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22 Jacobian Elliptic FunctionsProperties

§22.4 Periods, Poles, and Zeros


§22.4(i) Distribution

For each Jacobian function, Table 22.4.1 gives its periods in the z-plane in the left column, and the position of one of its poles in the second row. The other poles are at congruent points, which is the set of points obtained by making translations by 2mK+2nK, where m,n. For example, the poles of sn(z,k), abbreviated as sn in the following tables, are at z=2mK+(2n+1)K.

Table 22.4.1: Periods and poles of Jacobian elliptic functions.
Periods z-Poles
K K+K K 0
4K, 2K sn cd dc ns
4K, 2K+2K cn sd nc ds
2K, 4K dn nd sc cs

Three functions in the same column of Table 22.4.1 are copolar, and four functions in the same row are coperiodic.

Table 22.4.2 displays the periods and zeros of the functions in the z-plane in a similar manner to Table 22.4.1. Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form 2mK+2nK, where m,n.

Table 22.4.2: Periods and zeros of Jacobian elliptic functions.
Periods z-Zeros
0 K K+K K
4K, 2K sn cd dc ns
4K, 2K+2K sd cn ds nc
2K, 4K sc cs dn nd

Figure 22.4.1 illustrates the locations in the z-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices 0, 2K, 2K+2K, 2K. The other poles and zeros are at the congruent points.

See accompanying text See accompanying text See accompanying text
(a) sn(z,k) (b) cn(z,k) (c) dn(z,k)
Figure 22.4.1: z-plane. Poles ××× and zeros of the principal Jacobian elliptic functions. Magnify

For the distribution of the k-zeros of the Jacobian elliptic functions see Walker (2009).

§22.4(ii) Graphical Interpretation via Glaisher’s Notation

Figure 22.4.2 depicts the fundamental unit cell in the z-plane, with vertices s=0, c=K, d=K+K, n=K. The set of points z=mK+nK, m,n, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by mK+nK, where again m,n.

See accompanying text
Figure 22.4.2: z-plane. Fundamental unit cell. Magnify

Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. Let p,q be any two distinct letters from the set s,c,d,n which appear in counterclockwise orientation at the corners of all lattice unit cells. Then: (a) In any lattice unit cell pq(z,k) has a simple zero at z=p and a simple pole at z=q. (b) The difference between p and the nearest q is a half-period of pq(z,k). This half-period will be plus or minus a member of the triple K,K,K+K; the other two members of this triple are quarter periods of pq(z,k).

§22.4(iii) Translation by Half or Quarter Periods

See Table 22.4.3.

For example, sn(z+K,k)=cd(z,k). (The modulus k is suppressed throughout the table.)

Table 22.4.3: Half- or quarter-period shifts of variable for the Jacobian elliptic functions.
z+K z+K+K z+K z+2K z+2K+2K z+2K
snu cdz k-1dcz k-1nsz -snz -snz snz
cnu -ksdz -kk-1ncz -k-1dsz -cnz cnz -cnz
dnu kndz kscz -csz dnz -dnz -dnz
cdu -snz -k-1nsz k-1dcz -cdz -cdz cdz
sdu k-1cnz -(kk)-1dsz k-1ncz -sdz sdz -sdz
ndu k-1dnz -k-1csz scz ndz -ndz -ndz
dcu -nsz -ksnz kcdz -dcz -dcz dcz
ncu -k-1dsz kk-1cnz ksdz -ncz ncz -ncz
scu -k-1csz k-1dnz ndz scz -scz -scz
nsu dcz kcdz ksnz -nsz -nsz nsz
dsu kncz kksdz -kcnz -dsz dsz -dsz
csu -kscz -kndz -dnz csz -csz -csz