Digital Library of Mathematical Functions
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22 Jacobian Elliptic FunctionsProperties

§22.13 Derivatives and Differential Equations

Contents

§22.13(i) Derivatives

Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.
z(snz)= cnzdnz z(dcz) = k2sczncz
z(cnz)= -snzdnz z(ncz) = sczdcz
z(dnz)= -k2snzcnz z(scz) = dczncz
z(cdz)= -k2sdzndz z(nsz) = -dszcsz
z(sdz)= cdzndz z(dsz) = -csznsz
z(ndz)= k2sdzcdz z(csz) = -nszdsz

Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. (The modulus k is suppressed throughout the table.)

For alternative, and symmetric, formulations of these results see Carlson (2004, 2006a).

§22.13(ii) First-Order Differential Equations

22.13.1 (zsn(z,k))2 =(1-sn2(z,k))(1-k2sn2(z,k)),
22.13.2 (zcn(z,k))2 =(1-cn2(z,k))(k2+k2cn2(z,k)),
22.13.3 (zdn(z,k))2 =(1-dn2(z,k))(dn2(z,k)-k2).
22.13.4 (zcd(z,k))2 =(1-cd2(z,k))(1-k2cd2(z,k)),
22.13.5 (zsd(z,k))2 =(1-k2sd2(z,k))(1+k2sd2(z,k)),
22.13.6 (znd(z,k))2 =(nd2(z,k)-1)(1-k2nd2(z,k)),
22.13.7 (zdc(z,k))2 =(dc2(z,k)-1)(dc2(z,k)-k2),
22.13.8 (znc(z,k))2 =(k2+k2nc2(z,k))(nc2(z,k)-1),
22.13.9 (zsc(z,k))2 =(1+sc2(z,k))(1+k2sc2(z,k)),
22.13.10 (zns(z,k))2 =(ns2(z,k)-k2)(ns2(z,k)-1),
22.13.11 (zds(z,k))2 =(ds2(z,k)-k2)(k2+ds2(z,k)),
22.13.12 (zcs(z,k))2 =(1+cs2(z,k))(k2+cs2(z,k)).

For alternative, and symmetric, formulations of these results see Carlson (2006a).

§22.13(iii) Second-Order Differential Equations

22.13.13 2z2sn(z,k) =-(1+k2)sn(z,k)+2k2sn3(z,k),
22.13.14 2z2cn(z,k) =-(k2-k2)cn(z,k)-2k2cn3(z,k),
22.13.15 2z2dn(z,k) =(1+k2)dn(z,k)-2dn3(z,k).
22.13.16 2z2cd(z,k) =-(1+k2)cd(z,k)+2k2cd3(z,k),
22.13.17 2z2sd(z,k) =(k2-k2)sd(z,k)-2k2k2sd3(z,k),
22.13.18 2z2nd(z,k) =(1+k2)nd(z,k)-2k2nd3(z,k),
22.13.19 2z2dc(z,k) =-(1+k2)dc(z,k)+2dc3(z,k),
22.13.20 2z2nc(z,k) =(k2-k2)nc(z,k)+2k2nc3(z,k),
22.13.21 2z2sc(z,k) =(1+k2)sc(z,k)+2k2sc3(z,k),
22.13.22 2z2ns(z,k) =-(1+k2)ns(z,k)+2ns3(z,k),
22.13.23 2z2ds(z,k) =(k2-k2)ds(z,k)+2ds3(z,k),
22.13.24 2z2cs(z,k) =(1+k2)cs(z,k)+2cs3(z,k).

For alternative, and symmetric, formulations of these results see Carlson (2006a).