R_{C}\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})\right),

(1-\alpha^{2})(1-\omega^{2})=1-k^{2}

.

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Defines:: $\omega^{2}$ : change of variable (locally)
Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter and $c$ : change of variable
Permalink:: http://dlmf.nist.gov/19.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.10

(1-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(1-\omega^{2})\Pi\left(\phi,% \omega^{2},k\right)=F\left(\phi,k\right)+(1-\alpha^{2}-\omega^{2})\sqrt{c-k^{2% }}\*R_{C}\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})\right),

(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1)

.

ⓘ

Defines:: $\omega^{2}$ : change of variable (locally)
Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter and $c$ : change of variable
Permalink:: http://dlmf.nist.gov/19.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.7(iii), §19.7 and Ch.19

13: 22.13 Derivatives and Differential Equations

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22.13.2

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{% 2}={\left(1-{\operatorname{cn}}^{2}\left(z,k\right)\right)}{\left({k^{\prime}}% ^{2}+k^{2}{\operatorname{cn}}^{2}\left(z,k\right)\right)},

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : 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.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.3

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{dn}}^{2}\left(z,k\right)\right)\left({\operatorname{% dn}}^{2}\left(z,k\right)-{k^{\prime}}^{2}\right).

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Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : 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.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.5

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sd}\left(z,k\right)\right)^{% 2}={\left(1-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right)\right)}{% \left(1+k^{2}{\operatorname{sd}}^{2}\left(z,k\right)\right)},

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Symbols:: $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : 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.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.6

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nd}\left(z,k\right)\right)^{% 2}=\left({\operatorname{nd}}^{2}\left(z,k\right)-1\right)\left(1-{k^{\prime}}^% {2}{\operatorname{nd}}^{2}\left(z,k\right)\right),

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Symbols:: $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : 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.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.8

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nc}\left(z,k\right)\right)^{% 2}={\left(k^{2}+{k^{\prime}}^{2}{\operatorname{nc}}^{2}\left(z,k\right)\right)% }{\left({\operatorname{nc}}^{2}\left(z,k\right)-1\right)},

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Symbols:: $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : 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.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

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14: 22.3 Graphics

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See accompanying text — Figure 22.3.24: $\operatorname{sn}\left(x+iy,k\right)$ for $-4\leq x\leq 4$ , $0\leq y\leq 8$ , $k=1+\tfrac{1}{2}i$ . … Magnify 3D Help

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See accompanying text — Figure 22.3.25: $\operatorname{sn}\left(5,k\right)$ as a function of complex $k^{2}$ , $-1\leq\Re\left(k^{2}\right)\leq 3.5$ , $-1\leq\Im\left(k^{2}\right)\leq 1$ . … Magnify 3D Help

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See accompanying text — Figure 22.3.26: Density plot of $|\operatorname{sn}\left(5,k\right)|$ as a function of complex $k^{2}$ , $-10\leq\Re\left(k^{2}\right)\leq 20$ , $-10\leq\Im\left(k^{2}\right)\leq 10$ . … Magnify

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See accompanying text — Figure 22.3.27: Density plot of $|\operatorname{sn}\left(10,k\right)|$ as a function of complex $k^{2}$ , $-10\leq\Re\left(k^{2}\right)\leq 20$ , $-10\leq\Im\left(k^{2}\right)\leq 10$ . … Magnify

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See accompanying text — Figure 22.3.28: Density plot of $|\operatorname{sn}\left(20,k\right)|$ as a function of complex $k^{2}$ , $-10\leq\Re\left(k^{2}\right)\leq 20$ , $-10\leq\Im\left(k^{2}\right)\leq 10$ . … Magnify

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15: 19.14 Reduction of General Elliptic Integrals

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19.14.1

\int_{1}^{x}\frac{\,\mathrm{d}t}{\sqrt{t^{3}-1}}=3^{-1/4}F\left(\phi,k\right),

\cos\phi=\dfrac{\sqrt{3}+1-x}{\sqrt{3}-1+x}

,

k^{2}=\dfrac{2-\sqrt{3}}{4}

.

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Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.14(i)
Permalink:: http://dlmf.nist.gov/19.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

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19.14.2

\int_{x}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{3}}}=3^{-1/4}F\left(\phi,k\right),

\cos\phi=\dfrac{\sqrt{3}-1+x}{\sqrt{3}+1-x}

,

k^{2}=\dfrac{2+\sqrt{3}}{4}

.

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Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\phi$ : real or complex argument and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

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19.14.3

\int_{0}^{x}\frac{\,\mathrm{d}t}{\sqrt{1+t^{4}}}=\frac{\operatorname{sign}% \left(x\right)}{2}F\left(\phi,k\right),

\cos\phi=\dfrac{1-x^{2}}{1+x^{2}}

,

k^{2}=\dfrac{1}{2}

.

