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22 Jacobian Elliptic FunctionsProperties

§22.8 Addition Theorems

Contents
  1. §22.8(i) Sum of Two Arguments
  2. §22.8(ii) Alternative Forms for Sum of Two Arguments
  3. §22.8(iii) Special Relations Between Arguments

§22.8(i) Sum of Two Arguments

For u,v, and with the common modulus k suppressed:

22.8.1 sn(u+v) =snucnvdnv+snvcnudnu1k2sn2usn2v,
22.8.2 cn(u+v) =cnucnvsnudnusnvdnv1k2sn2usn2v,
22.8.3 dn(u+v) =dnudnvk2snucnusnvcnv1k2sn2usn2v.
22.8.4 cd(u+v)=cducdvk2sdundusdvndv1+k2k2sd2usd2v,
22.8.5 sd(u+v) =sducdvndv+sdvcdundu1+k2k2sd2usd2v,
22.8.6 nd(u+v) =ndundv+k2sducdusdvcdv1+k2k2sd2usd2v,
22.8.7 dc(u+v) =dcudcv+k2scuncuscvncv1k2sc2usc2v,
22.8.8 nc(u+v) =ncuncv+scudcuscvdcv1k2sc2usc2v,
22.8.9 sc(u+v) =scudcvncv+scvdcuncu1k2sc2usc2v,
22.8.10 ns(u+v) =nsudsvcsvnsvdsucsucs2vcs2u,
22.8.11 ds(u+v) =dsucsvnsvdsvcsunsucs2vcs2u,
22.8.12 cs(u+v) =csudsvnsvcsvdsunsucs2vcs2u.

See also Carlson (2004).

§22.8(ii) Alternative Forms for Sum of Two Arguments

For u,v, and with the common modulus k suppressed:

22.8.13 sn(u+v) =sn2usn2vsnucnvdnvsnvcnudnu,
22.8.14 sn(u+v) =snucnudnv+snvcnvdnucnucnv+snudnusnvdnv,
22.8.15 cn(u+v) =snucnudnvsnvcnvdnusnucnvdnvsnvcnudnu,
22.8.16 cn(u+v) =1sn2usn2v+k2sn2usn2vcnucnv+snudnusnvdnv,
22.8.17 dn(u+v) =snucnvdnusnvcnudnvsnucnvdnvsnvcnudnu,
22.8.18 dn(u+v) =cnudnucnvdnv+k2snusnvcnucnv+snudnusnvdnv.

See also Carlson (2004).

§22.8(iii) Special Relations Between Arguments

In the following equations the common modulus k is again suppressed.

Let

22.8.19 z1+z2+z3+z4=0.

Then

22.8.20 |snz1cnz1dnz11snz2cnz2dnz21snz3cnz3dnz31snz4cnz4dnz41|=0,

and

22.8.21 k2k2k2snz1snz2snz3snz4+k2cnz1cnz2cnz3cnz4dnz1dnz2dnz3dnz4=0.

A geometric interpretation of (22.8.20) analogous to that of (23.10.5) is given in Whittaker and Watson (1927, p. 530).

Next, let

22.8.22 z1+z2+z3+z4=2K(k).

Then

22.8.23 |snz1cnz1cnz1dnz1cnz1dnz1snz2cnz2cnz2dnz2cnz2dnz2snz3cnz3cnz3dnz3cnz3dnz3snz4cnz4cnz4dnz4cnz4dnz4|=0.

For these and related identities see Copson (1935, pp. 415–416).

If sums/differences of the zj’s are rational multiples of K(k), then further relations follow. For instance, if

22.8.24 z1z2=z2z3=23K(k),

then

22.8.25 (dnz2+dnz3)(dnz3+dnz1)(dnz1+dnz2)dnz1+dnz2+dnz3

is independent of z1, z2, z3. Similarly, if

22.8.26 z1z2=z2z3=z3z4=12K(k),

then

22.8.27 dnz1dnz3=dnz2dnz4=k.

Greenhill (1959, pp. 121–130) reviews these results in terms of the geometric poristic polygon constructions of Poncelet. Generalizations are given in §22.9.