Jacobian

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31: William P. Reinhardt

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22 Jacobian Elliptic Functions

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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: 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:

pMML, png, TeX

See also:

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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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: 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

Encodings:

pMML, png, TeX

See also:

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:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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37: 1.5 Calculus of Two or More Variables

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§1.5(vi) Jacobians and Change of Variables

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Jacobian

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1.5.38

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Symbols:

$det$: determinant, $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/1.5.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(vi), §1.5(vi), §1.5 and Ch.1

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38: 20.1 Special Notation

… ►Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21. … ►This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951). …

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: 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). …