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22 Jacobian Elliptic FunctionsNotation

§22.1 Special Notation

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

x,y real variables.
z complex variable.
k modulus. Except in §§22.3(iv), 22.17, and 22.19, 0k1.
k complementary modulus, k2+k2=1. If k[0,1], then k[0,1].
K, K K(k), K(k)=K(k) (complete elliptic integrals of the first kind (§19.2(ii))).
q nome. 0q<1 except in §22.17; see also §20.1.
τ iK/K.

All derivatives are denoted by differentials, not primes.

The functions treated in this chapter are the three principal Jacobian elliptic functions sn(z,k), cn(z,k), dn(z,k); the nine subsidiary Jacobian elliptic functions cd(z,k), sd(z,k), nd(z,k), dc(z,k), nc(z,k), sc(z,k), ns(z,k), ds(z,k), cs(z,k); the amplitude function am(x,k); Jacobi’s epsilon and zeta functions (x,k) and Z(x|k).

The notation sn(z,k), cn(z,k), dn(z,k) is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for sn(z,k) are sn(z|m) and sn(z,m) with m=k2; see Abramowitz and Stegun (1964) and Walker (1996). Similarly for the other functions.