Jacobi elliptic

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21: 22.20 Methods of Computation

… ►To compute $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ to 10D when $x=0.8$, $k=0.65$. … ►Then from (22.20.5), $\mathrm{sn}(0.8,0.65)=\mathrm{0.69506\hspace{0.33em}42165}$, $\mathrm{cn}(0.8,0.65)=\mathrm{0.71894\hspace{0.33em}76580}$, $\mathrm{dn}(0.8,0.65)=\mathrm{0.89212\hspace{0.33em}34349}$. … ►Then by using (22.7.4) we have $\mathrm{dn}(0.2,\sqrt{0.19})=0.996253$. ►If needed, the corresponding values of $\mathrm{sn}$ and $\mathrm{cn}$ can be found subsequently by applying (22.10.4) and (22.7.2), followed by (22.10.5) and (22.7.3). … ►

§22.20(vi) Related Functions

…

22: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.2 $$2Kk\mathrm{sn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-(n+\frac{1}{2})\tau }\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.8 $$2K\mathrm{dc}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t+\frac{1}{2}-m-n\tau }\right),$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.11 $$2K\mathrm{ns}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-n\tau }\right),$$

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Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

►

22.12.12 $$2K\mathrm{ds}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^{m+n}}{t-m-n\tau }\right),$$

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Symbols:

$\mathrm{ds}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

►

22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

ⓘ

Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

23: 29.17 Other Solutions

… ►They are algebraic functions of $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, and $\mathrm{dn}(z,k)$, and have primitive period $8K$. … ►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …

24: 29.12 Definitions

… ►These functions are polynomials in $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, and $\mathrm{dn}(z,k)$. … ►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). … ►

Table 29.12.1: Lamé polynomials.

► ►►►►

$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
…
$2n+2$	$b_{\nu }^{2m+2}\left(k^2\right)$	${\mathrm{𝑠𝑐𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}\mathrm{cn}P({\mathrm{sn}}^2)$	$2K$	$4\mathrm{i}{K}^{\prime }$	odd	odd	even
…
$2n+3$	$b_{\nu }^{2m+2}\left(k^2\right)$	${\mathrm{𝑠𝑐𝑑𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}\mathrm{cn}\mathrm{dn}P({\mathrm{sn}}^2)$	$2K$	$2\mathrm{i}{K}^{\prime }$	odd	odd	odd

►

… ►With the substitution $\xi ={\mathrm{sn}}^2(z,k)$ every Lamé polynomial in Table 29.12.1 can be written in the form …

25: 29.15 Fourier Series and Chebyshev Series

… ►Since (29.2.5) implies that $\cos \varphi =\mathrm{sn}(z,k)$, (29.15.1) can be rewritten in the form …This determines the polynomial $P$ of degree $n$ for which ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)=P({\mathrm{sn}}^2(z,k))$; compare Table 29.12.1. … ►

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

… ►

29.15.49 $${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

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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

►

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

…

26: 22.19 Physical Applications

… ►

§22.19(i) Classical Dynamics: The Pendulum

… ►

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

… ►Figure 22.19.1 shows the nature of the solutions $\theta (t)$ of (22.19.3) by graphing $\mathrm{am}(x,k)$ for both $0\le k\le 1$, as in Figure 22.16.1, and $k\ge 1$, where it is periodic. … ►

22.19.8 $$x(t)=a\mathrm{dn}(t\sqrt{\eta },k).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $k$: modulus, $x(t)$: coordinate, $t$: time, $a$: displacement and $\eta$: parameter

Referenced by:

Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

Permalink:

http://dlmf.nist.gov/22.19.E8

Encodings:

pMML, png, TeX

See also:

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

… ►Both the $\mathrm{dn}$ and $\mathrm{cn}$ solutions approach $a\mathrm{sech}t$ as $a\to \sqrt{2/\beta }$ from the appropriate directions. …

27: 31.2 Differential Equations

… ►

Jacobi’s Elliptic Form

… ►

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

… ►

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:

pMML, png, TeX

See also:

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

…

28: 22.9 Cyclic Identities

… ►

22.9.1 $$s_{m,p}^{(2)}=\mathrm{sn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

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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, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $s_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

►

22.9.2 $$c_{m,p}^{(2)}=\mathrm{cn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $c_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

►

22.9.3 $$d_{m,p}^{(2)}=\mathrm{dn}(z+2p^{-1}(m-1)K\left(k\right),k),$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

►

22.9.4 $$s_{m,p}^{(4)}=\mathrm{sn}(z+4p^{-1}(m-1)K\left(k\right),k),$$

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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, $z$: complex, $k$: modulus, $m$: positive integer, $p$: positive integer and $s_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(i), §22.9 and Ch.22

… ►

22.9.7 $$\kappa =\mathrm{dn}(2K\left(k\right)/3,k),$$

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Defines:

$\kappa$: change of variable (locally)

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $k$: modulus

Referenced by:

§22.9(iii), §22.9(iv)

Permalink:

http://dlmf.nist.gov/22.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

…

29: 19.25 Relations to Other Functions

… ►

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

… ►If ${\mathrm{cs}}^2(u,k)\ge 0$, then ►

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

►

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

… ►where we assume $0\le x^2\le 1$ if $x=\mathrm{sn}$, $\mathrm{cn}$, or $\mathrm{cd}$; $x^2\ge 1$ if $x=\mathrm{ns}$, $\mathrm{nc}$, or $\mathrm{dc}$; $x$ real if $x=\mathrm{cs}$ or $\mathrm{sc}$; $k^{\prime }\le x\le 1$ if $x=\mathrm{dn}$; $1\le x\le 1/k^{\prime }$ if $x=\mathrm{nd}$; $x^2\ge k_{}^{\prime }{}_{}{}^2$ if $x=\mathrm{ds}$; $0\le x^2\le 1/k_{}^{\prime }{}_{}{}^2$ if $x=\mathrm{sd}$. …

30: 29.10 Lamé Functions with Imaginary Periods

… ►

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

ⓘ

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

…