Results of Jacobian Elliptic Functions

DLMF	Formula	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
22.2#Ex1	${\displaystyle{\displaystyle k=\frac{{\theta_{2}^{2}}\left(0,q\right)}{{\theta_{3}^{2}}\left(0,q\right)}}}$	k =((JacobiTheta2(0, q))^(2))/((JacobiTheta3(0, q))^(2))	k =Divide[(EllipticTheta[2, 0, q])^(2),(EllipticTheta[3, 0, q])^(2)]	Failure	Failure	Error	Successful
22.2#Ex2	${\displaystyle{\displaystyle k^{\prime}=\frac{{\theta_{4}^{2}}\left(0,q\right)}{{\theta_{3}^{2}}\left(0,q\right)}}}$	sqrt(1 - (k)^(2))=((JacobiTheta4(0, q))^(2))/((JacobiTheta3(0, q))^(2))	Sqrt[1 - (k)^(2)]=Divide[(EllipticTheta[4, 0, q])^(2),(EllipticTheta[3, 0, q])^(2)]	Failure	Failure	Error	Successful
22.2#Ex3	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}{\theta_{3}^{2}}\left(0,q\right)}}$	EllipticK(k)=(Pi)/(2)*(JacobiTheta3(0, q))^(2)	EllipticK[(k)^2]=Divide[Pi,2]*(EllipticTheta[3, 0, q])^(2)	Failure	Failure	Error	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[q, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[q, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
22.2.E3	${\displaystyle{\displaystyle\zeta=\frac{\pi z}{2\!K\left(k\right)}}}$	zeta =(Pi*z)/(2*EllipticK(k))	\[zeta]=Divide[Pi*z,2*EllipticK[(k)^2]]	Failure	Failure	Error	Error
22.2.E4	${\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}}}$	JacobiSN(z, k)=(JacobiTheta3(0, q))/(JacobiTheta2(0, q))*(JacobiTheta1(zeta, q))/(JacobiTheta4(zeta, q))	JacobiSN[z, (k)^2]=Divide[EllipticTheta[3, 0, q],EllipticTheta[2, 0, q]]*Divide[EllipticTheta[1, \[zeta], q],EllipticTheta[4, \[zeta], q]]	Failure	Failure	Error	Error
22.2.E4	${\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}=\frac{1}{\operatorname{ns}\left(z,k\right)}}}$	(JacobiTheta3(0, q))/(JacobiTheta2(0, q))*(JacobiTheta1(zeta, q))/(JacobiTheta4(zeta, q))=(1)/(JacobiNS(z, k))	Divide[EllipticTheta[3, 0, q],EllipticTheta[2, 0, q]]*Divide[EllipticTheta[1, \[zeta], q],EllipticTheta[4, \[zeta], q]]=Divide[1,JacobiNS[z, (k)^2]]	Failure	Failure	Error	Error
22.2.E5	${\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}}}$	JacobiCN(z, k)=(JacobiTheta4(0, q))/(JacobiTheta2(0, q))*(JacobiTheta2(zeta, q))/(JacobiTheta4(zeta, q))	JacobiCN[z, (k)^2]=Divide[EllipticTheta[4, 0, q],EllipticTheta[2, 0, q]]*Divide[EllipticTheta[2, \[zeta], q],EllipticTheta[4, \[zeta], q]]	Failure	Failure	Error	Error
22.2.E5	${\displaystyle{\displaystyle\frac{\theta_{4}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}=\frac{1}{\operatorname{nc}\left(z,k\right)}}}$	(JacobiTheta4(0, q))/(JacobiTheta2(0, q))*(JacobiTheta2(zeta, q))/(JacobiTheta4(zeta, q))=(1)/(JacobiNC(z, k))	Divide[EllipticTheta[4, 0, q],EllipticTheta[2, 0, q]]*Divide[EllipticTheta[2, \[zeta], q],EllipticTheta[4, \[zeta], q]]=Divide[1,JacobiNC[z, (k)^2]]	Failure	Failure	Error	Error
22.2.E6	${\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{3}\left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}}}$	JacobiDN(z, k)=(JacobiTheta4(0, q))/(JacobiTheta3(0, q))*(JacobiTheta3(zeta, q))/(JacobiTheta4(zeta, q))	JacobiDN[z, (k)^2]=Divide[EllipticTheta[4, 0, q],EllipticTheta[3, 0, q]]*Divide[EllipticTheta[3, \[zeta], q],EllipticTheta[4, \[zeta], q]]	Failure	Failure	Error	Error
22.2.E6	${\displaystyle{\displaystyle\frac{\theta_{4}\left(0,q\right)}{\theta_{3}\left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}=\frac{1}{\operatorname{nd}\left(z,k\right)}}}$	(JacobiTheta4(0, q))/(JacobiTheta3(0, q))*(JacobiTheta3(zeta, q))/(JacobiTheta4(zeta, q))=(1)/(JacobiND(z, k))	Divide[EllipticTheta[4, 0, q],EllipticTheta[3, 0, q]]*Divide[EllipticTheta[3, \[zeta], q],EllipticTheta[4, \[zeta], q]]=Divide[1,JacobiND[z, (k)^2]]	Failure	Failure	Error	Error
22.2.E7	${\displaystyle{\displaystyle\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}^{2}}\left(0,q\right)}{\theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}}}$	JacobiSD(z, k)=((JacobiTheta3(0, q))^(2))/(JacobiTheta2(0, q)*JacobiTheta4(0, q))*(JacobiTheta1(zeta, q))/(JacobiTheta3(zeta, q))	JacobiSD[z, (k)^2]=Divide[(EllipticTheta[3, 0, q])^(2),EllipticTheta[2, 0, q]*EllipticTheta[4, 0, q]]*Divide[EllipticTheta[1, \[zeta], q],EllipticTheta[3, \[zeta], q]]	Failure	Failure	Error	Error
22.2.E7	${\displaystyle{\displaystyle\frac{{\theta_{3}^{2}}\left(0,q\right)}{\theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}\left(z,k\right)}}}$	((JacobiTheta3(0, q))^(2))/(JacobiTheta2(0, q)*JacobiTheta4(0, q))*(JacobiTheta1(zeta, q))/(JacobiTheta3(zeta, q))=(1)/(JacobiDS(z, k))	Divide[(EllipticTheta[3, 0, q])^(2),EllipticTheta[2, 0, q]*EllipticTheta[4, 0, q]]*Divide[EllipticTheta[1, \[zeta], q],EllipticTheta[3, \[zeta], q]]=Divide[1,JacobiDS[z, (k)^2]]	Failure	Failure	Error	Error
22.2.E8	${\displaystyle{\displaystyle\operatorname{cd}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}}}$	JacobiCD(z, k)=(JacobiTheta3(0, q))/(JacobiTheta2(0, q))*(JacobiTheta2(zeta, q))/(JacobiTheta3(zeta, q))	JacobiCD[z, (k)^2]=Divide[EllipticTheta[3, 0, q],EllipticTheta[2, 0, q]]*Divide[EllipticTheta[2, \[zeta], q],EllipticTheta[3, \[zeta], q]]	Failure	Failure	Error	Error
22.2.E8	${\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{dc}\left(z,k\right)}}}$	(JacobiTheta3(0, q))/(JacobiTheta2(0, q))*(JacobiTheta2(zeta, q))/(JacobiTheta3(zeta, q))=(1)/(JacobiDC(z, k))	Divide[EllipticTheta[3, 0, q],EllipticTheta[2, 0, q]]*Divide[EllipticTheta[2, \[zeta], q],EllipticTheta[3, \[zeta], q]]=Divide[1,JacobiDC[z, (k)^2]]	Failure	Failure	Error	Error
22.2.E9	${\displaystyle{\displaystyle\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q\right)}}}$	JacobiSC(z, k)=(JacobiTheta3(0, q))/(JacobiTheta4(0, q))*(JacobiTheta1(zeta, q))/(JacobiTheta2(zeta, q))	JacobiSC[z, (k)^2]=Divide[EllipticTheta[3, 0, q],EllipticTheta[4, 0, q]]*Divide[EllipticTheta[1, \[zeta], q],EllipticTheta[2, \[zeta], q]]	Failure	Failure	Error	Error
22.2.E9	${\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q\right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}}}$	(JacobiTheta3(0, q))/(JacobiTheta4(0, q))*(JacobiTheta1(zeta, q))/(JacobiTheta2(zeta, q))=(1)/(JacobiCS(z, k))	Divide[EllipticTheta[3, 0, q],EllipticTheta[4, 0, q]]*Divide[EllipticTheta[1, \[zeta], q],EllipticTheta[2, \[zeta], q]]=Divide[1,JacobiCS[z, (k)^2]]	Failure	Failure	Error	Error
22.2.E10	${\displaystyle{\displaystyle\operatorname{pq}\left(z,k\right)=\frac{\operatorname{pr}\left(z,k\right)}{\operatorname{qr}\left(z,k\right)}}}$	genJacobiellk(p)*q* z*k =(genJacobiellk(p)*r* z*k)/(genJacobiellk(q)*r* z*k)	genJacobiellk(p)*q* z*k =Divide[genJacobiellk(p)*r* z*k,genJacobiellk(q)*r* z*k]	Failure	Failure	Error	Skip
22.2.E10	${\displaystyle{\displaystyle\frac{\operatorname{pr}\left(z,k\right)}{\operatorname{qr}\left(z,k\right)}=\frac{1}{\operatorname{qp}\left(z,k\right)}}}$	(genJacobiellk(p)*r* z*k)/(genJacobiellk(q)*r* z*k)=(1)/(genJacobiellk(q)*p* z*k)	Divide[genJacobiellk(p)*r* z*k,genJacobiellk(q)*r* z*k]=Divide[1,genJacobiellk(q)*p* z*k]	Failure	Failure	Error	Skip
22.2.E11	${\displaystyle{\displaystyle\operatorname{pq}\left(z,k\right)=\ifrac{\theta_{p}\left(z\middle|\tau\right)}{\theta_{q}\left(z\middle|\tau\right)}}}$	genJacobiellk(p)*q* z*k =(JacobiThetap(z,exp(I*Pi*tau)))/(JacobiThetaq(z,exp(I*Pi*tau)))	genJacobiellk(p)*q* z*k =Divide[EllipticTheta[p, z, \[Tau]],EllipticTheta[q, z, \[Tau]]]	Failure	Failure	Error	Skip