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Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\operatorname{sign}\NVar{x}$ : sign of, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.14(i)
Permalink:: http://dlmf.nist.gov/19.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

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19.14.4

{\int_{y}^{x}\frac{\,\mathrm{d}t}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}}% =\frac{1}{\sqrt{\gamma-\alpha}}F\left(\phi,k\right)},

k^{2}=\ifrac{(\gamma-\beta)}{(\gamma-\alpha)}

.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.14(i), §19.14(i), §19.14(i), §19.15, §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

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16: 19.12 Asymptotic Approximations

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19.12.1

K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left% (\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{% \prime}}\right)+d(m)\right),

0<|k^{\prime}|<1

,

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

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19.12.2

E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2\right)_{m}}m!}{k^{\prime% }}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)-\frac{1}{(2m+1)(2m+% 2)}\right),

|k^{\prime}|<1

,

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

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19.12.4

(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\frac{4}{k^{\prime}}\right% )\left(1+O\left({k^{\prime}}^{2}\right)\right)-\alpha^{2}R_{C}\left(1,1-\alpha% ^{2}\right),

-\infty<\alpha^{2}<1

,

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

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19.12.5

(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\left(\frac{4}{k^{\prime}}% \right)-R_{C}\left(1,1-\alpha^{-2}\right)\right)\*\left(1+O\left({k^{\prime}}^% {2}\right)\right),

1<\alpha^{2}<\infty

.

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

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17: 22.7 Landen Transformations

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22.7.2

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.2
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

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22.7.3

\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.3
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

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22.7.4

\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}}^{2}\left(z/(1+k_{1}),k_% {1}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.4
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

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22.7.6

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.2
Permalink:: http://dlmf.nist.gov/22.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

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22.7.8

\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.4
Permalink:: http://dlmf.nist.gov/22.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

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18: 19.30 Lengths of Plane Curves

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19.30.2

s=a\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}\theta.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

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19.30.3

s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\right)},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

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19.30.5

L(a,b)=4aE\left(k\right)=8aR_{G}\left(0,b^{2}/a^{2},1\right)=8R_{G}\left(0,a^{% 2},b^{2}\right)=8abR_{G}\left(0,a^{-2},b^{-2}\right),

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $L(a,b)$ : length
Referenced by:: §19.15, §19.24(i), §19.30(i), §19.33(i), §19.9(i)
Permalink:: http://dlmf.nist.gov/19.30.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

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19.30.6

\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}}F\left(\phi,k\right)=\sqrt{% a^{2}-b^{2}}R_{F}\left(c-1,c-k^{2},c\right),

k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda)

,

c={\csc}^{2}\phi

.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\csc\NVar{z}$ : cosecant function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

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19.30.12

s=aF\left(\phi,1/\sqrt{2}\right),

\phi=\operatorname{arcsin}\sqrt{2/(q+1)}=\operatorname{arccos}\left(\tan\theta\right)

.

ⓘ

Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $q$ and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(iii), §19.30 and Ch.19

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19: 19.2 Definitions

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19.2.4

F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{\sqrt{1-k^{2}{% \sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{% 1-k^{2}t^{2}}},

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Defines:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.5, §19.6(ii)
Permalink:: http://dlmf.nist.gov/19.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.5

E\left(\phi,k\right)=\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}% \theta\\ =\int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\,\mathrm{d}t.

ⓘ

Defines:: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.6(iii), §22.16(ii)
Permalink:: http://dlmf.nist.gov/19.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.7

\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}(1-\alpha^{2}{\sin}^{2}\theta)}=\int_{0}^{\sin% \phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}.

ⓘ

Defines:: $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.5
Permalink:: http://dlmf.nist.gov/19.2.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.8_1

{K^{\prime}}\left(k\right)=\int_{0}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}% \sqrt{1-(1-k^{2})t^{2}}},

ⓘ

Defines:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $k$ : real or complex modulus
Referenced by:: §19.2(ii)
Permalink:: http://dlmf.nist.gov/19.2.E8_1
Encodings:: pMML, png, TeX
Addition (effective with 1.1.10):: This equation was added.
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.8_2

{E^{\prime}}\left(k\right)=\int_{0}^{1}\frac{\sqrt{1-(1-k^{2})t^{2}}}{\sqrt{1-% t^{2}}}\,\mathrm{d}t,

ⓘ

Defines:: ${E^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the second kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $k$ : real or complex modulus
Referenced by:: §19.2(ii)
Permalink:: http://dlmf.nist.gov/19.2.E8_2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.10):: This equation was added.
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

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20: 22.6 Elementary Identities

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22.6.3

{k^{\prime}}^{2}{\operatorname{sc}}^{2}\left(z,k\right)+1={\operatorname{dc}}^% {2}\left(z,k\right)={k^{\prime}}^{2}{\operatorname{nc}}^{2}\left(z,k\right)+k^% {2},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.9.3
Permalink:: http://dlmf.nist.gov/22.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(i), §22.6 and Ch.22

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22.6.4

-k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right)=k^{2}({% \operatorname{cd}}^{2}\left(z,k\right)-1)={k^{\prime}}^{2}(1-{\operatorname{nd% }}^{2}\left(z,k\right)).

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.9.2
Permalink:: http://dlmf.nist.gov/22.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(i), §22.6 and Ch.22

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22.6.8

\operatorname{cd}\left(2z,k\right)=\frac{{\operatorname{cd}}^{2}\left(z,k% \right)-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right){\operatorname{% nd}}^{2}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(% z,k\right)},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.9

\operatorname{sd}\left(2z,k\right)=\frac{2\operatorname{sd}\left(z,k\right)% \operatorname{cd}\left(z,k\right)\operatorname{nd}\left(z,k\right)}{1+k^{2}{k^% {\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.10

\operatorname{nd}\left(2z,k\right)=\frac{{\operatorname{nd}}^{2}\left(z,k% \right)+k^{2}{\operatorname{sd}}^{2}\left(z,k\right){\operatorname{cd}}^{2}% \left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k% \right)},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

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