general elliptic functions

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1: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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§19.2(ii) Legendre’s Integrals

… ►The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. … ►

§19.2(iii) Bulirsch’s Integrals

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§19.2(iv) A Related Function: $R_C(x,y)$

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2: 21.8 Abelian Functions

… ►In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

3: 23.2 Definitions and Periodic Properties

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23.2.9 $$\mathrm{\wp }\left(z+2{\omega }_j\right)=\mathrm{\wp }\left(z\right),$$ $j=1,2,3$.

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $𝕃$: lattice, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

A&S Ref:

18.2.18

Permalink:

http://dlmf.nist.gov/23.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(iii), §23.2 and Ch.23

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4: 23.6 Relations to Other 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(iii) General Elliptic Functions

►For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma \left(z\right)$ and $\mathrm{\wp }\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta \left(z\right)$ see Lawden (1989, §8.11). …

5: 22.2 Definitions

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22.2.10 $$\mathrm{p}\mathrm{q}(z,k)=\frac{\mathrm{p}\mathrm{r}(z,k)}{\mathrm{q}\mathrm{r}(z,k)}=\frac{1}{\mathrm{q}\mathrm{p}(z,k)},$$

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Defines:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function

Symbols:

$z$: complex and $k$: modulus

A&S Ref:

16.3.4

Referenced by:

§22.14(iv), §22.20(ii), §22.4(ii), §22.5(i)

Permalink:

http://dlmf.nist.gov/22.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2, §22.2 and Ch.22

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22.2.11 $$\mathrm{p}\mathrm{q}(z,k)={\theta }_p\left(z\right|\tau )/{\theta }_q\left(z\right|\tau ),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function, $q$: nome, $z$: complex, $k$: modulus and $\tau$: ratio

A&S Ref:

16.36.3

Referenced by:

§22.2

Permalink:

http://dlmf.nist.gov/22.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2, §22.2 and Ch.22

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6: 22.17 Moduli Outside the Interval [0,1]

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22.17.1 $$\mathrm{p}\mathrm{q}(z,k)=\mathrm{p}\mathrm{q}(z,-k),$$

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Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.17(i), §22.17 and Ch.22

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7: 22.7 Landen Transformations

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§22.7(iii) Generalized Landen Transformations

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8: 22.6 Elementary Identities

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22.6.22 $${\mathrm{p}\mathrm{q}}^2(\frac{1}{2}z,k)=\frac{\mathrm{p}\mathrm{s}(z,k)+\mathrm{r}\mathrm{s}(z,k)}{\mathrm{q}\mathrm{s}(z,k)+\mathrm{r}\mathrm{s}(z,k)}=\frac{\mathrm{p}\mathrm{q}(z,k)+\mathrm{r}\mathrm{q}(z,k)}{1+\mathrm{r}\mathrm{q}(z,k)}=\frac{\mathrm{p}\mathrm{r}(z,k)+1}{\mathrm{q}\mathrm{r}(z,k)+1}.$$

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Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function, $z$: complex and $k$: modulus

Referenced by:

§22.6(iii)

Permalink:

http://dlmf.nist.gov/22.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(iii), §22.6 and Ch.22

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9: 19.25 Relations to Other Functions

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19.25.28 $$\mathrm{\Delta }(\mathrm{p},\mathrm{q})={\mathrm{p}\mathrm{s}}^2(u,k)-{\mathrm{q}\mathrm{s}}^2(u,k)=-\mathrm{\Delta }(\mathrm{q},\mathrm{p}),$$

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Defines:

$\mathrm{\Delta }(\mathrm{p},\mathrm{q})$: function (locally)

Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.25.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(v), §19.25 and Ch.19

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19.25.31 $$u=R_F({\mathrm{p}\mathrm{s}}^2(u,k),{\mathrm{q}\mathrm{s}}^2(u,k),{\mathrm{r}\mathrm{s}}^2(u,k));$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function and $k$: real or complex modulus

Referenced by:

§19.25(v)

Permalink:

http://dlmf.nist.gov/19.25.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(v), §19.25 and Ch.19

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10: 22.14 Integrals

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22.14.2 $$\int \mathrm{cn}(x,k)dx=k^{-1}\mathrm{Arccos}\left(\mathrm{dn}(x,k)\right),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real and $k$: modulus

A&S Ref:

16.24.2

Permalink:

http://dlmf.nist.gov/22.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.3 $$\int \mathrm{dn}(x,k)dx=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right)=\mathrm{am}(x,k).$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real and $k$: modulus

A&S Ref:

16.24.3

Permalink:

http://dlmf.nist.gov/22.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.5 $$\int \mathrm{sd}(x,k)dx={(k{k}^{\prime })}^{-1}\mathrm{Arcsin}\left(-k\mathrm{cd}(x,k)\right),$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.5

Permalink:

http://dlmf.nist.gov/22.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.6 $$\int \mathrm{nd}(x,k)dx=k_{}^{\prime }{}_{}{}^{-1}\mathrm{Arccos}\left(\mathrm{cd}(x,k)\right).$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.6

Permalink:

http://dlmf.nist.gov/22.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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