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	${\displaystyle{\displaystyle{k^{\prime}}^{2}-{k^{\prime}}^{2}k^{2}\operatorname{sn}z_{1}\operatorname{sn}z_{2}\operatorname{sn}z_{3}\operatorname{sn}z_{4}+k^{2}\operatorname{cn}z_{1}\operatorname{cn}z_{2}\operatorname{cn}z_{3}\operatorname{cn}z_{4}-\operatorname{dn}z_{1}\operatorname{dn}z_{2}\operatorname{dn}z_{3}\operatorname{dn}z_{4}=0}}$	1 - (k)^(2)- 1 - (k)^(2)* (k)^(2)* JacobiSN(z[1], k)*JacobiSN(z[2], k)*JacobiSN(z[3], k)*JacobiSN(z[4], k)+ (k)^(2)* JacobiCN(z[1], k)*JacobiCN(z[2], k)*JacobiCN(z[3], k)*JacobiCN(z[4], k)- JacobiDN(z[1], k)*JacobiDN(z[2], k)*JacobiDN(z[3], k)*JacobiDN(z[4], k)= 0	1 - (k)^(2)- 1 - (k)^(2)* (k)^(2)* JacobiSN[Subscript[z, 1], (k)^2]*JacobiSN[Subscript[z, 2], (k)^2]*JacobiSN[Subscript[z, 3], (k)^2]*JacobiSN[Subscript[z, 4], (k)^2]+ (k)^(2)* JacobiCN[Subscript[z, 1], (k)^2]*JacobiCN[Subscript[z, 2], (k)^2]*JacobiCN[Subscript[z, 3], (k)^2]*JacobiCN[Subscript[z, 4], (k)^2]- JacobiDN[Subscript[z, 1], (k)^2]*JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 3], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2]= 0	Failure	Failure	Fail -2.551869041-.2283807357*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)+I*2^(1/2), k = 1} 43.87494853-8.870468766*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)+I*2^(1/2), k = 2} -1.106498767+4.008029613*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)+I*2^(1/2), k = 3} -2.564399564-.1144957915*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
22.8.E22	${\displaystyle{\displaystyle z_{1}+z_{2}+z_{3}+z_{4}=2\!K\left(k\right)}}$	z[1]+ z[2]+ z[3]+ z[4]= 2*EllipticK(k)	Subscript[z, 1]+ Subscript[z, 2]+ Subscript[z, 3]+ Subscript[z, 4]= 2*EllipticK[(k)^2]	Failure	Failure	Error	Skip
22.8.E24	${\displaystyle{\displaystyle z_{1}-z_{2}=z_{2}-z_{3}}}$	z[1]- z[2]= z[2]- z[3]	Subscript[z, 1]- Subscript[z, 2]= Subscript[z, 2]- Subscript[z, 3]	Failure	Failure	Fail -2.828427124*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)-I*2^(1/2)} -2.828427124-2.828427124*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = -2^(1/2)-I*2^(1/2)} -2.828427124 <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = -2^(1/2)+I*2^(1/2)} 5.656854248*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)-I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.0, -2.8284271247461903] <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} -2.8284271247461903 <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[0.0, 5.656854249492381] <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E24	${\displaystyle{\displaystyle z_{2}-z_{3}=\tfrac{2}{3}\!K\left(k\right)}}$	z[2]- z[3]=(2)/(3)*EllipticK(k)	Subscript[z, 2]- Subscript[z, 3]=Divide[2,3]*EllipticK[(k)^2]	Failure	Failure	Error	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.561916784937532, 0.7188385491665478] <- {Rule[k, 2], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.35941927458327383, 0.5619167849375319] <- {Rule[k, 3], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E26	${\displaystyle{\displaystyle z_{1}-z_{2}=z_{2}-z_{3}}}$	z[1]- z[2]= z[2]- z[3]	Subscript[z, 1]- Subscript[z, 2]= Subscript[z, 2]- Subscript[z, 3]	Failure	Failure	Fail -2.828427124*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)-I*2^(1/2)} -2.828427124-2.828427124*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = -2^(1/2)-I*2^(1/2)} -2.828427124 <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = -2^(1/2)+I*2^(1/2)} 5.656854248*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)-I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.0, -2.8284271247461903] <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} -2.8284271247461903 <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[0.0, 5.656854249492381] <- {Rule[Subscript[z, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E26	${\displaystyle{\displaystyle z_{2}-z_{3}=z_{3}-z_{4}}}$	z[2]- z[3]= z[3]- z[4]	Subscript[z, 2]- Subscript[z, 3]= Subscript[z, 3]- Subscript[z, 4]	Failure	Failure	Fail -2.828427124*I <- {z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)-I*2^(1/2)} -2.828427124-2.828427124*I <- {z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = -2^(1/2)-I*2^(1/2)} -2.828427124 <- {z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = -2^(1/2)+I*2^(1/2)} 5.656854248*I <- {z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)-I*2^(1/2), z[4] = 2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.0, -2.8284271247461903] <- {Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} -2.8284271247461903 <- {Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[0.0, 5.656854249492381] <- {Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E26	${\displaystyle{\displaystyle z_{3}-z_{4}=\tfrac{1}{2}\!K\left(k\right)}}$	z[3]- z[4]=(1)/(2)*EllipticK(k)	Subscript[z, 3]- Subscript[z, 4]=Divide[1,2]*EllipticK[(k)^2]	Failure	Failure	Skip	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.42143758870314907, 0.5391289118749109] <- {Rule[k, 2], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.26956445593745537, 0.421437588703149] <- {Rule[k, 3], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[Subscript[z, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.8.E27	${\displaystyle{\displaystyle\operatorname{dn}z_{1}\operatorname{dn}z_{3}=\operatorname{dn}z_{2}\operatorname{dn}z_{4}}}$	JacobiDN(z[1], k)*JacobiDN(z[3], k)= JacobiDN(z[2], k)*JacobiDN(z[4], k)	JacobiDN[Subscript[z, 1], (k)^2]*JacobiDN[Subscript[z, 3], (k)^2]= JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2]	Failure	Failure	Fail -.5144265506-.9141882921e-1*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)-I*2^(1/2), k = 1} -4.611092996-5.088311894*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)-I*2^(1/2), k = 2} -1.061172547+1.016431323*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)-I*2^(1/2), k = 3} -.5144265506-.9141882921e-1*I <- {z[1] = 2^(1/2)+I*2^(1/2), z[2] = 2^(1/2)+I*2^(1/2), z[3] = 2^(1/2)+I*2^(1/2), z[4] = -2^(1/2)+I*2^(1/2), k = 1} ... skip entries to safe data	Skip