22.2.E12	${\displaystyle{\displaystyle\tau=\ifrac{\mathrm{i}{K^{\prime}}\left(k\right)}{K\left(k\right)}}}$	tau =(I*EllipticCK(k))/(EllipticK(k))	\[Tau]=Divide[I*EllipticK[1-(k)^2],EllipticK[(k)^2]]	Failure	Failure	Error	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[k, 1], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.034924298733449, 0.929003383041147] <- {Rule[k, 2], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.1238725324006347, 0.9602938378765811] <- {Rule[k, 3], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[k, 1], Rule[τ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E1	${\displaystyle{\displaystyle{\operatorname{sn}^{2}}\left(z,k\right)+{\operatorname{cn}^{2}}\left(z,k\right)=k^{2}{\operatorname{sn}^{2}}\left(z,k\right)+{\operatorname{dn}^{2}}\left(z,k\right)}}$	(JacobiSN(z, k))^(2)+ (JacobiCN(z, k))^(2)= (k)^(2)* (JacobiSN(z, k))^(2)+ (JacobiDN(z, k))^(2)	(JacobiSN[z, (k)^2])^(2)+ (JacobiCN[z, (k)^2])^(2)= (k)^(2)* (JacobiSN[z, (k)^2])^(2)+ (JacobiDN[z, (k)^2])^(2)	Successful	Successful	-	-
22.6.E1	${\displaystyle{\displaystyle k^{2}{\operatorname{sn}^{2}}\left(z,k\right)+{\operatorname{dn}^{2}}\left(z,k\right)=1}}$	(k)^(2)* (JacobiSN(z, k))^(2)+ (JacobiDN(z, k))^(2)= 1	(k)^(2)* (JacobiSN[z, (k)^2])^(2)+ (JacobiDN[z, (k)^2])^(2)= 1	Successful	Successful	-	-
22.6.E2	${\displaystyle{\displaystyle 1+{\operatorname{cs}^{2}}\left(z,k\right)=k^{2}+{\operatorname{ds}^{2}}\left(z,k\right)}}$	1 + (JacobiCS(z, k))^(2)= (k)^(2)+ (JacobiDS(z, k))^(2)	1 + (JacobiCS[z, (k)^2])^(2)= (k)^(2)+ (JacobiDS[z, (k)^2])^(2)	Successful	Successful	-	-
22.6.E2	${\displaystyle{\displaystyle k^{2}+{\operatorname{ds}^{2}}\left(z,k\right)={\operatorname{ns}^{2}}\left(z,k\right)}}$	(k)^(2)+ (JacobiDS(z, k))^(2)= (JacobiNS(z, k))^(2)	(k)^(2)+ (JacobiDS[z, (k)^2])^(2)= (JacobiNS[z, (k)^2])^(2)	Successful	Successful	-	-
22.6.E3	${\displaystyle{\displaystyle{k^{\prime}}^{2}{\operatorname{sc}^{2}}\left(z,k\right)+1={\operatorname{dc}^{2}}\left(z,k\right)}}$	1 - (k)^(2)* (JacobiSC(z, k))^(2)+ 1 = (JacobiDC(z, k))^(2)	1 - (k)^(2)* (JacobiSC[z, (k)^2])^(2)+ 1 = (JacobiDC[z, (k)^2])^(2)	Failure	Failure	Fail 5.538045195-1.298501057*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 1.632868431-.533445486*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .8869726205+.142192957*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 5.538045195+1.298501057*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[5.538045200949385, -1.2985010548545433] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.6328684295963352, -0.5334454854262538] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.8869726205411628, 0.14219295744217275] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[5.538045200949385, 1.2985010548545433] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E3	${\displaystyle{\displaystyle{\operatorname{dc}^{2}}\left(z,k\right)={k^{\prime}}^{2}{\operatorname{nc}^{2}}\left(z,k\right)+k^{2}}}$	(JacobiDC(z, k))^(2)= 1 - (k)^(2)* (JacobiNC(z, k))^(2)+ (k)^(2)	(JacobiDC[z, (k)^2])^(2)= 1 - (k)^(2)* (JacobiNC[z, (k)^2])^(2)+ (k)^(2)	Failure	Failure	Fail -4.538045196+1.298501057*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -.632868431+.533445486*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .113027376-.142192958*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -4.538045196-1.298501057*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-4.5380452009493855, 1.2985010548545435] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.6328684295963332, 0.5334454854262538] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.11302737945883834, -0.14219295744217342] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-4.5380452009493855, -1.2985010548545435] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E4	${\displaystyle{\displaystyle-k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{2}}\left(z,k\right)=k^{2}({\operatorname{cd}^{2}}\left(z,k\right)-1)}}$	- (k)^(2)* 1 - (k)^(2)* (JacobiSD(z, k))^(2)= (k)^(2)*((JacobiCD(z, k))^(2)- 1)	- (k)^(2)* 1 - (k)^(2)* (JacobiSD[z, (k)^2])^(2)= (k)^(2)*((JacobiCD[z, (k)^2])^(2)- 1)	Failure	Failure	Fail 3.538045195-1.298501057*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -.767955680e-1-.7785401828*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .3808337301+8.838098584*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 3.538045195+1.298501057*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[3.5380452009493846, -1.2985010548545433] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.07679556965128587, -0.7785401828344382] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.3808337424857964, 8.838098611812974] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.5380452009493846, 1.2985010548545433] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E4	${\displaystyle{\displaystyle k^{2}({\operatorname{cd}^{2}}\left(z,k\right)-1)={k^{\prime}}^{2}(1-{\operatorname{nd}^{2}}\left(z,k\right))}}$	(k)^(2)*((JacobiCD(z, k))^(2)- 1)= 1 - (k)^(2)*(1 - (JacobiND(z, k))^(2))	(k)^(2)*((JacobiCD[z, (k)^2])^(2)- 1)= 1 - (k)^(2)*(1 - (JacobiND[z, (k)^2])^(2))	Failure	Failure	Fail 3.538045196-1.298501057*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -.1919889172e-1-.1946350456*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .423148605e-1+.982010952*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 3.538045196+1.298501057*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[3.538045200949385, -1.2985010548545435] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.01919889241282169, -0.1946350457086099] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.04231486027619802, 0.9820109568681117] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.538045200949385, 1.2985010548545435] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E5	${\displaystyle{\displaystyle\operatorname{sn}\left(2z,k\right)=\frac{2\operatorname{sn}\left(z,k\right)\operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,k\right)}}}$	JacobiSN(2*z, k)=(2*JacobiSN(z, k)*JacobiCN(z, k)*JacobiDN(z, k))/(1 - (k)^(2)* (JacobiSN(z, k))^(4))	JacobiSN[2*z, (k)^2]=Divide[2*JacobiSN[z, (k)^2]*JacobiCN[z, (k)^2]*JacobiDN[z, (k)^2],1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4)]	Failure	Failure	Successful	Successful
22.6.E6	${\displaystyle{\displaystyle\operatorname{cn}\left(2z,k\right)=\frac{{\operatorname{cn}^{2}}\left(z,k\right)-{\operatorname{sn}^{2}}\left(z,k\right){\operatorname{dn}^{2}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,k\right)}}}$	JacobiCN(2*z, k)=((JacobiCN(z, k))^(2)- (JacobiSN(z, k))^(2)* (JacobiDN(z, k))^(2))/(1 - (k)^(2)* (JacobiSN(z, k))^(4))	JacobiCN[2*z, (k)^2]=Divide[(JacobiCN[z, (k)^2])^(2)- (JacobiSN[z, (k)^2])^(2)* (JacobiDN[z, (k)^2])^(2),1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4)]	Failure	Failure	Successful	Successful
