Jacobian elliptic functions

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31: 29.8 Integral Equations

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29.8.1 $$x=k^2\mathrm{sn}(z,k)\mathrm{sn}(z_1,k)\mathrm{sn}(z_2,k)\mathrm{sn}(z_3,k)-\frac{k^2}{k_{}^{\prime }{}_{}{}^2}\mathrm{cn}(z,k)\mathrm{cn}(z_1,k)\mathrm{cn}(z_2,k)\mathrm{cn}(z_3,k)+\frac{1}{k_{}^{\prime }{}_{}{}^2}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k)\mathrm{dn}(z_2,k)\mathrm{dn}(z_3,k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $x$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E1

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Annotations for §29.8 and Ch.29

►where $z,z_1,z_2,z_3$ are real, and $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ are the Jacobian elliptic functions (§22.2). … ►

29.8.6 $$y=\frac{1}{k^{\prime }}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $y$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E6

Encodings:

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Annotations for §29.8 and Ch.29

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29.8.7 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z_1,k^2)\frac{w_2(K)+w_2(-K)}{w_2(0)}=-k^2\mathrm{sn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)dz,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: 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

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http://dlmf.nist.gov/29.8.E7

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See also:

Annotations for §29.8 and Ch.29

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29.8.9 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}-{d{w}_2(z)/dz|}_{z=-K}}{w_2(0)}=-\frac{k^4}{k^{\prime }}\mathrm{sn}(z_1,k)\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\mathrm{cn}(z,k)\frac{d^2{𝖯}_{\nu }\left(y\right)}{{dy}^2}{\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)dz.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: 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.E9

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See also:

Annotations for §29.8 and Ch.29

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32: 20.9 Relations to Other Functions

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§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. …

33: 19.25 Relations to Other Functions

… ►Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … ►

§19.25(v) Jacobian Elliptic Functions

… ►For the use of $R$-functions with $\mathrm{\Delta }(\mathrm{p},\mathrm{q})$ in unifying other properties of Jacobian elliptic functions, see Carlson (2004, 2006a, 2006b, 2008). ►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_F(x,y,z)$. … …

34: 23.6 Relations to Other Functions

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§23.6(ii) Jacobian Elliptic Functions

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23.6.21 $$\mathrm{\wp }\left(z\right)-e_1=\frac{K^2}{{\omega }_1^2}{\mathrm{cs}}^2(\frac{Kz}{{\omega }_1},k),$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E21

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Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.22 $$\mathrm{\wp }\left(z\right)-e_2=\frac{K^2}{{\omega }_1^2}{\mathrm{ds}}^2(\frac{Kz}{{\omega }_1},k),$$

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Symbols:

$\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

Permalink:

http://dlmf.nist.gov/23.6.E22

Encodings:

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See also:

Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.23 $$\mathrm{\wp }\left(z\right)-e_3=\frac{K^2}{{\omega }_1^2}{\mathrm{ns}}^2(\frac{Kz}{{\omega }_1},k).$$

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Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.9.11

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E23

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See also:

Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.24 $$\mathrm{\wp }\left(z+{\omega }_1\right)-e_1={\left(\frac{K{k}^{\prime }}{{\omega }_1}\right)}^2{\mathrm{sc}}^2(\frac{Kz}{{\omega }_1},k),$$

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Symbols:

$\mathrm{sc}(z,k)$: Jacobian elliptic function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E24

Encodings:

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See also:

Annotations for §23.6(ii), §23.6 and Ch.23

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35: 29.10 Lamé Functions with Imaginary Periods

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29.10.3 $$\frac{d^{2}w}{{d{z}^{\prime }}^2}+(h^{\prime }-\nu (\nu +1)k_{}^{\prime }{}_{}{}^2{\mathrm{sn}}^2(z^{\prime },k^{\prime }))w=0.$$

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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 and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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36: 29.2 Differential Equations

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29.2.1 $$\frac{d^{2}w}{{dz}^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(z,k))w=0,$$

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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 and $w(z)$: solution

Referenced by:

§29.10, §29.11, §29.12(i), §29.17(i), §29.17(ii), §29.17(iii), §29.18(i), §29.2(i), §29.20(i), §29.3(i), §29.7(ii), §29.8, §29.9

Permalink:

http://dlmf.nist.gov/29.2.E1

Encodings:

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Annotations for §29.2(i), §29.2 and Ch.29

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§29.2(ii) Other Forms

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29.2.3 $$\xi ={\mathrm{sn}}^2(z,k).$$

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Defines:

$\xi$: variable (locally)

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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37: 31.2 Differential Equations

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31.2.8 $$\frac{d^{2}w}{{d\zeta }^2}+\left((2\gamma -1)\frac{\mathrm{cn}\zeta \mathrm{dn}\zeta }{\mathrm{sn}\zeta }-(2\delta -1)\frac{\mathrm{sn}\zeta \mathrm{dn}\zeta }{\mathrm{cn}\zeta }-(2ϵ-1)k^2\frac{\mathrm{sn}\zeta \mathrm{cn}\zeta }{\mathrm{dn}\zeta }\right)\frac{dw}{d\zeta }+4k^2(\alpha \beta {\mathrm{sn}}^2\zeta -q)w=0.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\zeta$: change of variable and $k$: modulus

Referenced by:

§31.7(ii)

Permalink:

http://dlmf.nist.gov/31.2.E8

Encodings:

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See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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38: 29.12 Definitions

… ►In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). …

39: 29.7 Asymptotic Expansions

… ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu \to \mathrm{\infty }$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …

40: Bibliography C

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Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.

Encodings:

BibTeX

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Jacobian elliptic functions as inverses of an integral. J. Comput. Appl. Math. 174 (2), pp. 355–359.

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External Links:

ISSN 0377-0427, Document, MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

BibTeX

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Some reformulated properties of Jacobian elliptic functions. J. Math. Anal. Appl. 323 (1), pp. 522–529.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.13(ii), §22.13(iii), §22.14(i), §22.14(ii).

Encodings:

BibTeX

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

Encodings:

BibTeX

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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:

ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

BibTeX

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