# §22.1 Special Notation

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

 real variables. complex variable. modulus. Except in §§22.3(iv), 22.17, and 22.19, . complementary modulus, . If , then . , (complete elliptic integrals of the first kind (§19.2(ii))). nome. except in §22.17; see also §20.1. .

All derivatives are denoted by differentials, not primes.

The functions treated in this chapter are the three principal Jacobian elliptic functions , , ; the nine subsidiary Jacobian elliptic functions , , , , , , , , ; the amplitude function ; Jacobi’s epsilon and zeta functions and .

The notation , , is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for are and with ; see Abramowitz and Stegun (1964) and Walker (1996). Similarly for the other functions.