22.6.E6	${\displaystyle{\displaystyle\frac{{\operatorname{cn}^{2}}\left(z,k\right)-{\operatorname{sn}^{2}}\left(z,k\right){\operatorname{dn}^{2}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,k\right)}=\frac{{\operatorname{cn}^{4}}\left(z,k\right)-{k^{\prime}}^{2}{\operatorname{sn}^{4}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,k\right)}}}$	((JacobiCN(z, k))^(2)- (JacobiSN(z, k))^(2)* (JacobiDN(z, k))^(2))/(1 - (k)^(2)* (JacobiSN(z, k))^(4))=((JacobiCN(z, k))^(4)- 1 - (k)^(2)* (JacobiSN(z, k))^(4))/(1 - (k)^(2)* (JacobiSN(z, k))^(4))	Divide[(JacobiCN[z, (k)^2])^(2)- (JacobiSN[z, (k)^2])^(2)* (JacobiDN[z, (k)^2])^(2),1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4)]=Divide[(JacobiCN[z, (k)^2])^(4)- 1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4),1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4)]	Failure	Failure	Fail -4.094154376+1.280458226*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -1.389457565+.616517316e-1*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .8632946284-.6529631058*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -4.094154376-1.280458226*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-4.0941543674510195, 1.2804582127704043] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.3894575644075957, 0.06165173015688402] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.8632946317597533, -0.6529631059321507] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-4.0941543674510195, -1.2804582127704043] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E7	${\displaystyle{\displaystyle\operatorname{dn}\left(2z,k\right)=\frac{{\operatorname{dn}^{2}}\left(z,k\right)-k^{2}{\operatorname{sn}^{2}}\left(z,k\right){\operatorname{cn}^{2}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,k\right)}}}$	JacobiDN(2*z, k)=((JacobiDN(z, k))^(2)- (k)^(2)* (JacobiSN(z, k))^(2)* (JacobiCN(z, k))^(2))/(1 - (k)^(2)* (JacobiSN(z, k))^(4))	JacobiDN[2*z, (k)^2]=Divide[(JacobiDN[z, (k)^2])^(2)- (k)^(2)* (JacobiSN[z, (k)^2])^(2)* (JacobiCN[z, (k)^2])^(2),1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4)]	Failure	Failure	Successful	Successful
22.6.E7	${\displaystyle{\displaystyle\frac{{\operatorname{dn}^{2}}\left(z,k\right)-k^{2}{\operatorname{sn}^{2}}\left(z,k\right){\operatorname{cn}^{2}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,k\right)}=\frac{{\operatorname{dn}^{4}}\left(z,k\right)+k^{2}{k^{\prime}}^{2}{\operatorname{sn}^{4}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,k\right)}}}$	((JacobiDN(z, k))^(2)- (k)^(2)* (JacobiSN(z, k))^(2)* (JacobiCN(z, k))^(2))/(1 - (k)^(2)* (JacobiSN(z, k))^(4))=((JacobiDN(z, k))^(4)+ (k)^(2)* 1 - (k)^(2)* (JacobiSN(z, k))^(4))/(1 - (k)^(2)* (JacobiSN(z, k))^(4))	Divide[(JacobiDN[z, (k)^2])^(2)- (k)^(2)* (JacobiSN[z, (k)^2])^(2)* (JacobiCN[z, (k)^2])^(2),1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4)]=Divide[(JacobiDN[z, (k)^2])^(4)+ (k)^(2)* 1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4),1 - (k)^(2)* (JacobiSN[z, (k)^2])^(4)]	Failure	Failure	Fail -.9999999995-.2e-10*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 1.213361962-.134512870*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -8.242862557+3.616411048*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -.9999999995+.2e-10*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-0.9999999999999996, -8.326672684688674*^-17] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.213361958707478, -0.1345128657968393] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-8.24286257590017, 3.6164110482396037] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.9999999999999996, 8.326672684688674*^-17] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E8	${\displaystyle{\displaystyle\operatorname{cd}\left(2z,k\right)=\frac{{\operatorname{cd}^{2}}\left(z,k\right)-{k^{\prime}}^{2}{\operatorname{sd}^{2}}\left(z,k\right){\operatorname{nd}^{2}}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{4}}\left(z,k\right)}}}$	JacobiCD(2*z, k)=((JacobiCD(z, k))^(2)- 1 - (k)^(2)* (JacobiSD(z, k))^(2)* (JacobiND(z, k))^(2))/(1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD(z, k))^(4))	JacobiCD[2*z, (k)^2]=Divide[(JacobiCD[z, (k)^2])^(2)- 1 - (k)^(2)* (JacobiSD[z, (k)^2])^(2)* (JacobiND[z, (k)^2])^(2),1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD[z, (k)^2])^(4)]	Failure	Failure	Fail .1370541185+.1873251287e-1*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} .5364817078-.4234624245*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -.1981753675-.1199254751*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} .1370541185-.1873251287e-1*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.13705411883745122, 0.018732512731960915] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.5364817078278756, -0.42346242319671296] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.19817536922951154, -0.1199254747103525] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.13705411883745144, -0.018732512731960915] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E9	${\displaystyle{\displaystyle\operatorname{sd}\left(2z,k\right)=\frac{2\operatorname{sd}\left(z,k\right)\operatorname{cd}\left(z,k\right)\operatorname{nd}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{4}}\left(z,k\right)}}}$	JacobiSD(2*z, k)=(2*JacobiSD(z, k)*JacobiCD(z, k)*JacobiND(z, k))/(1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD(z, k))^(4))	JacobiSD[2*z, (k)^2]=Divide[2*JacobiSD[z, (k)^2]*JacobiCD[z, (k)^2]*JacobiND[z, (k)^2],1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD[z, (k)^2])^(4)]	Failure	Failure	Fail -8.411649228+2.496794499*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} .4171906607e-1-.5404871752*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .4672486370e-1-.3344443629*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -8.411649228-2.496794499*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-8.411649235958867, 2.496794495227415] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.041719065906232006, -0.5404871748442961] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.046724863833304056, -0.33444436276903133] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-8.411649235958867, -2.496794495227415] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E10	${\displaystyle{\displaystyle\operatorname{nd}\left(2z,k\right)=\frac{{\operatorname{nd}^{2}}\left(z,k\right)+k^{2}{\operatorname{sd}^{2}}\left(z,k\right){\operatorname{cd}^{2}}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{4}}\left(z,k\right)}}}$	JacobiND(2*z, k)=((JacobiND(z, k))^(2)+ (k)^(2)* (JacobiSD(z, k))^(2)* (JacobiCD(z, k))^(2))/(1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD(z, k))^(4))	JacobiND[2*z, (k)^2]=Divide[(JacobiND[z, (k)^2])^(2)+ (k)^(2)* (JacobiSD[z, (k)^2])^(2)* (JacobiCD[z, (k)^2])^(2),1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD[z, (k)^2])^(4)]	Failure	Failure	Fail -8.469613315+2.476300734*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -.2446575645+.5929818787*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -.5059779542e-1-.4472642739*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -8.469613315-2.476300734*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-8.469613322356564, 2.4763007298746587] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.24465756431597135, 0.5929818773302299] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.05059779615055053, -0.4472642742544212] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-8.469613322356564, -2.4763007298746587] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E11	${\displaystyle{\displaystyle\operatorname{dc}\left(2z,k\right)=\frac{{\operatorname{dc}^{2}}\left(z,k\right)+{k^{\prime}}^{2}{\operatorname{sc}^{2}}\left(z,k\right){\operatorname{nc}^{2}}\left(z,k\right)}{1-{k^{\prime}}^{2}{\operatorname{sc}^{4}}\left(z,k\right)}}}$	JacobiDC(2*z, k)=((JacobiDC(z, k))^(2)+ 1 - (k)^(2)* (JacobiSC(z, k))^(2)* (JacobiNC(z, k))^(2))/(1 - 1 - (k)^(2)* (JacobiSC(z, k))^(4))	JacobiDC[2*z, (k)^2]=Divide[(JacobiDC[z, (k)^2])^(2)+ 1 - (k)^(2)* (JacobiSC[z, (k)^2])^(2)* (JacobiNC[z, (k)^2])^(2),1 - 1 - (k)^(2)* (JacobiSC[z, (k)^2])^(4)]	Failure	Failure	Fail .2798628459+.1057645812*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 2.155279764+3.336838966*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -10.41618961+.723801634*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} .2798628459-.1057645812*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.27986284597445743, 0.1057645806458628] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.1552797720040004, 3.3368389687939786] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-10.416189608158701, 0.7238016559320513] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.27986284597445743, -0.1057645806458628] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E12	${\displaystyle{\displaystyle\operatorname{nc}\left(2z,k\right)=\frac{{\operatorname{nc}^{2}}\left(z,k\right)+{\operatorname{sc}^{2}}\left(z,k\right){\operatorname{dc}^{2}}\left(z,k\right)}{1-{k^{\prime}}^{2}{\operatorname{sc}^{4}}\left(z,k\right)}}}$	JacobiNC(2*z, k)=((JacobiNC(z, k))^(2)+ (JacobiSC(z, k))^(2)* (JacobiDC(z, k))^(2))/(1 - 1 - (k)^(2)* (JacobiSC(z, k))^(4))	JacobiNC[2*z, (k)^2]=Divide[(JacobiNC[z, (k)^2])^(2)+ (JacobiSC[z, (k)^2])^(2)* (JacobiDC[z, (k)^2])^(2),1 - 1 - (k)^(2)* (JacobiSC[z, (k)^2])^(4)]	Failure	Failure	Fail -8.445366052+2.504181595*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -2.348000820+.4644873082*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .3919060714+4.559323678*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -8.445366052-2.504181595*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-8.4453660597032, 2.504181591384576] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.3480008192225705, 0.46448731013438893] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.39190608798513005, 4.559323684618953] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-8.4453660597032, -2.504181591384576] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E13	${\displaystyle{\displaystyle\operatorname{sc}\left(2z,k\right)=\frac{2\operatorname{sc}\left(z,k\right)\operatorname{dc}\left(z,k\right)\operatorname{nc}\left(z,k\right)}{1-{k^{\prime}}^{2}{\operatorname{sc}^{4}}\left(z,k\right)}}}$	JacobiSC(2*z, k)=(2*JacobiSC(z, k)*JacobiDC(z, k)*JacobiNC(z, k))/(1 - 1 - (k)^(2)* (JacobiSC(z, k))^(4))	JacobiSC[2*z, (k)^2]=Divide[2*JacobiSC[z, (k)^2]*JacobiDC[z, (k)^2]*JacobiNC[z, (k)^2],1 - 1 - (k)^(2)* (JacobiSC[z, (k)^2])^(4)]	Failure	Failure	Fail -8.387425493+2.524419001*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 1.765721394-.9866914128*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -.5663184000+3.386135413*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -8.387425493-2.524419001*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-8.387425500158386, 2.5244189972251565] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.7657213943311842, -0.9866914167974857] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.5663183953417591, 3.3861354179416785] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-8.387425500158386, -2.5244189972251565] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E14	${\displaystyle{\displaystyle\operatorname{ns}\left(2z,k\right)=\frac{{\operatorname{ns}^{4}}\left(z,k\right)-k^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k\right)\operatorname{ns}\left(z,k\right)}}}$	JacobiNS(2*z, k)=((JacobiNS(z, k))^(4)- (k)^(2))/(2*JacobiCS(z, k)*JacobiDS(z, k)*JacobiNS(z, k))	JacobiNS[2*z, (k)^2]=Divide[(JacobiNS[z, (k)^2])^(4)- (k)^(2),2*JacobiCS[z, (k)^2]*JacobiDS[z, (k)^2]*JacobiNS[z, (k)^2]]	Failure	Failure	Successful	Successful
22.6.E15	${\displaystyle{\displaystyle\operatorname{ds}\left(2z,k\right)=\frac{k^{2}{k^{\prime}}^{2}+{\operatorname{ds}^{4}}\left(z,k\right)}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k\right)\operatorname{ns}\left(z,k\right)}}}$	JacobiDS(2*z, k)=((k)^(2)* 1 - (k)^(2)+ (JacobiDS(z, k))^(4))/(2*JacobiCS(z, k)*JacobiDS(z, k)*JacobiNS(z, k))	JacobiDS[2*z, (k)^2]=Divide[(k)^(2)* 1 - (k)^(2)+ (JacobiDS[z, (k)^2])^(4),2*JacobiCS[z, (k)^2]*JacobiDS[z, (k)^2]*JacobiNS[z, (k)^2]]	Failure	Failure	Fail -2.958056429-.877828454*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} 1.088429910+2.208840486*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -2.958056429+.877828454*I <- {z = 2^(1/2)-I*2^(1/2), k = 2} 1.088429910-2.208840486*I <- {z = 2^(1/2)-I*2^(1/2), k = 3} ... skip entries to safe data	Fail Complex[-2.9580564228983786, -0.8778284568736507] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.0884299045387233, 2.2088404891294435] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.958056422898387, 0.8778284568736494] <- {Rule[k, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[1.088429904538725, -2.2088404891294466] <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E16	${\displaystyle{\displaystyle\operatorname{cs}\left(2z,k\right)=\frac{{\operatorname{cs}^{4}}\left(z,k\right)-{k^{\prime}}^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k\right)\operatorname{ns}\left(z,k\right)}}}$	JacobiCS(2*z, k)=((JacobiCS(z, k))^(4)- 1 - (k)^(2))/(2*JacobiCS(z, k)*JacobiDS(z, k)*JacobiNS(z, k))	JacobiCS[2*z, (k)^2]=Divide[(JacobiCS[z, (k)^2])^(4)- 1 - (k)^(2),2*JacobiCS[z, (k)^2]*JacobiDS[z, (k)^2]*JacobiNS[z, (k)^2]]	Failure	Failure	Fail -5.128303818+1.266734153*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 1.972037619+.5852189693*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -.2721074781-.5522101210*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -5.128303818-1.266734153*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-5.12830382438052, 1.2667341514547281] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.9720376152655872, 0.5852189712491013] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.2721074761346811, -0.5522101222823614] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-5.12830382438052, -1.2667341514547281] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E17	${\displaystyle{\displaystyle\frac{1-\operatorname{cn}\left(2z,k\right)}{1+\operatorname{cn}\left(2z,k\right)}=\frac{{\operatorname{sn}^{2}}\left(z,k\right){\operatorname{dn}^{2}}\left(z,k\right)}{{\operatorname{cn}^{2}}\left(z,k\right)}}}$	(1 - JacobiCN(2*z, k))/(1 + JacobiCN(2*z, k))=((JacobiSN(z, k))^(2)* (JacobiDN(z, k))^(2))/((JacobiCN(z, k))^(2))	Divide[1 - JacobiCN[2*z, (k)^2],1 + JacobiCN[2*z, (k)^2]]=Divide[(JacobiSN[z, (k)^2])^(2)* (JacobiDN[z, (k)^2])^(2),(JacobiCN[z, (k)^2])^(2)]	Failure	Failure	Successful	Successful
22.6.E18	${\displaystyle{\displaystyle\frac{1-\operatorname{dn}\left(2z,k\right)}{1+\operatorname{dn}\left(2z,k\right)}=\frac{k^{2}{\operatorname{sn}^{2}}\left(z,k\right){\operatorname{cn}^{2}}\left(z,k\right)}{{\operatorname{dn}^{2}}\left(z,k\right)}}}$	(1 - JacobiDN(2*z, k))/(1 + JacobiDN(2*z, k))=((k)^(2)* (JacobiSN(z, k))^(2)* (JacobiCN(z, k))^(2))/((JacobiDN(z, k))^(2))	Divide[1 - JacobiDN[2*z, (k)^2],1 + JacobiDN[2*z, (k)^2]]=Divide[(k)^(2)* (JacobiSN[z, (k)^2])^(2)* (JacobiCN[z, (k)^2])^(2),(JacobiDN[z, (k)^2])^(2)]	Failure	Failure	Successful	Successful
22.6.E19	${\displaystyle{\displaystyle{\operatorname{sn}^{2}}\left(\tfrac{1}{2}z,k\right)=\frac{1-\operatorname{cn}\left(z,k\right)}{1+\operatorname{dn}\left(z,k\right)}}}$	(JacobiSN((1)/(2)*z, k))^(2)=(1 - JacobiCN(z, k))/(1 + JacobiDN(z, k))	(JacobiSN[Divide[1,2]*z, (k)^2])^(2)=Divide[1 - JacobiCN[z, (k)^2],1 + JacobiDN[z, (k)^2]]	Failure	Failure	Successful	Successful
