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22 Jacobian Elliptic FunctionsProperties

§22.5 Special Values

Contents
  1. §22.5(i) Special Values of z
  2. §22.5(ii) Limiting Values of k

§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.
z
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 ,k1 k1,0 1,0 1,0 1,0
sdz 0,1 k1,0 ,i(kk)1 ik1,0 0,1 0,1 0,1
ndz 1,0 k1,0 ,ik1 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 ,k1 ikk1,0 0,ik 1,0 1,0 1,0
scz 0,1 ,k1 ik1,0 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.
z
12K 12(K+iK) 12iK
snz (1+k)1/2 ((1+k)1/2+i(1k)1/2)/(2k)1/2 ik1/2
cnz (k/(1+k))1/2 (1i)k1/2/(2k)1/2 (1+k)1/2k1/2
dnz k1/2 1i2k1/2((1+k)1/2+i(1k)1/2) (1+k)1/2
z
32K 32(K+iK) 32iK
snz (1+k)1/2 ((1+k)1/2+i(1k)1/2)/(2k)1/2 ik1/2
cnz (k/(1+k))1/2 (1i)k1/2/(2k)1/2 (1+k)1/2k1/2
dnz k1/2 i12k1/2((1+k)1/2+i(1k)1/2) (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).