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11: 22.6 Elementary Identities

§22.6 Elementary Identities

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§22.6(ii) Double Argument

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§22.6(iii) Half Argument

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§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

… ►See §22.17.

12: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

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§22.17(i) Real or Purely Imaginary Moduli

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§22.17(ii) Complex Moduli

►When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^2$. …For proofs of these results and further information see Walker (2003).

13: 22.21 Tables

§22.21 Tables

►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. … ►Zhang and Jin (1996, p. 678) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. … ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

14: 22.1 Special Notation

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$x,y$	real variables.
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$K$, $K^{\prime }$	$K\left(k\right)$, $K^{\prime }\left(k\right)=K\left(k^{\prime }\right)$ (complete elliptic integrals of the first kind (§19.2(ii))).
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… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. ►The notation $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\mathrm{sn}(z,k)$ are $\mathrm{sn}(z|m)$ and $\mathrm{sn}(z,m)$ with $m=k^2$; see Abramowitz and Stegun (1964) and Walker (1996). …

15: 22.13 Derivatives and Differential Equations

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§22.13(i) Derivatives

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Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

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$\frac{d}{dz}(\mathrm{sn}z)$ $=$	$\mathrm{cn}z\mathrm{dn}z$	$\frac{d}{dz}(\mathrm{dc}z)$ =	$k_{}^{\prime }{}_{}{}^2\mathrm{sc}z\mathrm{nc}z$
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§22.13(ii) First-Order Differential Equations

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22.13.7 $${\left(\frac{d}{dz}\mathrm{dc}(z,k)\right)}^2=\left({\mathrm{dc}}^2(z,k)-1\right)\left({\mathrm{dc}}^2(z,k)-k^2\right),$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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§22.13(iii) Second-Order Differential Equations

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16: 22.14 Integrals

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§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

… ►See §22.16(i) for $\mathrm{am}(z,k)$. … ►

§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions

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§22.14(iv) Definite Integrals

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17: 22.5 Special Values

§22.5 Special Values

… ►Table 22.5.2 gives $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ for other special values of $z$. … ►

§22.5(ii) Limiting Values of $k$

… ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. … ►

18: 22.3 Graphics

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§22.3(i) Real Variables: Line Graphs

►Line graphs of the functions $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $\mathrm{cd}(x,k)$, $\mathrm{sd}(x,k)$, $\mathrm{nd}(x,k)$, $\mathrm{dc}(x,k)$, $\mathrm{nc}(x,k)$, $\mathrm{sc}(x,k)$, $\mathrm{ns}(x,k)$, $\mathrm{ds}(x,k)$, and $\mathrm{cs}(x,k)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ($k=0$), and near hyperbolic ($k=1$) limits. … ► $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, and $\mathrm{dn}(x,k)$ as functions of real arguments $x$ and $k$. … ►

§22.3(iii) Complex $z$; Real $k$

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§22.3(iv) Complex $k$

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19: 22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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22.7.3 $$\mathrm{cn}(z,k)=\frac{\mathrm{cn}(z/(1+k_1),k_1)\mathrm{dn}(z/(1+k_1),k_1)}{1+k_1{\mathrm{sn}}^2(z/(1+k_1),k_1)},$$

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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, $z$: complex, $k$: modulus and $k_1$: change of variable

A&S Ref:

16.12.3

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(i), §22.7 and Ch.22

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§22.7(ii) Ascending Landen Transformation

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22.7.6 $$\mathrm{sn}(z,k)=\frac{(1+k_2^{\prime })\mathrm{sn}(z/(1+k_2^{\prime }),k_2)\mathrm{cn}(z/(1+k_2^{\prime }),k_2)}{\mathrm{dn}(z/(1+k_2^{\prime }),k_2)},$$

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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, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.2

Permalink:

http://dlmf.nist.gov/22.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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§22.7(iii) Generalized Landen Transformations

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20: 22.18 Mathematical Applications

§22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

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