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

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.