22 Jacobian Elliptic Functions Notation 22 Jacobian Elliptic Functions 22.2 Definitions

§22.1 Special Notation

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Keywords:: Glaisher’s notation, Jacobian elliptic functions, elliptic functions, epsilon function, notation, principal, subsidiary
Referenced by:: §23.15(i)
Permalink:: http://dlmf.nist.gov/22.1
See also:: Annotations for Ch.22

(For other notation see Notation for the Special Functions.)

$x, y$	real variables.
$z$	complex variable.
$k$	modulus. Except in §§22.3(iv), 22.17, and 22.19, $0\leq k\leq 1$ .
$k^{\prime}$	complementary modulus, $k^{2}+{k^{\prime}}^{2}=1$ . If $k\in[0,1]$ , then $k^{\prime}\in[0,1]$ .
$K$ , ${K^{\prime}}$	$K\left(k\right)$ , ${K^{\prime}}\left(k\right)=K\left(k^{\prime}\right)$ (complete elliptic integrals of the first kind (§19.2(ii))).
$q$	nome. $0\leq q<1$ except in §22.17; see also §20.1.
$\tau$	$i{K^{\prime}}/K$ .

All derivatives are denoted by differentials, not primes.

The functions treated in this chapter are the three principal Jacobian elliptic functions $\operatorname{sn}\left(z,k\right)$ , $\operatorname{cn}\left(z,k\right)$ , $\operatorname{dn}\left(z,k\right)$ ; the nine subsidiary Jacobian elliptic functions $\operatorname{cd}\left(z,k\right)$ , $\operatorname{sd}\left(z,k\right)$ , $\operatorname{nd}\left(z,k\right)$ , $\operatorname{dc}\left(z,k\right)$ , $\operatorname{nc}\left(z,k\right)$ , $\operatorname{sc}\left(z,k\right)$ , $\operatorname{ns}\left(z,k\right)$ , $\operatorname{ds}\left(z,k\right)$ , $\operatorname{cs}\left(z,k\right)$ ; the amplitude function $\operatorname{am}\left(x,k\right)$ ; Jacobi’s epsilon and zeta functions $\mathcal{E}\left(x,k\right)$ and $\mathrm{Z}\left(x|k\right)$ .

The notation $\operatorname{sn}\left(z,k\right)$ , $\operatorname{cn}\left(z,k\right)$ , $\operatorname{dn}\left(z,k\right)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\operatorname{sn}\left(z,k\right)$ are $\mathrm{sn}(z\mathpunct{|}m)$ and $\mathrm{sn}(z,m)$ with $m=k^{2}$ ; see Abramowitz and Stegun (1964) and Walker (1996). Similarly for the other functions.

22 Jacobian Elliptic Functions 22.2 Definitions

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