§22.1 Special Notation

Keywords:: Glaisher’s notation, Jacobian elliptic functions, elliptic functions, epsilon function
Referenced by:: §23.15(i)
Permalink:: http://dlmf.nist.gov/22.1

(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 $k^{{\prime}}\in[0,1]$ .
$\mathop{K\/}\nolimits$ , $\mathop{{K^{{\prime}}}\/}\nolimits$	$\mathop{K\/}\nolimits\!\left(k\right)$ , $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)=\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right)$ (complete elliptic integrals of the first kind (§19.2(ii))).
$q$	nome. $0\leq q<1$ except in §22.17; see also §20.1.
$\tau$	$i\mathop{{K^{{\prime}}}\/}\nolimits/\mathop{K\/}\nolimits$ .

All derivatives are denoted by differentials, not primes.

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

The notation $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ , $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ , $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ 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.