22.8.E27	${\displaystyle{\displaystyle\operatorname{dn}z_{2}\operatorname{dn}z_{4}=k^{\prime}}}$	JacobiDN(z[2], k)*JacobiDN(z[4], k)=sqrt(1 - (k)^(2))	JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2]=Sqrt[1 - (k)^(2)]	Failure	Failure	Fail -.2490902475-.9141882921e-1*I <- {z[2] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)+I*2^(1/2), k = 1} .5019134627-6.820362702*I <- {z[2] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)+I*2^(1/2), k = 2} -.4379803287e-1-1.811995802*I <- {z[2] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)+I*2^(1/2), k = 3} .2653363031+0.*I <- {z[2] = 2^(1/2)+I*2^(1/2), z[4] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-0.2490902473433826, -0.0914188289178922] <- {Rule[k, 1], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.5019134769388622, -6.820362696879865] <- {Rule[k, 2], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.04379803397253301, -1.8119958055932632] <- {Rule[k, 3], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.26533630283530063, 0.0] <- {Rule[k, 1], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 4], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.9.E1	${\displaystyle{\displaystyle s_{m,p}^{(2)}=\operatorname{sn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right)}}$	(s[m , p])^(2)= JacobiSN(z + 2*(p)^(- 1)*(m - 1)* EllipticK(k), k)	(Subscript[s, m , p])^(2)= JacobiSN[z + 2*(p)^(- 1)*(m - 1)* EllipticK[(k)^2], (k)^2]	Failure	Failure	Error	Skip
22.9.E2	${\displaystyle{\displaystyle c_{m,p}^{(2)}=\operatorname{cn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right)}}$	(c[m , p])^(2)= JacobiCN(z + 2*(p)^(- 1)*(m - 1)* EllipticK(k), k)	(Subscript[c, m , p])^(2)= JacobiCN[z + 2*(p)^(- 1)*(m - 1)* EllipticK[(k)^2], (k)^2]	Failure	Failure	Error	Skip
22.9.E3	${\displaystyle{\displaystyle d_{m,p}^{(2)}=\operatorname{dn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right)}}$	(d[m , p])^(2)= JacobiDN(z + 2*(p)^(- 1)*(m - 1)* EllipticK(k), k)	(Subscript[d, m , p])^(2)= JacobiDN[z + 2*(p)^(- 1)*(m - 1)* EllipticK[(k)^2], (k)^2]	Failure	Failure	Error	Skip
22.9.E4	${\displaystyle{\displaystyle s_{m,p}^{(4)}=\operatorname{sn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right)}}$	(s[m , p])^(4)= JacobiSN(z + 4*(p)^(- 1)*(m - 1)* EllipticK(k), k)	(Subscript[s, m , p])^(4)= JacobiSN[z + 4*(p)^(- 1)*(m - 1)* EllipticK[(k)^2], (k)^2]	Failure	Failure	Error	Skip
22.9.E5	${\displaystyle{\displaystyle c_{m,p}^{(4)}=\operatorname{cn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right)}}$	(c[m , p])^(4)= JacobiCN(z + 4*(p)^(- 1)*(m - 1)* EllipticK(k), k)	(Subscript[c, m , p])^(4)= JacobiCN[z + 4*(p)^(- 1)*(m - 1)* EllipticK[(k)^2], (k)^2]	Failure	Failure	Error	Skip
22.9.E6	${\displaystyle{\displaystyle d_{m,p}^{(4)}=\operatorname{dn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right)}}$	(d[m , p])^(4)= JacobiDN(z + 4*(p)^(- 1)*(m - 1)* EllipticK(k), k)	(Subscript[d, m , p])^(4)= JacobiDN[z + 4*(p)^(- 1)*(m - 1)* EllipticK[(k)^2], (k)^2]	Failure	Failure	Error	Skip
22.9.E7	${\displaystyle{\displaystyle\kappa=\operatorname{dn}\left(2\!K\left(k\right)/3,k\right)}}$	kappa = JacobiDN(2*EllipticK(k)/ 3, k)	\[Kappa]= JacobiDN[2*EllipticK[(k)^2]/ 3, (k)^2]	Failure	Failure	Error	Fail Complex[0.8378491740658316, -0.012111700392196889] <- {Rule[k, 2], Rule[κ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.5354591881471004, -0.6433228556445936] <- {Rule[k, 3], Rule[κ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.8378491740658316, -2.8405388251383874] <- {Rule[k, 2], Rule[κ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.5354591881471004, -3.4717499803907836] <- {Rule[k, 3], Rule[κ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.11.E1	${\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}}}}$	JacobiSN(z, k)=(2*Pi)/(K*k)*sum(((q)^(n +(1)/(2))* sin((2*n + 1)* zeta))/(1 - (q)^(2*n + 1)), n = 0..infinity)	JacobiSN[z, (k)^2]=Divide[2*Pi,K*k]*Sum[Divide[(q)^(n +Divide[1,2])* Sin[(2*n + 1)* \[zeta]],1 - (q)^(2*n + 1)], {n, 0, Infinity}]	Error	Failure	-	Error
22.11.E2	${\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}}}}$	JacobiCN(z, k)=(2*Pi)/(K*k)*sum(((q)^(n +(1)/(2))* cos((2*n + 1)* zeta))/(1 + (q)^(2*n + 1)), n = 0..infinity)	JacobiCN[z, (k)^2]=Divide[2*Pi,K*k]*Sum[Divide[(q)^(n +Divide[1,2])* Cos[(2*n + 1)* \[zeta]],1 + (q)^(2*n + 1)], {n, 0, Infinity}]	Failure	Failure	Skip	Error
22.11.E3	${\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}}}$	JacobiDN(z, k)=(Pi)/(2*EllipticK(k))+(2*Pi)/(EllipticK(k))*sum(((q)^(n)* cos(2*n*zeta))/(1 + (q)^(2*n)), n = 1..infinity)	JacobiDN[z, (k)^2]=Divide[Pi,2*EllipticK[(k)^2]]+Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(q)^(n)* Cos[2*n*\[zeta]],1 + (q)^(2*n)], {n, 1, Infinity}]	Failure	Failure	Skip	Error
