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22 Jacobian Elliptic FunctionsProperties

§22.4 Periods, Poles, and Zeros

Contents
  1. §22.4(i) Distribution
  2. §22.4(ii) Graphical Interpretation via Glaisher’s Notation
  3. §22.4(iii) Translation by Half or Quarter Periods

§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+2niK, where m,n. For example, the poles of sn(z,k), abbreviated as sn in the following tables, are at z=2mK+(2n+1)iK.

Table 22.4.1: Periods and poles of Jacobian elliptic functions.
Periods z-Poles
iK K+iK K 0
4K, 2iK sn cd dc ns
4K, 2K+2iK cn sd nc ds
2K, 4iK 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+2niK, where m,n.

Table 22.4.2: Periods and zeros of Jacobian elliptic functions.
Periods z-Zeros
0 K K+iK iK
4K, 2iK sn cd dc ns
4K, 2K+2iK sd cn ds nc
2K, 4iK 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+2iK, 2iK. 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+iK, n=iK. The set of points z=mK+niK, 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+niK, 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,iK,K+iK; 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.
u
z+K z+K+iK z+iK z+2K z+2K+2iK z+2iK
snu cdz k1dcz k1nsz snz snz snz
cnu ksdz ikk1ncz ik1dsz cnz cnz cnz
dnu kndz ikscz icsz dnz dnz dnz
cdu snz k1nsz k1dcz cdz cdz cdz
sdu k1cnz i(kk)1dsz ik1ncz sdz sdz sdz
ndu k1dnz ik1csz iscz ndz ndz ndz
dcu nsz ksnz kcdz dcz dcz dcz
ncu k1dsz ikk1cnz iksdz ncz ncz ncz
scu k1csz ik1dnz indz scz scz scz
nsu dcz kcdz ksnz nsz nsz nsz
dsu kncz ikksdz ikcnz dsz dsz dsz
csu kscz ikndz idnz csz csz csz