22.6.E19	${\displaystyle{\displaystyle\frac{1-\operatorname{cn}\left(z,k\right)}{1+\operatorname{dn}\left(z,k\right)}=\frac{1-\operatorname{dn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}\left(z,k\right))}}}$	(1 - JacobiCN(z, k))/(1 + JacobiDN(z, k))=(1 - JacobiDN(z, k))/((k)^(2)*(1 + JacobiCN(z, k)))	Divide[1 - JacobiCN[z, (k)^2],1 + JacobiDN[z, (k)^2]]=Divide[1 - JacobiDN[z, (k)^2],(k)^(2)*(1 + JacobiCN[z, (k)^2])]	Successful	Successful	-	-
22.6.E19	${\displaystyle{\displaystyle\frac{1-\operatorname{dn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}\left(z,k\right))}=\frac{\operatorname{dn}\left(z,k\right)-k^{2}\operatorname{cn}\left(z,k\right)-{k^{\prime}}^{2}}{k^{2}(\operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z,k\right))}}}$	(1 - JacobiDN(z, k))/((k)^(2)*(1 + JacobiCN(z, k)))=(JacobiDN(z, k)- (k)^(2)* JacobiCN(z, k)- 1 - (k)^(2))/((k)^(2)*(JacobiDN(z, k)- JacobiCN(z, k)))	Divide[1 - JacobiDN[z, (k)^2],(k)^(2)*(1 + JacobiCN[z, (k)^2])]=Divide[JacobiDN[z, (k)^2]- (k)^(2)* JacobiCN[z, (k)^2]- 1 - (k)^(2),(k)^(2)*(JacobiDN[z, (k)^2]- JacobiCN[z, (k)^2])]	Failure	Failure	Skip	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.45878538673953206, 0.3465757753125938] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.9906706508509473, -0.4762646080392824] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E20	${\displaystyle{\displaystyle{\operatorname{cn}^{2}}\left(\tfrac{1}{2}z,k\right)=\frac{-{k^{\prime}}^{2}+\operatorname{dn}\left(z,k\right)+k^{2}\operatorname{cn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}\left(z,k\right))}}}$	(JacobiCN((1)/(2)*z, k))^(2)=(- 1 - (k)^(2)+ JacobiDN(z, k)+ (k)^(2)* JacobiCN(z, k))/((k)^(2)*(1 + JacobiCN(z, k)))	(JacobiCN[Divide[1,2]*z, (k)^2])^(2)=Divide[- 1 - (k)^(2)+ JacobiDN[z, (k)^2]+ (k)^(2)* JacobiCN[z, (k)^2],(k)^(2)*(1 + JacobiCN[z, (k)^2])]	Failure	Failure	Fail 1.508209580+.7016668416*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -.5803980617-3.358394228*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} 16.67395347+17.27038148*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 1.508209580-.7016668416*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[1.5082095810878147, 0.7016668414711776] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.5803980702877369, -3.3583942301798655] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[16.673953528328266, 17.27038147971548] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.5082095810878147, -0.7016668414711776] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E20	${\displaystyle{\displaystyle\frac{-{k^{\prime}}^{2}+\operatorname{dn}\left(z,k\right)+k^{2}\operatorname{cn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}\left(z,k\right))}=\frac{{k^{\prime}}^{2}(1-\operatorname{dn}\left(z,k\right))}{k^{2}(\operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z,k\right))}}}$	(- 1 - (k)^(2)+ JacobiDN(z, k)+ (k)^(2)* JacobiCN(z, k))/((k)^(2)*(1 + JacobiCN(z, k)))=(1 - (k)^(2)*(1 - JacobiDN(z, k)))/((k)^(2)*(JacobiDN(z, k)- JacobiCN(z, k)))	Divide[- 1 - (k)^(2)+ JacobiDN[z, (k)^2]+ (k)^(2)* JacobiCN[z, (k)^2],(k)^(2)*(1 + JacobiCN[z, (k)^2])]=Divide[1 - (k)^(2)*(1 - JacobiDN[z, (k)^2]),(k)^(2)*(JacobiDN[z, (k)^2]- JacobiCN[z, (k)^2])]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} .4185282685+3.372883816*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -16.73162628-17.29201076*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} Float(infinity)+Float(infinity)*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.41852827726622005, 3.3728838175892157] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-16.731626338940483, -17.292010762136766] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E20	${\displaystyle{\displaystyle\frac{{k^{\prime}}^{2}(1-\operatorname{dn}\left(z,k\right))}{k^{2}(\operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z,k\right))}=\frac{{k^{\prime}}^{2}(1+\operatorname{cn}\left(z,k\right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,k\right)-k^{2}\operatorname{cn}\left(z,k\right)}}}$	(1 - (k)^(2)*(1 - JacobiDN(z, k)))/((k)^(2)*(JacobiDN(z, k)- JacobiCN(z, k)))=(1 - (k)^(2)*(1 + JacobiCN(z, k)))/(1 - (k)^(2)+ JacobiDN(z, k)- (k)^(2)* JacobiCN(z, k))	Divide[1 - (k)^(2)*(1 - JacobiDN[z, (k)^2]),(k)^(2)*(JacobiDN[z, (k)^2]- JacobiCN[z, (k)^2])]=Divide[1 - (k)^(2)*(1 + JacobiCN[z, (k)^2]),1 - (k)^(2)+ JacobiDN[z, (k)^2]- (k)^(2)* JacobiCN[z, (k)^2]]	Failure	Failure	Skip	Fail Complex[-0.07006090347484534, -0.11541357157973622] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.33824294612571393, 0.3972114524620808] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.07006090347484517, 0.11541357157973614] <- {Rule[k, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.33824294612571454, -0.3972114524620811] <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E21	${\displaystyle{\displaystyle{\operatorname{dn}^{2}}\left(\tfrac{1}{2}z,k\right)=\frac{k^{2}\operatorname{cn}\left(z,k\right)+\operatorname{dn}\left(z,k\right)+{k^{\prime}}^{2}}{1+\operatorname{dn}\left(z,k\right)}}}$	(JacobiDN((1)/(2)*z, k))^(2)=((k)^(2)* JacobiCN(z, k)+ JacobiDN(z, k)+ 1 - (k)^(2))/(1 + JacobiDN(z, k))	(JacobiDN[Divide[1,2]*z, (k)^2])^(2)=Divide[(k)^(2)* JacobiCN[z, (k)^2]+ JacobiDN[z, (k)^2]+ 1 - (k)^(2),1 + JacobiDN[z, (k)^2]]	Failure	Failure	Successful	Successful
22.6.E21	${\displaystyle{\displaystyle\frac{k^{2}\operatorname{cn}\left(z,k\right)+\operatorname{dn}\left(z,k\right)+{k^{\prime}}^{2}}{1+\operatorname{dn}\left(z,k\right)}=\frac{{k^{\prime}}^{2}(1-\operatorname{cn}\left(z,k\right))}{\operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z,k\right)}}}$	((k)^(2)* JacobiCN(z, k)+ JacobiDN(z, k)+ 1 - (k)^(2))/(1 + JacobiDN(z, k))=(1 - (k)^(2)*(1 - JacobiCN(z, k)))/(JacobiDN(z, k)- JacobiCN(z, k))	Divide[(k)^(2)* JacobiCN[z, (k)^2]+ JacobiDN[z, (k)^2]+ 1 - (k)^(2),1 + JacobiDN[z, (k)^2]]=Divide[1 - (k)^(2)*(1 - JacobiCN[z, (k)^2]),JacobiDN[z, (k)^2]- JacobiCN[z, (k)^2]]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} .352520828+.579583496e-1*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} .480944700-.194663538*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} Float(infinity)+Float(infinity)*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.3525208279139327, 0.05795834963740132] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.4809447044900228, -0.19466354179159007] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E21	${\displaystyle{\displaystyle\frac{{k^{\prime}}^{2}(1-\operatorname{cn}\left(z,k\right))}{\operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z,k\right)}=\frac{{k^{\prime}}^{2}(1+\operatorname{dn}\left(z,k\right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,k\right)-k^{2}\operatorname{cn}\left(z,k\right)}}}$	(1 - (k)^(2)*(1 - JacobiCN(z, k)))/(JacobiDN(z, k)- JacobiCN(z, k))=(1 - (k)^(2)*(1 + JacobiDN(z, k)))/(1 - (k)^(2)+ JacobiDN(z, k)- (k)^(2)* JacobiCN(z, k))	Divide[1 - (k)^(2)*(1 - JacobiCN[z, (k)^2]),JacobiDN[z, (k)^2]- JacobiCN[z, (k)^2]]=Divide[1 - (k)^(2)*(1 + JacobiDN[z, (k)^2]),1 - (k)^(2)+ JacobiDN[z, (k)^2]- (k)^(2)* JacobiCN[z, (k)^2]]	Failure	Failure	Skip	Fail Complex[0.10198361655247856, 0.014552344162486408] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.10205512091915558, 0.18553829333350658] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.10198361655248034, -0.014552344162487074] <- {Rule[k, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.10205512091914493, -0.18553829333351368] <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.6.E22	${\displaystyle{\displaystyle{\operatorname{pq}^{2}}\left(\tfrac{1}{2}z,k\right)=\frac{\operatorname{ps}\left(z,k\right)+\operatorname{rs}\left(z,k\right)}{\operatorname{qs}\left(z,k\right)+\operatorname{rs}\left(z,k\right)}}}$	genJacobiellk(p)*(q)^(2)* (1)/(2)*z*k =(genJacobiellk(p)*s* z*k + genJacobiellk(r)*s* z*k)/(genJacobiellk(q)*s* z*k + genJacobiellk(r)*s* z*k)	genJacobiellk(p)*(q)^(2)* Divide[1,2]*z*k =Divide[genJacobiellk(p)*s* z*k + genJacobiellk(r)*s* z*k,genJacobiellk(q)*s* z*k + genJacobiellk(r)*s* z*k]	Failure	Failure	Skip	Skip