22.11.E4	${\displaystyle{\displaystyle\operatorname{cd}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}}}}$	JacobiCD(z, k)=(2*Pi)/(K*k)*sum(((- 1)^(n)* (q)^(n +(1)/(2))* cos((2*n + 1)* zeta))/(1 - (q)^(2*n + 1)), n = 0..infinity)	JacobiCD[z, (k)^2]=Divide[2*Pi,K*k]*Sum[Divide[(- 1)^(n)* (q)^(n +Divide[1,2])* Cos[(2*n + 1)* \[zeta]],1 - (q)^(2*n + 1)], {n, 0, Infinity}]	Failure	Failure	Skip	Error
22.11.E5	${\displaystyle{\displaystyle\operatorname{sd}\left(z,k\right)=\frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}}}}$	JacobiSD(z, k)=(2*Pi)/(K*k*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)* (q)^(n +(1)/(2))* sin((2*n + 1)* zeta))/(1 + (q)^(2*n + 1)), n = 0..infinity)	JacobiSD[z, (k)^2]=Divide[2*Pi,K*k*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)* (q)^(n +Divide[1,2])* Sin[(2*n + 1)* \[zeta]],1 + (q)^(2*n + 1)], {n, 0, Infinity}]	Failure	Failure	Skip	Error
22.11.E6	${\displaystyle{\displaystyle\operatorname{nd}\left(z,k\right)=\frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}}}$	JacobiND(z, k)=(Pi)/(2*K*sqrt(1 - (k)^(2)))+(2*Pi)/(K*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)* (q)^(n)* cos(2*n*zeta))/(1 + (q)^(2*n)), n = 1..infinity)	JacobiND[z, (k)^2]=Divide[Pi,2*K*Sqrt[1 - (k)^(2)]]+Divide[2*Pi,K*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)* (q)^(n)* Cos[2*n*\[zeta]],1 + (q)^(2*n)], {n, 1, Infinity}]	Failure	Failure	Skip	Error
22.11.E7	${\displaystyle{\displaystyle\operatorname{ns}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta=\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}}}}$	JacobiNS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta)=(2*Pi)/(EllipticK(k))*sum(((q)^(2*n + 1)* sin((2*n + 1)* zeta))/(1 - (q)^(2*n + 1)), n = 0..infinity)	JacobiNS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Csc[\[zeta]]=Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(q)^(2*n + 1)* Sin[(2*n + 1)* \[zeta]],1 - (q)^(2*n + 1)], {n, 0, Infinity}]	Failure	Failure	Skip	Error
22.11.E8	${\displaystyle{\displaystyle\operatorname{ds}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta=-\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}}}}$	JacobiDS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta)= -(2*Pi)/(EllipticK(k))*sum(((q)^(2*n + 1)* sin((2*n + 1)* zeta))/(1 + (q)^(2*n + 1)), n = 0..infinity)	JacobiDS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Csc[\[zeta]]= -Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(q)^(2*n + 1)* Sin[(2*n + 1)* \[zeta]],1 + (q)^(2*n + 1)], {n, 0, Infinity}]	Failure	Failure	Skip	Error
22.11.E9	${\displaystyle{\displaystyle\operatorname{cs}\left(z,k\right)-\frac{\pi}{2K}\cot\zeta=-\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{2n}\sin\left(2n\zeta\right)}{1+q^{2n}}}}$	JacobiCS(z, k)-(Pi)/(2*EllipticK(k))*cot(zeta)= -(2*Pi)/(EllipticK(k))*sum(((q)^(2*n)* sin(2*n*zeta))/(1 + (q)^(2*n)), n = 1..infinity)	JacobiCS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Cot[\[zeta]]= -Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(q)^(2*n)* Sin[2*n*\[zeta]],1 + (q)^(2*n)], {n, 1, Infinity}]	Failure	Failure	Skip	Error
22.11.E10	${\displaystyle{\displaystyle\operatorname{dc}\left(z,k\right)-\frac{\pi}{2K}\sec\zeta=\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}}}}$	JacobiDC(z, k)-(Pi)/(2*EllipticK(k))*sec(zeta)=(2*Pi)/(EllipticK(k))*sum(((- 1)^(n)* (q)^(2*n + 1)* cos((2*n + 1)* zeta))/(1 - (q)^(2*n + 1)), n = 0..infinity)	JacobiDC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Sec[\[zeta]]=Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(- 1)^(n)* (q)^(2*n + 1)* Cos[(2*n + 1)* \[zeta]],1 - (q)^(2*n + 1)], {n, 0, Infinity}]	Failure	Failure	Skip	Error
22.11.E11	${\displaystyle{\displaystyle\operatorname{nc}\left(z,k\right)-\frac{\pi}{2Kk^{\prime}}\sec\zeta=-\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}}}}$	JacobiNC(z, k)-(Pi)/(2*K*sqrt(1 - (k)^(2)))*sec(zeta)= -(2*Pi)/(K*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)* (q)^(2*n + 1)* cos((2*n + 1)* zeta))/(1 + (q)^(2*n + 1)), n = 0..infinity)	JacobiNC[z, (k)^2]-Divide[Pi,2*K*Sqrt[1 - (k)^(2)]]*Sec[\[zeta]]= -Divide[2*Pi,K*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)* (q)^(2*n + 1)* Cos[(2*n + 1)* \[zeta]],1 + (q)^(2*n + 1)], {n, 0, Infinity}]	Failure	Failure	Skip	Error
22.11.E12	${\displaystyle{\displaystyle\operatorname{sc}\left(z,k\right)-\frac{\pi}{2Kk^{\prime}}\tan\zeta=\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin\left(2n\zeta\right)}{1+q^{2n}}}}$	JacobiSC(z, k)-(Pi)/(2*K*sqrt(1 - (k)^(2)))*tan(zeta)=(2*Pi)/(K*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)* (q)^(2*n)* sin(2*n*zeta))/(1 + (q)^(2*n)), n = 1..infinity)	JacobiSC[z, (k)^2]-Divide[Pi,2*K*Sqrt[1 - (k)^(2)]]*Tan[\[zeta]]=Divide[2*Pi,K*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)* (q)^(2*n)* Sin[2*n*\[zeta]],1 + (q)^(2*n)], {n, 1, Infinity}]	Failure	Failure	Skip	Error
22.11.E13	${\displaystyle{\displaystyle{\operatorname{sn}^{2}}\left(z,k\right)=\frac{1}{k^{2}}\left(1-\frac{E}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}\cos\left(2n\zeta\right)}}$	(JacobiSN(z, k))^(2)=(1)/((k)^(2))*(1 -(EllipticE(k))/(EllipticK(k)))-(2*(Pi)^(2))/((k)^(2)* (EllipticK(k))^(2))*sum((n*(q)^(n))/(1 - (q)^(2*n))*cos(2*n*zeta), n = 1..infinity)	(JacobiSN[z, (k)^2])^(2)=Divide[1,(k)^(2)]*(1 -Divide[EllipticE[(k)^2],EllipticK[(k)^2]])-Divide[2*(Pi)^(2),(k)^(2)* (EllipticK[(k)^2])^(2)]*Sum[Divide[n*(q)^(n),1 - (q)^(2*n)]*Cos[2*n*\[zeta]], {n, 1, Infinity}]	Failure	Failure	Skip	Error
