Jacobi’s

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11: 22.18 Mathematical Applications

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Ellipse

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22.18.3 $$l(u)=aℰ(u,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $k$: modulus, $a$: real, $u$: variable and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

►where $ℰ(u,k)$ is Jacobi’s epsilon function (§22.16(ii)). …

12: 20.11 Generalizations and Analogs

… ►This is Jacobi’s inversion problem of §20.9(ii). … ►Each provides an extension of Jacobi’s inversion problem. …

13: 20.12 Mathematical Applications

… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …

14: 22.20 Methods of Computation

… ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. Jacobi’s zeta function can then be found by use of (22.16.32). …

15: 31.2 Differential Equations

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Jacobi’s Elliptic Form

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16: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Jacobi’s Triple Product

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17: 20.5 Infinite Products and Related Results

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Jacobi’s Triple Product

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18: 22.19 Physical Applications

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right)),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $\theta (t)$: angular displacement and $\alpha$: initial displacement

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.2)

Permalink:

http://dlmf.nist.gov/22.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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22.19.3 $$\theta (t)=2\mathrm{am}(t\sqrt{E/2},\sqrt{2/E}),$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\theta (t)$: angular displacement and $E$: energy

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.3)

Permalink:

http://dlmf.nist.gov/22.19.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{am}$ was given incorrectly as $t$. The correct argument is $t\sqrt{E/2}$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.5,0.9999,1.0001,2$. … Magnify

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19: 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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19.25.30 $$\mathrm{am}(u,k)=R_C({\mathrm{cs}}^2(u,k),{\mathrm{ns}}^2(u,k)),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function and $k$: real or complex modulus

Referenced by:

§19.25(v)

Permalink:

http://dlmf.nist.gov/19.25.E30

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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20: 22.2 Definitions

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22.2.9 $$\mathrm{sc}(z,k)=\frac{{\theta }_3(0,q)}{{\theta }_4(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_2(\zeta ,q)}=\frac{1}{\mathrm{cs}(z,k)}.$$

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Defines:

$\mathrm{cs}(z,k)$: Jacobian elliptic function and $\mathrm{sc}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/22.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

… ► $\mathrm{s}\mathrm{s}(z,k)=1$. …