22.6.E22	${\displaystyle{\displaystyle\frac{\operatorname{ps}\left(z,k\right)+\operatorname{rs}\left(z,k\right)}{\operatorname{qs}\left(z,k\right)+\operatorname{rs}\left(z,k\right)}=\frac{\operatorname{pq}\left(z,k\right)+\operatorname{rq}\left(z,k\right)}{1+\operatorname{rq}\left(z,k\right)}}}$	(genJacobiellk(p)*s* z*k + genJacobiellk(r)*s* z*k)/(genJacobiellk(q)*s* z*k + genJacobiellk(r)*s* z*k)=(genJacobiellk(p)*q* z*k + genJacobiellk(r)*q* z*k)/(1 + genJacobiellk(r)*q* z*k)	Divide[genJacobiellk(p)*s* z*k + genJacobiellk(r)*s* z*k,genJacobiellk(q)*s* z*k + genJacobiellk(r)*s* z*k]=Divide[genJacobiellk(p)*q* z*k + genJacobiellk(r)*q* z*k,1 + genJacobiellk(r)*q* z*k]	Failure	Failure	Skip	Skip
22.6.E22	${\displaystyle{\displaystyle\frac{\operatorname{pq}\left(z,k\right)+\operatorname{rq}\left(z,k\right)}{1+\operatorname{rq}\left(z,k\right)}=\frac{\operatorname{pr}\left(z,k\right)+1}{\operatorname{qr}\left(z,k\right)+1}}}$	(genJacobiellk(p)*q* z*k + genJacobiellk(r)*q* z*k)/(1 + genJacobiellk(r)*q* z*k)=(genJacobiellk(p)*r* z*k + 1)/(genJacobiellk(q)*r* z*k + 1)	Divide[genJacobiellk(p)*q* z*k + genJacobiellk(r)*q* z*k,1 + genJacobiellk(r)*q* z*k]=Divide[genJacobiellk(p)*r* z*k + 1,genJacobiellk(q)*r* z*k + 1]	Failure	Failure	Skip	Skip
22.7.E2	${\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}^{2}}\left(z/(1+k_{1}),k_{1}\right)}}}$	JacobiSN(z, k)=((1 + k[1])* JacobiSN(z/(1 + k[1]), k[1]))/(1 + k[1]*(JacobiSN(z/(1 + k[1]), k[1]))^(2))	JacobiSN[z, (k)^2]=Divide[(1 + Subscript[k, 1])* JacobiSN[z/(1 + Subscript[k, 1]), (Subscript[k, 1])^2],1 + Subscript[k, 1]*(JacobiSN[z/(1 + Subscript[k, 1]), (Subscript[k, 1])^2])^(2)]	Failure	Failure	Fail .55864218e-1-.8664119331e-1*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 1} -.2250495657+.6319729879*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 2} -1.435242225+.239860557e-1*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 3} -.17665263e-1+.9799998387e-1*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
22.7.E3	${\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}^{2}}\left(z/(1+k_{1}),k_{1}\right)}}}$	JacobiCN(z, k)=(JacobiCN(z/(1 + k[1]), k[1])*JacobiDN(z/(1 + k[1]), k[1]))/(1 + k[1]*(JacobiSN(z/(1 + k[1]), k[1]))^(2))	JacobiCN[z, (k)^2]=Divide[JacobiCN[z/(1 + Subscript[k, 1]), (Subscript[k, 1])^2]*JacobiDN[z/(1 + Subscript[k, 1]), (Subscript[k, 1])^2],1 + Subscript[k, 1]*(JacobiSN[z/(1 + Subscript[k, 1]), (Subscript[k, 1])^2])^(2)]	Failure	Failure	Fail -.2105289453-.565375351e-1*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 1} -1.400590390+1.028876610*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 2} -1.242789928+.3906876874*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 3} .2083630894+.417473069e-1*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
22.7.E4	${\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}^{2}}\left(z/(1+k_{1}),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}^{2}}\left(z/(1+k_{1}),k_{1}\right)}}}$	JacobiDN(z, k)=((JacobiDN(z/(1 + k[1]), k[1]))^(2)-(1 - k[1]))/(1 + k[1]- (JacobiDN(z/(1 + k[1]), k[1]))^(2))	JacobiDN[z, (k)^2]=Divide[(JacobiDN[z/(1 + Subscript[k, 1]), (Subscript[k, 1])^2])^(2)-(1 - Subscript[k, 1]),1 + Subscript[k, 1]- (JacobiDN[z/(1 + Subscript[k, 1]), (Subscript[k, 1])^2])^(2)]	Failure	Failure	Fail .1402363057-.923635469e-1*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 1} 1.725656124-1.103604180*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 2} .7478105788+1.143211966*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 3} -.1058691384+.765115293e-1*I <- {z = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
22.7.E6	${\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{(1+k^{\prime}_{2})\operatorname{sn}\left(z/(1+k^{\prime}_{2}),k_{2}\right)\operatorname{cn}\left(z/(1+k^{\prime}_{2}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k^{\prime}_{2}),k_{2}\right)}}}$	JacobiSN(z, k)=((1 +sqrt(1 - (k)^(2))[2])* JacobiSN(z/(1 +sqrt(1 - (k)^(2))[2]), k[2])*JacobiCN(z/(1 +sqrt(1 - (k)^(2))[2]), k[2]))/(JacobiDN(z/(1 +sqrt(1 - (k)^(2))[2]), k[2]))	JacobiSN[z, (k)^2]=Divide[(1 +Subscript[Sqrt[1 - (k)^(2)], 2])* JacobiSN[z/(1 +Subscript[Sqrt[1 - (k)^(2)], 2]), (Subscript[k, 2])^2]*JacobiCN[z/(1 +Subscript[Sqrt[1 - (k)^(2)], 2]), (Subscript[k, 2])^2],JacobiDN[z/(1 +Subscript[Sqrt[1 - (k)^(2)], 2]), (Subscript[k, 2])^2]]	Failure	Failure	Error	Successful
22.7.E7	${\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{(1+k^{\prime}_{2})({\operatorname{dn}^{2}}\left(z/(1+k^{\prime}_{2}),k_{2}\right)-k^{\prime}_{2})}{k_{2}^{2}\operatorname{dn}\left(z/(1+k^{\prime}_{2}),k_{2}\right)}}}$	JacobiCN(z, k)((1 +sqrt(1 - (k)^(2))[2])*((JacobiDN(z/(1 +sqrt(1 - (k)^(2))[2]), k[2]))^(2)-sqrt(1 - (k)^(2))[2]))/(k(k[2])^(2)*JacobiDN(z/(1 +sqrt(1 - (k)^(2))[2]), k[2]))	JacobiCN[z, (k)^2]Divide[(1 +Subscript[Sqrt[1 - (k)^(2)], 2])*((JacobiDN[z/(1 +Subscript[Sqrt[1 - (k)^(2)], 2]), (Subscript[k, 2])^2])^(2)-Subscript[Sqrt[1 - (k)^(2)], 2]),k(Subscript[k, 2])^(2)*JacobiDN[z/(1 +Subscript[Sqrt[1 - (k)^(2)], 2]), (Subscript[k, 2])^2]]	Failure	Failure	Error	Successful
22.7.E8	${\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{(1-k^{\prime}_{2})({\operatorname{dn}^{2}}\left(z/(1+k^{\prime}_{2}),k_{2}\right)+k^{\prime}_{2})}{k_{2}^{2}\operatorname{dn}\left(z/(1+k^{\prime}_{2}),k_{2}\right)}}}$	JacobiDN(z, k)((1 -sqrt(1 - (k)^(2))[2])*((JacobiDN(z/(1 +sqrt(1 - (k)^(2))[2]), k[2]))^(2)+sqrt(1 - (k)^(2))[2]))/(k(k[2])^(2)*JacobiDN(z/(1 +sqrt(1 - (k)^(2))[2]), k[2]))	JacobiDN[z, (k)^2]Divide[(1 -Subscript[Sqrt[1 - (k)^(2)], 2])*((JacobiDN[z/(1 +Subscript[Sqrt[1 - (k)^(2)], 2]), (Subscript[k, 2])^2])^(2)+Subscript[Sqrt[1 - (k)^(2)], 2]),k(Subscript[k, 2])^(2)*JacobiDN[z/(1 +Subscript[Sqrt[1 - (k)^(2)], 2]), (Subscript[k, 2])^2]]	Failure	Failure	Error	Successful
22.8.E1	${\displaystyle{\displaystyle\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v+\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}{1-k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}}}$	JacobiSN(u + v, k)=(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)+ JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))/(1 - (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))	JacobiSN[u + v, (k)^2]=Divide[JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]+ JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2],1 - (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2)]	Successful	Failure	-	Skip
22.8.E2	${\displaystyle{\displaystyle\operatorname{cn}(u+v)=\frac{\operatorname{cn}u\operatorname{cn}v-\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}{1-k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}}}$	JacobiCN(u + v, k)=(JacobiCN(u, k)*JacobiCN(v, k)- JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))/(1 - (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))	JacobiCN[u + v, (k)^2]=Divide[JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]- JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2],1 - (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2)]	Successful	Failure	-	Successful