22.11.E14	${\displaystyle{\displaystyle k^{2}{\operatorname{sn}^{2}}\left(z,k\right)=\frac{{E^{\prime}}}{{K^{\prime}}}-\left(\frac{\pi}{2\!{K^{\prime}}}\right)^{2}\sum_{n=-\infty}^{\infty}\left({\operatorname{sech}^{2}}\left(\frac{\pi}{2\!{K^{\prime}}}(z-2n\!K)\right)\right)}}$	(k)^(2)* (JacobiSN(z, k))^(2)=(EllipticCE(k))/(EllipticCK(k))-((Pi)/(2*EllipticCK($0)))^(2)* sum((sech((Pi)/(2*EllipticCK(k))*(z - 2*n*EllipticK(k))))^(2), n = - infinity..infinity)	(k)^(2)* (JacobiSN[z, (k)^2])^(2)=Divide[EllipticE[1-(k)^2],EllipticK[1-(k)^2]]-(Divide[Pi,2*EllipticK[1-($0)^2]])^(2)* Sum[(Sech[Divide[Pi,2*EllipticK[1-(k)^2]]*(z - 2*n*EllipticK[(k)^2])])^(2), {n, - Infinity, Infinity}]	Error	Failure	-	Error
22.12.E1	${\displaystyle{\displaystyle\tau=i{K^{\prime}}\left(k\right)/K\left(k\right)}}$	tau = I*EllipticCK(k)/ EllipticK(k)	\[Tau]= 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
22.12.E2	${\displaystyle{\displaystyle 2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}}}$	2*K*k*JacobiSN(2*K*t, k)= sum((Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity)	2*K*k*JacobiSN[2*K*t, (k)^2]= Sum[Divide[Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}]	Failure	Failure	Skip	Error
22.12.E2	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right)}}$	sum((Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity)= sum(sum(((- 1)^(m))/(t - m -(n +(1)/(2))* tau), m = - infinity..infinity), n = - infinity..infinity)	Sum[Divide[Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}]= Sum[Sum[Divide[(- 1)^(m),t - m -(n +Divide[1,2])* \[Tau]], {m, - Infinity, Infinity}], {n, - Infinity, Infinity}]	Error	Failure	-	Error
22.12.E3	${\displaystyle{\displaystyle 2iKk\operatorname{cn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}}}$	2*I*K*k*JacobiCN(2*K*t, k)= sum(((- 1)^(n)* Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity)	2*I*K*k*JacobiCN[2*K*t, (k)^2]= Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}]	Failure	Failure	Skip	Skip
22.12.E3	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-(n+\frac{1}{2})\tau}\right)}}$	sum(((- 1)^(n)* Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity)= sum(sum(((- 1)^(m + n))/(t - m -(n +(1)/(2))* tau), m = - infinity..infinity), n = - infinity..infinity)	Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}]= Sum[Sum[Divide[(- 1)^(m + n),t - m -(n +Divide[1,2])* \[Tau]], {m, - Infinity, Infinity}], {n, - Infinity, Infinity}]	Error	Failure	-	Error
22.12.E8	${\displaystyle{\displaystyle 2K\operatorname{dc}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)\right)}}}$	2*EllipticK(k)*JacobiDC(2*K*t, k)= sum((Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity)	2*EllipticK[(k)^2]*JacobiDC[2*K*t, (k)^2]= Sum[Divide[Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}]	Failure	Failure	Skip	Error
22.12.E8	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right)}}$	sum((Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity)= sum(sum(((- 1)^(m))/(t +(1)/(2)- m - n*tau), m = - infinity..infinity), n = - infinity..infinity)	Sum[Divide[Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}]= Sum[Sum[Divide[(- 1)^(m),t +Divide[1,2]- m - n*\[Tau]], {m, - Infinity, Infinity}], {n, - Infinity, Infinity}]	Error	Failure	-	Error
22.12.E9	${\displaystyle{\displaystyle 2Kk^{\prime}\operatorname{nc}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)\right)}}}$	2*K*sqrt(1 - (k)^(2))*JacobiNC(2*K*t, k)= sum(((- 1)^(n)* Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity)	2*K*Sqrt[1 - (k)^(2)]*JacobiNC[2*K*t, (k)^2]= Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}]	Failure	Failure	Skip	Skip
22.12.E9	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-n\tau}\right)}}$	sum(((- 1)^(n)* Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity)= sum(sum(((- 1)^(m + n))/(t +(1)/(2)- m - n*tau), m = - infinity..infinity), n = - infinity..infinity)	Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}]= Sum[Sum[Divide[(- 1)^(m + n),t +Divide[1,2]- m - n*\[Tau]], {m, - Infinity, Infinity}], {n, - Infinity, Infinity}]	Error	Failure	-	Error
22.12.E11	${\displaystyle{\displaystyle 2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-n\tau)\right)}}}$	2*EllipticK(k)*JacobiNS(2*K*t, k)= sum((Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity)	2*EllipticK[(k)^2]*JacobiNS[2*K*t, (k)^2]= Sum[Divide[Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}]	Failure	Failure	Skip	Skip
22.12.E11	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-n\tau}\right)}}$	sum((Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity)= sum(sum(((- 1)^(m))/(t - m - n*tau), m = - infinity..infinity), n = - infinity..infinity)	Sum[Divide[Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}]= Sum[Sum[Divide[(- 1)^(m),t - m - n*\[Tau]], {m, - Infinity, Infinity}], {n, - Infinity, Infinity}]	Error	Failure	-	Error
22.12.E12	${\displaystyle{\displaystyle 2K\operatorname{ds}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-n\tau)\right)}}}$	2*EllipticK(k)*JacobiDS(2*K*t, k)= sum(((- 1)^(n)* Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity)	2*EllipticK[(k)^2]*JacobiDS[2*K*t, (k)^2]= Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}]	Failure	Failure	Skip	Skip
22.12.E12	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-n\tau}\right)}}$	sum(((- 1)^(n)* Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity)= sum(sum(((- 1)^(m + n))/(t - m - n*tau), m = - infinity..infinity), n = - infinity..infinity)	Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}]= Sum[Sum[Divide[(- 1)^(m + n),t - m - n*\[Tau]], {m, - Infinity, Infinity}], {n, - Infinity, Infinity}]	Error	Failure	-	Error
