# §22.1 Special Notation

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

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

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.