22.8.E3	${\displaystyle{\displaystyle\operatorname{dn}(u+v)=\frac{\operatorname{dn}u\operatorname{dn}v-k^{2}\operatorname{sn}u\operatorname{cn}u\operatorname{sn}v\operatorname{cn}v}{1-k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}}}$	JacobiDN(u + v, k)=(JacobiDN(u, k)*JacobiDN(v, k)- (k)^(2)* JacobiSN(u, k)*JacobiCN(u, k)*JacobiSN(v, k)*JacobiCN(v, k))/(1 - (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))	JacobiDN[u + v, (k)^2]=Divide[JacobiDN[u, (k)^2]*JacobiDN[v, (k)^2]- (k)^(2)* JacobiSN[u, (k)^2]*JacobiCN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiCN[v, (k)^2],1 - (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2)]	Successful	Failure	-	Skip
22.8.E4	${\displaystyle{\displaystyle\operatorname{cd}(u+v)=\frac{\operatorname{cd}u\operatorname{cd}v-{k^{\prime}}^{2}\operatorname{sd}u\operatorname{nd}u\operatorname{sd}v\operatorname{nd}v}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{2}}u{\operatorname{sd}^{2}}v}}}$	JacobiCD(u + v, k)=(JacobiCD(u, k)*JacobiCD(v, k)- 1 - (k)^(2)* JacobiSD(u, k)*JacobiND(u, k)*JacobiSD(v, k)*JacobiND(v, k))/(1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD(u, k))^(2)* (JacobiSD(v, k))^(2))	JacobiCD[u + v, (k)^2]=Divide[JacobiCD[u, (k)^2]*JacobiCD[v, (k)^2]- 1 - (k)^(2)* JacobiSD[u, (k)^2]*JacobiND[u, (k)^2]*JacobiSD[v, (k)^2]*JacobiND[v, (k)^2],1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD[u, (k)^2])^(2)* (JacobiSD[v, (k)^2])^(2)]	Failure	Failure	Fail .1370541185+.1873251287e-1*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 1} .5364817078-.4234624245*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 2} -.1981753675-.1199254751*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 3} .1228104592+.1366601567e-11*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
22.8.E5	${\displaystyle{\displaystyle\operatorname{sd}(u+v)=\frac{\operatorname{sd}u\operatorname{cd}v\operatorname{nd}v+\operatorname{sd}v\operatorname{cd}u\operatorname{nd}u}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{2}}u{\operatorname{sd}^{2}}v}}}$	JacobiSD(u + v, k)=(JacobiSD(u, k)*JacobiCD(v, k)*JacobiND(v, k)+ JacobiSD(v, k)*JacobiCD(u, k)*JacobiND(u, k))/(1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD(u, k))^(2)* (JacobiSD(v, k))^(2))	JacobiSD[u + v, (k)^2]=Divide[JacobiSD[u, (k)^2]*JacobiCD[v, (k)^2]*JacobiND[v, (k)^2]+ JacobiSD[v, (k)^2]*JacobiCD[u, (k)^2]*JacobiND[u, (k)^2],1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD[u, (k)^2])^(2)* (JacobiSD[v, (k)^2])^(2)]	Failure	Failure	Fail -8.411649228+2.496794499*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 1} .4171906607e-1-.5404871752*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 2} .4672486370e-1-.3344443629*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 3} 8.845535938-0.*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
22.8.E6	${\displaystyle{\displaystyle\operatorname{nd}(u+v)=\frac{\operatorname{nd}u\operatorname{nd}v+k^{2}\operatorname{sd}u\operatorname{cd}u\operatorname{sd}v\operatorname{cd}v}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{2}}u{\operatorname{sd}^{2}}v}}}$	JacobiND(u + v, k)=(JacobiND(u, k)*JacobiND(v, k)+ (k)^(2)* JacobiSD(u, k)*JacobiCD(u, k)*JacobiSD(v, k)*JacobiCD(v, k))/(1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD(u, k))^(2)* (JacobiSD(v, k))^(2))	JacobiND[u + v, (k)^2]=Divide[JacobiND[u, (k)^2]*JacobiND[v, (k)^2]+ (k)^(2)* JacobiSD[u, (k)^2]*JacobiCD[u, (k)^2]*JacobiSD[v, (k)^2]*JacobiCD[v, (k)^2],1 + (k)^(2)* 1 - (k)^(2)* (JacobiSD[u, (k)^2])^(2)* (JacobiSD[v, (k)^2])^(2)]	Failure	Failure	Fail -8.469613315+2.476300734*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 1} -.2446575645+.5929818787*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 2} -.5059779542e-1-.4472642739*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 3} 8.907556175-0.*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-8.469613322356564, 2.4763007298746587] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.24465756431597135, 0.5929818773302299] <- {Rule[k, 2], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.05059779615055053, -0.4472642742544212] <- {Rule[k, 3], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[8.907556180814666, 0.0] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E7	${\displaystyle{\displaystyle\operatorname{dc}(u+v)=\frac{\operatorname{dc}u\operatorname{dc}v+{k^{\prime}}^{2}\operatorname{sc}u\operatorname{nc}u\operatorname{sc}v\operatorname{nc}v}{1-{k^{\prime}}^{2}{\operatorname{sc}^{2}}u{\operatorname{sc}^{2}}v}}}$	JacobiDC(u + v, k)=(JacobiDC(u, k)*JacobiDC(v, k)+ 1 - (k)^(2)* JacobiSC(u, k)*JacobiNC(u, k)*JacobiSC(v, k)*JacobiNC(v, k))/(1 - 1 - (k)^(2)* (JacobiSC(u, k))^(2)* (JacobiSC(v, k))^(2))	JacobiDC[u + v, (k)^2]=Divide[JacobiDC[u, (k)^2]*JacobiDC[v, (k)^2]+ 1 - (k)^(2)* JacobiSC[u, (k)^2]*JacobiNC[u, (k)^2]*JacobiSC[v, (k)^2]*JacobiNC[v, (k)^2],1 - 1 - (k)^(2)* (JacobiSC[u, (k)^2])^(2)* (JacobiSC[v, (k)^2])^(2)]	Failure	Failure	Fail .2798628459+.1057645812*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 1} 2.155279764+3.336838966*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 2} -10.41618961+.723801634*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 3} .2913197033+.1243926156e-11*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.27986284597445743, 0.1057645806458628] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.1552797720040004, 3.3368389687939786] <- {Rule[k, 2], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-10.416189608158701, 0.7238016559320513] <- {Rule[k, 3], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.2913197027505876, 0.0] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E8	${\displaystyle{\displaystyle\operatorname{nc}(u+v)=\frac{\operatorname{nc}u\operatorname{nc}v+\operatorname{sc}u\operatorname{dc}u\operatorname{sc}v\operatorname{dc}v}{1-{k^{\prime}}^{2}{\operatorname{sc}^{2}}u{\operatorname{sc}^{2}}v}}}$	JacobiNC(u + v, k)=(JacobiNC(u, k)*JacobiNC(v, k)+ JacobiSC(u, k)*JacobiDC(u, k)*JacobiSC(v, k)*JacobiDC(v, k))/(1 - 1 - (k)^(2)* (JacobiSC(u, k))^(2)* (JacobiSC(v, k))^(2))	JacobiNC[u + v, (k)^2]=Divide[JacobiNC[u, (k)^2]*JacobiNC[v, (k)^2]+ JacobiSC[u, (k)^2]*JacobiDC[u, (k)^2]*JacobiSC[v, (k)^2]*JacobiDC[v, (k)^2],1 - 1 - (k)^(2)* (JacobiSC[u, (k)^2])^(2)* (JacobiSC[v, (k)^2])^(2)]	Failure	Failure	Fail -8.445366052+2.504181595*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 1} -2.348000820+.4644873082*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 2} .3919060714+4.559323678*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 3} 8.869980794+0.