22.13.E1	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k\right)\right)^{2}=\left(1-{\operatorname{sn}^{2}}\left(z,k\right)\right)\left(1-k^{2}{\operatorname{sn}^{2}}\left(z,k\right)\right)}}$	(diff(JacobiSN(z, k), z))^(2)=(1 - (JacobiSN(z, k))^(2))*(1 - (k)^(2)* (JacobiSN(z, k))^(2))	(D[JacobiSN[z, (k)^2], z])^(2)=(1 - (JacobiSN[z, (k)^2])^(2))*(1 - (k)^(2)* (JacobiSN[z, (k)^2])^(2))	Successful	Successful	-	-
22.13.E2	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{2}={\left(1-{\operatorname{cn}^{2}}\left(z,k\right)\right)}{\left({k^{\prime}}^{2}+k^{2}{\operatorname{cn}^{2}}\left(z,k\right)\right)}}}$	(diff(JacobiCN(z, k), z))^(2)=(1 - (JacobiCN(z, k))^(2))*(1 - (k)^(2)+ (k)^(2)* (JacobiCN(z, k))^(2))	(D[JacobiCN[z, (k)^2], z])^(2)=(1 - (JacobiCN[z, (k)^2])^(2))*(1 - (k)^(2)+ (k)^(2)* (JacobiCN[z, (k)^2])^(2))	Successful	Successful	-	-
22.13.E3	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dn}\left(z,k\right)\right)^{2}=\left(1-{\operatorname{dn}^{2}}\left(z,k\right)\right)\left({\operatorname{dn}^{2}}\left(z,k\right)-{k^{\prime}}^{2}\right)}}$	(diff(JacobiDN(z, k), z))^(2)=(1 - (JacobiDN(z, k))^(2))*((JacobiDN(z, k))^(2)- 1 - (k)^(2))	(D[JacobiDN[z, (k)^2], z])^(2)=(1 - (JacobiDN[z, (k)^2])^(2))*((JacobiDN[z, (k)^2])^(2)- 1 - (k)^(2))	Failure	Failure	Fail 2.498180497+.1828376586*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 3.98469228+40.70649515*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} 18.78836459-18.29576381*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 2.498180497-.1828376586*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[2.4981804946867654, 0.18283765783578443] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.9846921844891163, 40.70649511448791] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[18.78836461150559, -18.29576374475269] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.4981804946867654, -0.18283765783578443] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E4	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cd}\left(z,k\right)\right)^{2}=\left(1-{\operatorname{cd}^{2}}\left(z,k\right)\right)\left(1-k^{2}{\operatorname{cd}^{2}}\left(z,k\right)\right)}}$	(diff(JacobiCD(z, k), z))^(2)=(1 - (JacobiCD(z, k))^(2))*(1 - (k)^(2)* (JacobiCD(z, k))^(2))	(D[JacobiCD[z, (k)^2], z])^(2)=(1 - (JacobiCD[z, (k)^2])^(2))*(1 - (k)^(2)* (JacobiCD[z, (k)^2])^(2))	Successful	Successful	-	-
22.13.E5	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sd}\left(z,k\right)\right)^{2}={\left(1-{k^{\prime}}^{2}{\operatorname{sd}^{2}}\left(z,k\right)\right)}{\left(1+k^{2}{\operatorname{sd}^{2}}\left(z,k\right)\right)}}}$	(diff(JacobiSD(z, k), z))^(2)=(1 - 1 - (k)^(2)* (JacobiSD(z, k))^(2))*(1 + (k)^(2)* (JacobiSD(z, k))^(2))	(D[JacobiSD[z, (k)^2], z])^(2)=(1 - 1 - (k)^(2)* (JacobiSD[z, (k)^2])^(2))*(1 + (k)^(2)* (JacobiSD[z, (k)^2])^(2))	Failure	Failure	Fail 10.83165880-9.188310853*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -.8004902522e-1-.1328975640*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -1.780546172+1.029879100*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 10.83165880+9.188310853*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[10.831658854502608, -9.188310851111659] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.08004902551881446, -0.13289756372332284] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.7805461824378226, 1.0298791081722538] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[10.831658854502608, 9.188310851111659] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E6	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nd}\left(z,k\right)\right)^{2}=\left({\operatorname{nd}^{2}}\left(z,k\right)-1\right)\left(1-{k^{\prime}}^{2}{\operatorname{nd}^{2}}\left(z,k\right)\right)}}$	(diff(JacobiND(z, k), z))^(2)=((JacobiND(z, k))^(2)- 1)*(1 - 1 - (k)^(2)* (JacobiND(z, k))^(2))	(D[JacobiND[z, (k)^2], z])^(2)=((JacobiND[z, (k)^2])^(2)- 1)*(1 - 1 - (k)^(2)* (JacobiND[z, (k)^2])^(2))	Failure	Failure	Fail 9.831658819-9.188310855*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -1.377792776-1.115495393*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -16.68639667+17.12499953*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 9.831658819+9.188310855*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[9.831658854502614, -9.188310851111664] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.3777927793137241, -1.115495392019118] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-16.686396759730812, 17.124999628495146] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[9.831658854502614, 9.188310851111664] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E7	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{2}=\left({\operatorname{dc}^{2}}\left(z,k\right)-1\right)\left({\operatorname{dc}^{2}}\left(z,k\right)-k^{2}\right)}}$	(diff(JacobiDC(z, k), z))^(2)=((JacobiDC(z, k))^(2)- 1)*((JacobiDC(z, k))^(2)- (k)^(2))	(D[JacobiDC[z, (k)^2], z])^(2)=((JacobiDC[z, (k)^2])^(2)- 1)*((JacobiDC[z, (k)^2])^(2)- (k)^(2))	Successful	Successful	-	-
