Jacobian

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21: 22.20 Methods of Computation

§22.20 Methods of Computation

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§22.20(iii) Landen Transformations

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§22.20(iv) Lattice Calculations

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§22.20(v) Inverse Functions

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22: 22.21 Tables

§22.21 Tables

… ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

… ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

24: Peter L. Walker

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22 Jacobian Elliptic Functions

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25: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(i) Notation

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26: 29.18 Mathematical Applications

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29.18.6

\frac{{\mathrm{d}}^{2}u_{2}}{{\mathrm{d}\beta}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\beta,k\right))u_{2}=0,

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\beta$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

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29.18.7

\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\gamma,k\right))u_{3}=0,

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

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27: 29.11 Lamé Wave Equation

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29.11.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(h-\nu(\nu+1)k^{2}{\operatorname{% sn}}^{2}\left(z,k\right)+k^{2}\omega^{2}{\operatorname{sn}}^{4}\left(z,k\right% ))w=0,

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\omega$ : parameter and $w(z)$ : solution
Referenced by:: §29.11, §29.11, §29.18(ii), §29.7(ii)
Permalink:: http://dlmf.nist.gov/29.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §29.11 and Ch.29

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28: 29.15 Fourier Series and Chebyshev Series

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29.15.45

\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \sum_{p=0}^{n}B_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)\right),

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E45
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.46

\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}T_{2p}\left(\operatorname{sn}\left% (z,k\right)\right)\right),

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E46
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.47

\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \sum_{p=0}^{n}B_{2p+2}U_{2p+1}\left(\operatorname{sn}\left(z,k\right)\right),

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E47
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.49

\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+1}U_{2p}\left(% \operatorname{sn}\left(z,k\right)\right),

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : 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, $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E49
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.50

\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(% \operatorname{sn}\left(z,k\right)\right).

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : 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, $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E50
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

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29: 19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …

30: 29.8 Integral Equations

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29.8.1

x=k^{2}\operatorname{sn}\left(z,k\right)\operatorname{sn}\left(z_{1},k\right)% \operatorname{sn}\left(z_{2},k\right)\operatorname{sn}\left(z_{3},k\right)-% \frac{k^{2}}{{k^{\prime}}^{2}}\operatorname{cn}\left(z,k\right)\operatorname{% cn}\left(z_{1},k\right)\operatorname{cn}\left(z_{2},k\right)\operatorname{cn}% \left(z_{3},k\right)+\frac{1}{{k^{\prime}}^{2}}\operatorname{dn}\left(z,k% \right)\operatorname{dn}\left(z_{1},k\right)\operatorname{dn}\left(z_{2},k% \right)\operatorname{dn}\left(z_{3},k\right),

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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, $x$ : real variable, $z$ : complex variable and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §29.8 and Ch.29

►where

z,z_{1},z_{2},z_{3}

are real, and

\operatorname{sn}

\operatorname{cn}

\operatorname{dn}

are the Jacobian elliptic functions (§22.2). … ►

29.8.6

y=\frac{1}{k^{\prime}}\operatorname{dn}\left(z,k\right)\operatorname{dn}\left(% z_{1},k\right).

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Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $y$ : real variable, $z$ : complex variable and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §29.8 and Ch.29

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29.8.7

\mathit{Ec}^{2m+1}_{\nu}\left(z_{1},k^{2}\right)\frac{w_{2}(K)+w_{2}(-K)}{w_{2% }(0)}=-k^{2}\operatorname{sn}\left(z_{1},k\right)\int_{-K}^{K}\operatorname{sn% }\left(z,k\right)\frac{\mathrm{d}\mathsf{P}_{\nu}\left(y\right)}{\mathrm{d}y}% \mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)\,\mathrm{d}z,

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $m$ : nonnegative integer, $y$ : real variable, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution
Permalink:: http://dlmf.nist.gov/29.8.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §29.8 and Ch.29

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29.8.9

\mathit{Es}^{2m+2}_{\nu}\left(z_{1},k^{2}\right)\frac{\left.\ifrac{\mathrm{d}w% _{2}(z)}{\mathrm{d}z}\right|_{z=K}-\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}% z}\right|_{z=-K}}{w_{2}(0)}=-\frac{k^{4}}{k^{\prime}}\operatorname{sn}\left(z_% {1},k\right)\operatorname{cn}\left(z_{1},k\right)\int_{-K}^{K}\operatorname{sn% }\left(z,k\right)\operatorname{cn}\left(z,k\right)\frac{{\mathrm{d}}^{2}% \mathsf{P}_{\nu}\left(y\right)}{{\mathrm{d}y}^{2}}\mathit{Es}^{2m+2}_{\nu}% \left(z,k^{2}\right)\,\mathrm{d}z.

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