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22 Jacobian Elliptic FunctionsProperties

§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. For example, at z=K+iK, sn(z,k)=1/k, dsn(z,k)/dz=0. (The modulus k 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 K K+iK iK 2K 2K+2iK 2iK
snz 0,1 1,0 1/k,0 , 1/k 0,-1 0,-1 0,1
cnz 1,0 0,-k -ik/k,0 , -i/k -1,0 1,0 -1,0
dnz 1,0 k,0 0,ik , -i 1,0 -1,0 -1,0
cdz 1,0 0,-1 ,-k-1 k-1,0 -1,0 -1,0 1,0
sdz 0,1 k-1,0 ,-i(kk)-1 ik-1,0 0,-1 0,1 0,-1
ndz 1,0 k-1,0 ,-ik-1 0,i 1,0 -1,0 -1,0
dcz 1,0 ,-1 0,k k,0 -1,0 -1,0 1,0
ncz 1,0 ,-k-1 ikk-1,0 0,ik -1,0 1,0 -1,0
scz 0,1 ,-k-1 ik-1,0 i,0 0,1 0,-1 0,-1
nsz ,1 1,0 k,0 0,k ,-1 ,-1 ,1
dsz ,1 k,0 0,ikk -ik,0 ,-1 ,1 ,-1
csz ,1 0,-k -ik,0 -i,0 ,1 ,-1 ,-1

Table 22.5.2 gives sn(z,k), cn(z,k), dn(z,k) for other special values of z. For example, sn(12K,k)=(1+k)-1/2. For the other nine functions ratios can be taken; compare (22.2.10).

Table 22.5.2: Other special values of Jacobian elliptic functions.
12K 12(K+iK) 12iK
snz (1+k)-1/2 ((1+k)1/2+i(1-k)1/2)/(2k)1/2 ik-1/2
cnz (k/(1+k))1/2 (1-i)k1/2/(2k)1/2 (1+k)1/2k-1/2
dnz k1/2 k1/2((1+k)1/2-i(1-k)1/2)/21/2 (1+k)1/2
32K 32(K+iK) 32iK
snz (1+k)-1/2 (1+i)((1+k)1/2-i(1-k)1/2)/(2k1/2) -ik-1/2
cnz -(k/(1+k))1/2 (1-i)k1/2/(2k)1/2 -(1+k)1/2k-1/2
dnz k1/2 (-1+i)k1/2((1+k)1/2+i(1-k)1/2)/2 -(1+k)1/2

§22.5(ii) Limiting Values of k

If k0+, then Kπ/2 and K; if k1-, then K and Kπ/2. 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 k0.
sn(z,k) sinz cd(z,k) cosz dc(z,k) secz ns(z,k) cscz
cn(z,k) cosz sd(z,k) sinz nc(z,k) secz ds(z,k) cscz
dn(z,k) 1 nd(z,k) 1 sc(z,k) tanz cs(z,k) cotz
Table 22.5.4: Limiting forms of Jacobian elliptic functions as k1.
sn(z,k) tanhz cd(z,k) 1 dc(z,k) 1 ns(z,k) cothz
cn(z,k) sechz sd(z,k) sinhz nc(z,k) coshz ds(z,k) cschz
dn(z,k) sechz nd(z,k) coshz sc(z,k) sinhz cs(z,k) cschz

Expansions for K,K as k0 or 1 are given in §§19.5, 19.12.

For values of K,K when k2=12 (lemniscatic case) see §23.5(iii), and for k2=eiπ/3 (equianharmonic case) see §23.5(v).