22.13.E8	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nc}\left(z,k\right)\right)^{2}={\left(k^{2}+{k^{\prime}}^{2}{\operatorname{nc}^{2}}\left(z,k\right)\right)}{\left({\operatorname{nc}^{2}}\left(z,k\right)-1\right)}}}$	(diff(JacobiNC(z, k), z))^(2)=((k)^(2)+ 1 - (k)^(2)* (JacobiNC(z, k))^(2))*((JacobiNC(z, k))^(2)- 1)	(D[JacobiNC[z, (k)^2], z])^(2)=((k)^(2)+ 1 - (k)^(2)* (JacobiNC[z, (k)^2])^(2))*((JacobiNC[z, (k)^2])^(2)- 1)	Failure	Failure	Fail 18.90774921-11.78531296*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} .1159583621-.675201614*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -.74436492e-2-.321433941e-1*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 18.90774921+11.78531296*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[18.907749256401388, -11.78531296082075] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.11595836325807751, -0.6752016132739329] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.007443648638819578, -0.032143394714381934] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[18.907749256401388, 11.78531296082075] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E9	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sc}\left(z,k\right)\right)^{2}=\left(1+{\operatorname{sc}^{2}}\left(z,k\right)\right)\left(1+{k^{\prime}}^{2}{\operatorname{sc}^{2}}\left(z,k\right)\right)}}$	(diff(JacobiSC(z, k), z))^(2)=(1 + (JacobiSC(z, k))^(2))*(1 + 1 - (k)^(2)* (JacobiSC(z, k))^(2))	(D[JacobiSC[z, (k)^2], z])^(2)=(1 + (JacobiSC[z, (k)^2])^(2))*(1 + 1 - (k)^(2)* (JacobiSC[z, (k)^2])^(2))	Failure	Failure	Fail 17.90774919-11.78531296*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -.884041635-.6752016142*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -1.007443648-.32143393e-1*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 17.90774919+11.78531296*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[17.90774925640138, -11.785312960820747] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.8840416367419195, -0.6752016132739348] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.0074436486388192, -0.032143394714381435] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[17.90774925640138, 11.785312960820747] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E10	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ns}\left(z,k\right)\right)^{2}=\left({\operatorname{ns}^{2}}\left(z,k\right)-k^{2}\right)\left({\operatorname{ns}^{2}}\left(z,k\right)-1\right)}}$	(diff(JacobiNS(z, k), z))^(2)=((JacobiNS(z, k))^(2)- (k)^(2))*((JacobiNS(z, k))^(2)- 1)	(D[JacobiNS[z, (k)^2], z])^(2)=((JacobiNS[z, (k)^2])^(2)- (k)^(2))*((JacobiNS[z, (k)^2])^(2)- 1)	Successful	Successful	-	-
22.13.E11	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ds}\left(z,k\right)\right)^{2}=\left({\operatorname{ds}^{2}}\left(z,k\right)-{k^{\prime}}^{2}\right)\left(k^{2}+{\operatorname{ds}^{2}}\left(z,k\right)\right)}}$	(diff(JacobiDS(z, k), z))^(2)=((JacobiDS(z, k))^(2)- 1 - (k)^(2))*((k)^(2)+ (JacobiDS(z, k))^(2))	(D[JacobiDS[z, (k)^2], z])^(2)=((JacobiDS[z, (k)^2])^(2)- 1 - (k)^(2))*((k)^(2)+ (JacobiDS[z, (k)^2])^(2))	Failure	Failure	Fail 1.592634334-.1165622473*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} .6097694629-6.229233331*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} 79.66246105+77.57383910*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 1.592634334+.1165622473*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[1.5926343353666734, -0.11656224691795349] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.6097694563192116, -6.229233337330877] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[79.66246130818148, 77.57383899860325] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.5926343353666734, 0.11656224691795349] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E12	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cs}\left(z,k\right)\right)^{2}=\left(1+{\operatorname{cs}^{2}}\left(z,k\right)\right)\left({k^{\prime}}^{2}+{\operatorname{cs}^{2}}\left(z,k\right)\right)}}$	(diff(JacobiCS(z, k), z))^(2)=(1 + (JacobiCS(z, k))^(2))*(1 - (k)^(2)+ (JacobiCS(z, k))^(2))	(D[JacobiCS[z, (k)^2], z])^(2)=(1 + (JacobiCS[z, (k)^2])^(2))*(1 - (k)^(2)+ (JacobiCS[z, (k)^2])^(2))	Successful	Successful	-	-
22.13.E13	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sn}\left(z,k\right)=-(1+k^{2})\operatorname{sn}\left(z,k\right)+2k^{2}{\operatorname{sn}^{3}}\left(z,k\right)}}$	diff(JacobiSN(z, k), [z$(2)])= -(1 + (k)^(2))* JacobiSN(z, k)+ 2*(k)^(2)* (JacobiSN(z, k))^(3)	D[JacobiSN[z, (k)^2], {z, 2}]= -(1 + (k)^(2))* JacobiSN[z, (k)^2]+ 2*(k)^(2)* (JacobiSN[z, (k)^2])^(3)	Successful	Successful	-	-
22.13.E14	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cn}\left(z,k\right)=-({k^{\prime}}^{2}-k^{2})\operatorname{cn}\left(z,k\right)-2k^{2}{\operatorname{cn}^{3}}\left(z,k\right)}}$	diff(JacobiCN(z, k), [z$(2)])= -(1 - (k)^(2)- (k)^(2))* JacobiCN(z, k)- 2*(k)^(2)* (JacobiCN(z, k))^(3)	D[JacobiCN[z, (k)^2], {z, 2}]= -(1 - (k)^(2)- (k)^(2))* JacobiCN[z, (k)^2]- 2*(k)^(2)* (JacobiCN[z, (k)^2])^(3)	Successful	Successful	-	-
22.13.E15	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dn}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{dn}\left(z,k\right)-2{\operatorname{dn}^{3}}\left(z,k\right)}}$	diff(JacobiDN(z, k), [z$(2)])=(1 + 1 - (k)^(2))* JacobiDN(z, k)- 2*(JacobiDN(z, k))^(3)	D[JacobiDN[z, (k)^2], {z, 2}]=(1 + 1 - (k)^(2))* JacobiDN[z, (k)^2]- 2*(JacobiDN[z, (k)^2])^(3)	Successful	Successful	-	-
