# §22.5 Special Values

## §22.5(i) Special Values of

Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its -derivative (or at a pole, the residue), for values of that are integer multiples of , . For example, at , , . (The modulus is suppressed throughout the table.)

Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.
0
,
,
,

Table 22.5.2 gives , , for other special values of . For example, . For the other nine functions ratios can be taken; compare (22.2.10).

## §22.5(ii) Limiting Values of

If , then and ; if , then and . In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. See Tables 22.5.3 and 22.5.4.

Table 22.5.3: Limiting forms of Jacobian elliptic functions as .
 1 1

Expansions for as or 1 are given in §§19.5, 19.12.

For values of when (lemniscatic case) see §23.5(iii), and for (equianharmonic case) see §23.5(v).