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