22.13.E16	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cd}\left(z,k\right)=-(1+k^{2})\operatorname{cd}\left(z,k\right)+2k^{2}{\operatorname{cd}^{3}}\left(z,k\right)}}$	diff(JacobiCD(z, k), [z$(2)])= -(1 + (k)^(2))* JacobiCD(z, k)+ 2*(k)^(2)* (JacobiCD(z, k))^(3)	D[JacobiCD[z, (k)^2], {z, 2}]= -(1 + (k)^(2))* JacobiCD[z, (k)^2]+ 2*(k)^(2)* (JacobiCD[z, (k)^2])^(3)	Successful	Successful	-	-
22.13.E17	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sd}\left(z,k\right)=(k^{2}-{k^{\prime}}^{2})\operatorname{sd}\left(z,k\right)-2k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{3}}\left(z,k\right)}}$	diff(JacobiSD(z, k), [z$(2)])=((k)^(2)- 1 - (k)^(2))* JacobiSD(z, k)- 2*(k)^(2)* 1 - (k)^(2)* (JacobiSD(z, k))^(3)	D[JacobiSD[z, (k)^2], {z, 2}]=((k)^(2)- 1 - (k)^(2))* JacobiSD[z, (k)^2]- 2*(k)^(2)* 1 - (k)^(2)* (JacobiSD[z, (k)^2])^(3)	Failure	Failure	Fail -1.559655423-5.068856850*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 7.377533436+.6310733818*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} 24.14827883+2.560535027*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -1.559655423+5.068856850*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-1.5596554204154063, -5.068856868430466] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[7.377533435916555, 0.631073384002977] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[24.148278865809726, 2.560535034009612] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.5596554204154063, 5.068856868430466] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E18	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{nd}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{nd}\left(z,k\right)-2{k^{\prime}}^{2}{\operatorname{nd}^{3}}\left(z,k\right)}}$	diff(JacobiND(z, k), [z$(2)])=(1 + 1 - (k)^(2))* JacobiND(z, k)- 21 - (k)^(2)* (JacobiND(z, k))^(3)	D[JacobiND[z, (k)^2], {z, 2}]=(1 + 1 - (k)^(2))* JacobiND[z, (k)^2]- 21 - (k)^(2)* (JacobiND[z, (k)^2])^(3)	Failure	Failure	Fail 17.31627209-6.321528171*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 20.48491028+.6948396114*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} 2.697171893-16.07884790*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} 17.31627209+6.321528171*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[17.316272094007225, -6.32152818402195] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[20.48491028791516, 0.6948396156603991] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.6971717784427156, -16.07884795469416] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[17.316272094007225, 6.32152818402195] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E19	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dc}\left(z,k\right)=-(1+k^{2})\operatorname{dc}\left(z,k\right)+2{\operatorname{dc}^{3}}\left(z,k\right)}}$	diff(JacobiDC(z, k), [z$(2)])= -(1 + (k)^(2))* JacobiDC(z, k)+ 2*(JacobiDC(z, k))^(3)	D[JacobiDC[z, (k)^2], {z, 2}]= -(1 + (k)^(2))* JacobiDC[z, (k)^2]+ 2*(JacobiDC[z, (k)^2])^(3)	Successful	Successful	-	-
22.13.E20	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{nc}\left(z,k\right)=(k^{2}-{k^{\prime}}^{2})\operatorname{nc}\left(z,k\right)+2{k^{\prime}}^{2}{\operatorname{nc}^{3}}\left(z,k\right)}}$	diff(JacobiNC(z, k), [z$(2)])=((k)^(2)- 1 - (k)^(2))* JacobiNC(z, k)+ 21 - (k)^(2)* (JacobiNC(z, k))^(3)	D[JacobiNC[z, (k)^2], {z, 2}]=((k)^(2)- 1 - (k)^(2))* JacobiNC[z, (k)^2]+ 21 - (k)^(2)* (JacobiNC[z, (k)^2])^(3)	Failure	Failure	Fail -24.00437993-2.498741953*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -26.57479322-1.960795880*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -31.85910661-.365650073*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -24.00437993+2.498741953*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-24.004379922603327, -2.4987419636935297] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-26.574793224361827, -1.9607958726774548] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-31.85910661323699, -0.36565007186275755] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-24.004379922603327, 2.4987419636935297] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E21	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sc}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{sc}\left(z,k\right)+2{k^{\prime}}^{2}{\operatorname{sc}^{3}}\left(z,k\right)}}$	diff(JacobiSC(z, k), [z$(2)])=(1 + 1 - (k)^(2))* JacobiSC(z, k)+ 21 - (k)^(2)* (JacobiSC(z, k))^(3)	D[JacobiSC[z, (k)^2], {z, 2}]=(1 + 1 - (k)^(2))* JacobiSC[z, (k)^2]+ 21 - (k)^(2)* (JacobiSC[z, (k)^2])^(3)	Failure	Failure	Fail -25.16317836-9.371927930*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} -22.30678436-.748664075*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -21.11933094+.5286047673*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} -25.16317836+9.371927930*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-25.16317836015093, -9.371927951109038] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-22.30678435695311, -0.7486640736526304] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-21.119330946363334, 0.5286047645456822] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-25.16317836015093, 9.371927951109038] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E22	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ns}\left(z,k\right)=-(1+k^{2})\operatorname{ns}\left(z,k\right)+2{\operatorname{ns}^{3}}\left(z,k\right)}}$	diff(JacobiNS(z, k), [z$(2)])= -(1 + (k)^(2))* JacobiNS(z, k)+ 2*(JacobiNS(z, k))^(3)	D[JacobiNS[z, (k)^2], {z, 2}]= -(1 + (k)^(2))* JacobiNS[z, (k)^2]+ 2*(JacobiNS[z, (k)^2])^(3)	Successful	Successful	-	-