*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-8.4453660597032, 2.504181591384576] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.3480008192225705, 0.46448731013438893] <- {Rule[k, 2], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.39190608798513005, 4.559323684618953] <- {Rule[k, 3], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[8.869980800731279, 0.0] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E9	${\displaystyle{\displaystyle\operatorname{sc}(u+v)=\frac{\operatorname{sc}u\operatorname{dc}v\operatorname{nc}v+\operatorname{sc}v\operatorname{dc}u\operatorname{nc}u}{1-{k^{\prime}}^{2}{\operatorname{sc}^{2}}u{\operatorname{sc}^{2}}v}}}$	JacobiSC(u + v, k)=(JacobiSC(u, k)*JacobiDC(v, k)*JacobiNC(v, k)+ JacobiSC(v, k)*JacobiDC(u, k)*JacobiNC(u, k))/(1 - 1 - (k)^(2)* (JacobiSC(u, k))^(2)* (JacobiSC(v, k))^(2))	JacobiSC[u + v, (k)^2]=Divide[JacobiSC[u, (k)^2]*JacobiDC[v, (k)^2]*JacobiNC[v, (k)^2]+ JacobiSC[v, (k)^2]*JacobiDC[u, (k)^2]*JacobiNC[u, (k)^2],1 - 1 - (k)^(2)* (JacobiSC[u, (k)^2])^(2)* (JacobiSC[v, (k)^2])^(2)]	Failure	Failure	Fail -8.387425493+2.524419001*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 1} 1.765721394-.9866914128*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 2} -.5663184000+3.386135413*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 3} 8.808222182+0.*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-8.387425500158386, 2.5244189972251565] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.7657213943311842, -0.9866914167974857] <- {Rule[k, 2], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.5663183953417591, 3.3861354179416785] <- {Rule[k, 3], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[8.808222188159213, 0.0] <- {Rule[k, 1], Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[v, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E10	${\displaystyle{\displaystyle\operatorname{ns}(u+v)=\frac{\operatorname{ns}u\operatorname{ds}v\operatorname{cs}v-\operatorname{ns}v\operatorname{ds}u\operatorname{cs}u}{{\operatorname{cs}^{2}}v-{\operatorname{cs}^{2}}u}}}$	JacobiNS(u + v, k)=(JacobiNS(u, k)*JacobiDS(v, k)*JacobiCS(v, k)- JacobiNS(v, k)*JacobiDS(u, k)*JacobiCS(u, k))/((JacobiCS(v, k))^(2)- (JacobiCS(u, k))^(2))	JacobiNS[u + v, (k)^2]=Divide[JacobiNS[u, (k)^2]*JacobiDS[v, (k)^2]*JacobiCS[v, (k)^2]- JacobiNS[v, (k)^2]*JacobiDS[u, (k)^2]*JacobiCS[u, (k)^2],(JacobiCS[v, (k)^2])^(2)- (JacobiCS[u, (k)^2])^(2)]	Successful	Failure	-	Successful
22.8.E11	${\displaystyle{\displaystyle\operatorname{ds}(u+v)=\frac{\operatorname{ds}u\operatorname{cs}v\operatorname{ns}v-\operatorname{ds}v\operatorname{cs}u\operatorname{ns}u}{{\operatorname{cs}^{2}}v-{\operatorname{cs}^{2}}u}}}$	JacobiDS(u + v, k)=(JacobiDS(u, k)*JacobiCS(v, k)*JacobiNS(v, k)- JacobiDS(v, k)*JacobiCS(u, k)*JacobiNS(u, k))/((JacobiCS(v, k))^(2)- (JacobiCS(u, k))^(2))	JacobiDS[u + v, (k)^2]=Divide[JacobiDS[u, (k)^2]*JacobiCS[v, (k)^2]*JacobiNS[v, (k)^2]- JacobiDS[v, (k)^2]*JacobiCS[u, (k)^2]*JacobiNS[u, (k)^2],(JacobiCS[v, (k)^2])^(2)- (JacobiCS[u, (k)^2])^(2)]	Successful	Failure	-	Successful
22.8.E12	${\displaystyle{\displaystyle\operatorname{cs}(u+v)=\frac{\operatorname{cs}u\operatorname{ds}v\operatorname{ns}v-\operatorname{cs}v\operatorname{ds}u\operatorname{ns}u}{{\operatorname{cs}^{2}}v-{\operatorname{cs}^{2}}u}}}$	JacobiCS(u + v, k)=(JacobiCS(u, k)*JacobiDS(v, k)*JacobiNS(v, k)- JacobiCS(v, k)*JacobiDS(u, k)*JacobiNS(u, k))/((JacobiCS(v, k))^(2)- (JacobiCS(u, k))^(2))	JacobiCS[u + v, (k)^2]=Divide[JacobiCS[u, (k)^2]*JacobiDS[v, (k)^2]*JacobiNS[v, (k)^2]- JacobiCS[v, (k)^2]*JacobiDS[u, (k)^2]*JacobiNS[u, (k)^2],(JacobiCS[v, (k)^2])^(2)- (JacobiCS[u, (k)^2])^(2)]	Successful	Failure	-	Successful
22.8.E13	${\displaystyle{\displaystyle\operatorname{sn}(u+v)=\frac{{\operatorname{sn}^{2}}u-{\operatorname{sn}^{2}}v}{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}}}$	JacobiSN(u + v, k)=((JacobiSN(u, k))^(2)- (JacobiSN(v, k))^(2))/(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))	JacobiSN[u + v, (k)^2]=Divide[(JacobiSN[u, (k)^2])^(2)- (JacobiSN[v, (k)^2])^(2),JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]]	Successful	Failure	-	Successful
22.8.E14	${\displaystyle{\displaystyle\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v+\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn}u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}}}$	JacobiSN(u + v, k)=(JacobiSN(u, k)*JacobiCN(u, k)*JacobiDN(v, k)+ JacobiSN(v, k)*JacobiCN(v, k)*JacobiDN(u, k))/(JacobiCN(u, k)*JacobiCN(v, k)+ JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))	JacobiSN[u + v, (k)^2]=Divide[JacobiSN[u, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[v, (k)^2]+ JacobiSN[v, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[u, (k)^2],JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]+ JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2]]	Successful	Failure	-	Successful
22.8.E15	${\displaystyle{\displaystyle\operatorname{cn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}}}$	JacobiCN(u + v, k)=(JacobiSN(u, k)*JacobiCN(u, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(v, k)*JacobiDN(u, k))/(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))	JacobiCN[u + v, (k)^2]=Divide[JacobiSN[u, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[u, (k)^2],JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]]	Successful	Failure	-	Successful
22.8.E16	${\displaystyle{\displaystyle\operatorname{cn}(u+v)=\frac{1-{\operatorname{sn}^{2}}u-{\operatorname{sn}^{2}}v+k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}{\operatorname{cn}u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}}}$	JacobiCN(u + v, k)=(1 - (JacobiSN(u, k))^(2)- (JacobiSN(v, k))^(2)+ (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))/(JacobiCN(u, k)*JacobiCN(v, k)+ JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))	JacobiCN[u + v, (k)^2]=Divide[1 - (JacobiSN[u, (k)^2])^(2)- (JacobiSN[v, (k)^2])^(2)+ (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2),JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]+ JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2]]	Successful	Failure	-	Successful
22.8.E17	${\displaystyle{\displaystyle\operatorname{dn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}u-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}v}{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}}}$	JacobiDN(u + v, k)=(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(u, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(v, k))/(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))	JacobiDN[u + v, (k)^2]=Divide[JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[u, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[v, (k)^2],JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]]	Successful	Failure	-	Successful
22.8.E18	${\displaystyle{\displaystyle\operatorname{dn}(u+v)=\frac{\operatorname{cn}u\operatorname{dn}u\operatorname{cn}v\operatorname{dn}v+{k^{\prime}}^{2}\operatorname{sn}u\operatorname{sn}v}{\operatorname{cn}u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}}}$	JacobiDN(u + v, k)=(JacobiCN(u, k)*JacobiDN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)+ 1 - (k)^(2)* JacobiSN(u, k)*JacobiSN(v, k))/(JacobiCN(u, k)*JacobiCN(v, k)+ JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))	JacobiDN[u + v, (k)^2]=Divide[JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]+ 1 - (k)^(2)* JacobiSN[u, (k)^2]*JacobiSN[v, (k)^2],JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]+ JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2]]	Failure	Failure	Fail -.4438908315+.1804284132e-1*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 1} -.1432992406+.147150302*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 2} -.8677161564+.92219262e-1*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), k = 3} .4223716725-.4114809072e-11*I <- {u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
22.8.E21	$k^{\prime }{}^2-k^{\prime }{}^2{k}^2\mathrm{sn}{z}_1\mathrm{sn}{z}_2\mathrm{sn}{z}_3\mathrm{sn}{z}_4+k^2\mathrm{cn}{z}_1\mathrm{cn}{z}_2\mathrm{cn}{z}_{}$