# §22.5 Special Values

## §22.5(i) Special Values of $z$

Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $iK^{\prime}$. For example, at $z=K+iK^{\prime}$, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=1/k$, $\ifrac{\mathrm{d}\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)}{\mathrm{d}z}=0$. (The modulus $k$ is suppressed throughout the table.)

Table 22.5.2 gives $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ for other special values of $z$. For example, $\mathop{\mathrm{sn}\/}\nolimits\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}$. For the other nine functions ratios can be taken; compare (22.2.10).

## §22.5(ii) Limiting Values of $k$

If $k\to 0+$, then $K\to\pi/2$ and $K^{\prime}\to\infty$; if $k\to 1-$, then $K\to\infty$ and $K^{\prime}\to\pi/2$. In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. See Tables 22.5.3 and 22.5.4.

Expansions for $K,K^{\prime}$ as $k\to 0$ or $1$ are given in §§19.5, 19.12.

For values of $K,K^{\prime}$ when $k^{2}=\frac{1}{2}$ (lemniscatic case) see §23.5(iii), and for $k^{2}=e^{\mathrm{i}\pi/3}$ (equianharmonic case) see §23.5(v).