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22 Jacobian Elliptic FunctionsProperties

§22.8 Addition Theorems


§22.8(i) Sum of Two Arguments

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

22.8.1 sn(u+v) =snucnvdnv+snvcnudnu1-k2sn2usn2v,
22.8.2 cn(u+v) =cnucnv-snudnusnvdnv1-k2sn2usn2v,
22.8.3 dn(u+v) =dnudnv-k2snucnusnvcnv1-k2sn2usn2v.
22.8.4 cd(u+v)=cducdv-k2sdundusdvndv1+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+k2scuncuscvncv1-k2sc2usc2v,
22.8.8 nc(u+v) =ncuncv+scudcuscvdcv1-k2sc2usc2v,
22.8.9 sc(u+v) =scudcvncv+scvdcuncu1-k2sc2usc2v,
22.8.10 ns(u+v) =nsudsvcsv-nsvdsucsucs2v-cs2u,
22.8.11 ds(u+v) =dsucsvnsv-dsvcsunsucs2v-cs2u,
22.8.12 cs(u+v) =csudsvnsv-csvdsunsucs2v-cs2u.

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) =sn2u-sn2vsnucnvdnv-snvcnudnu,
22.8.14 sn(u+v) =snucnudnv+snvcnvdnucnucnv+snudnusnvdnv,
22.8.15 cn(u+v) =snucnudnv-snvcnvdnusnucnvdnv-snvcnudnu,
22.8.16 cn(u+v) =1-sn2u-sn2v+k2sn2usn2vcnucnv+snudnusnvdnv,
22.8.17 dn(u+v) =snucnvdnu-snvcnudnvsnucnvdnv-snvcnudnu,
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.


22.8.19 z1+z2+z3+z4=0.


22.8.20 |snz1cnz1dnz11snz2cnz2dnz21snz3cnz3dnz31snz4cnz4dnz41|=0,


22.8.21 k2-k2k2snz1snz2snz3snz4+k2cnz1cnz2cnz3cnz4-dnz1dnz2dnz3dnz4=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).


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 z1-z2=z2-z3=23K(k),


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

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

22.8.26 z1-z2=z2-z3=z3-z4=12K(k),


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.