Jacobian elliptic functions

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21: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

… ►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. …

22: Peter L. Walker

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22 Jacobian Elliptic Functions

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23: 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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24: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(i) Notation

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25: 29.18 Mathematical Applications

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29.18.6 $$\frac{d^2{u}_2}{{d\beta }^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(\beta ,k))u_2=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: 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 $$\frac{d^2{u}_3}{{d\gamma }^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(\gamma ,k))u_3=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: 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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26: Bille C. Carlson

… ►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. …

27: 29.11 Lamé Wave Equation

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29.11.1 $$\frac{d^{2}w}{{dz}^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(z,k)+k^2{\omega }^2{\mathrm{sn}}^4(z,k))w=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: 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: 19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

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29: 29.15 Fourier Series and Chebyshev Series

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29.15.45 $${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)=\mathrm{cn}(z,k)\sum _{p=0}^n{B}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: 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 $${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)=\mathrm{dn}(z,k)\left(\frac{1}{2}{C}_0+\sum _{p=1}^n{C}_{2p}{T}_{2p}\left(\mathrm{sn}(z,k)\right)\right),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: 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 $${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)=\mathrm{cn}(z,k)\sum _{p=0}^n{B}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right),$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$: 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 $${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$: 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 $${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right).$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$: 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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30: William P. Reinhardt

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22 Jacobian Elliptic Functions

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