22.13.E23	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ds}\left(z,k\right)=(k^{2}-{k^{\prime}}^{2})\operatorname{ds}\left(z,k\right)+2{\operatorname{ds}^{3}}\left(z,k\right)}}$	diff(JacobiDS(z, k), [z$(2)])=((k)^(2)- 1 - (k)^(2))* JacobiDS(z, k)+ 2*(JacobiDS(z, k))^(3)	D[JacobiDS[z, (k)^2], {z, 2}]=((k)^(2)- 1 - (k)^(2))* JacobiDS[z, (k)^2]+ 2*(JacobiDS[z, (k)^2])^(3)	Failure	Failure	Fail .1278605556-.9116357015*I <- {z = 2^(1/2)+I*2^(1/2), k = 1} 1.564751601-15.92389060*I <- {z = 2^(1/2)+I*2^(1/2), k = 2} -16.64582166-41.94233041*I <- {z = 2^(1/2)+I*2^(1/2), k = 3} .1278605556+.9116357015*I <- {z = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.1278605552889328, -0.9116357007409521] <- {Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.5647516031657338, -15.92389060277217] <- {Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-16.645821659513462, -41.94233034981555] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.1278605552889328, 0.9116357007409521] <- {Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.13.E24	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cs}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{cs}\left(z,k\right)+2{\operatorname{cs}^{3}}\left(z,k\right)}}$	diff(JacobiCS(z, k), [z$(2)])=(1 + 1 - (k)^(2))* JacobiCS(z, k)+ 2*(JacobiCS(z, k))^(3)	D[JacobiCS[z, (k)^2], {z, 2}]=(1 + 1 - (k)^(2))* JacobiCS[z, (k)^2]+ 2*(JacobiCS[z, (k)^2])^(3)	Successful	Successful	-	-
22.14.E1	${\displaystyle{\displaystyle\int\operatorname{sn}\left(x,k\right)\mathrm{d}x=k^{-1}\ln\left(\operatorname{dn}\left(x,k\right)-k\operatorname{cn}\left(x,k\right)\right)}}$	int(JacobiSN(x, k), x)= (k)^(- 1)* ln(JacobiDN(x, k)- k*JacobiCN(x, k))	Integrate[JacobiSN[x, (k)^2], x]= (k)^(- 1)* Log[JacobiDN[x, (k)^2]- k*JacobiCN[x, (k)^2]]	Successful	Failure	-	Successful
22.14.E2	${\displaystyle{\displaystyle\int\operatorname{cn}\left(x,k\right)\mathrm{d}x=k^{-1}\operatorname{Arccos}\left(\operatorname{dn}\left(x,k\right)\right)}}$	Error	Integrate[JacobiCN[x, (k)^2], x]= (k)^(- 1)* Integrate[Divide[1, (1-t^2)^(1/2)], {t, JacobiDN[x, (k)^2], 1}]	Error	Failure	-	Skip
22.14.E3	${\displaystyle{\displaystyle\int\operatorname{dn}\left(x,k\right)\mathrm{d}x=\operatorname{Arcsin}\left(\operatorname{sn}\left(x,k\right)\right)}}$	Error	Integrate[JacobiDN[x, (k)^2], x]= Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, JacobiSN[x, (k)^2]}]	Error	Failure	-	Successful
22.14.E4	${\displaystyle{\displaystyle\int\operatorname{cd}\left(x,k\right)\mathrm{d}x=k^{-1}\ln\left(\operatorname{nd}\left(x,k\right)+k\operatorname{sd}\left(x,k\right)\right)}}$	int(JacobiCD(x, k), x)= (k)^(- 1)* ln(JacobiND(x, k)+ k*JacobiSD(x, k))	Integrate[JacobiCD[x, (k)^2], x]= (k)^(- 1)* Log[JacobiND[x, (k)^2]+ k*JacobiSD[x, (k)^2]]	Successful	Failure	-	Successful
22.14.E5	${\displaystyle{\displaystyle\int\operatorname{sd}\left(x,k\right)\mathrm{d}x=(kk^{\prime})^{-1}\operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right)}}$	Error	Integrate[JacobiSD[x, (k)^2], x]=(k*Sqrt[1 - (k)^(2)])^(- 1)* Integrate[Divide[1, (1-t^2)^(1/2)], {t, 0, - k*JacobiCD[x, (k)^2]}]	Error	Failure	-	Skip
22.14.E6	${\displaystyle{\displaystyle\int\operatorname{nd}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)\right)}}$	Error	Integrate[JacobiND[x, (k)^2], x]=(Sqrt[1 - (k)^(2)])^(- 1)* Integrate[Divide[1, (1-t^2)^(1/2)], {t, JacobiCD[x, (k)^2], 1}]	Error	Failure	-	Skip
22.14.E7	${\displaystyle{\displaystyle\int\operatorname{dc}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{nc}\left(x,k\right)+\operatorname{sc}\left(x,k\right)\right)}}$	int(JacobiDC(x, k), x)= ln(JacobiNC(x, k)+ JacobiSC(x, k))	Integrate[JacobiDC[x, (k)^2], x]= Log[JacobiNC[x, (k)^2]+ JacobiSC[x, (k)^2]]	Successful	Successful	-	-
22.14.E8	${\displaystyle{\displaystyle\int\operatorname{nc}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\ln\left(\operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{sc}\left(x,k\right)\right)}}$	int(JacobiNC(x, k), x)=(sqrt(1 - (k)^(2)))^(- 1)* ln(JacobiDC(x, k)+sqrt(1 - (k)^(2))*JacobiSC(x, k))	Integrate[JacobiNC[x, (k)^2], x]=(Sqrt[1 - (k)^(2)])^(- 1)* Log[JacobiDC[x, (k)^2]+Sqrt[1 - (k)^(2)]*JacobiSC[x, (k)^2]]	Successful	Successful	-	-
22.14.E9	${\displaystyle{\displaystyle\int\operatorname{sc}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\ln\left(\operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{nc}\left(x,k\right)\right)}}$	int(JacobiSC(x, k), x)=(sqrt(1 - (k)^(2)))^(- 1)* ln(JacobiDC(x, k)+sqrt(1 - (k)^(2))*JacobiNC(x, k))	Integrate[JacobiSC[x, (k)^2], x]=(Sqrt[1 - (k)^(2)])^(- 1)* Log[JacobiDC[x, (k)^2]+Sqrt[1 - (k)^(2)]*JacobiNC[x, (k)^2]]	Successful	Successful	-	-
22.14.E10	${\displaystyle{\displaystyle\int\operatorname{ns}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{ds}\left(x,k\right)-\operatorname{cs}\left(x,k\right)\right)}}$	int(JacobiNS(x, k), x)= ln(JacobiDS(x, k)- JacobiCS(x, k))	Integrate[JacobiNS[x, (k)^2], x]= Log[JacobiDS[x, (k)^2]- JacobiCS[x, (k)^2]]	Successful	Successful	-	-
22.14.E11	${\displaystyle{\displaystyle\int\operatorname{ds}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{ns}\left(x,k\right)-\operatorname{cs}\left(x,k\right)\right)}}$	int(JacobiDS(x, k), x)= ln(JacobiNS(x, k)- JacobiCS(x, k))	Integrate[JacobiDS[x, (k)^2], x]= Log[JacobiNS[x, (k)^2]- JacobiCS[x, (k)^2]]	Successful	Successful	-	-
22.14.E12	${\displaystyle{\displaystyle\int\operatorname{cs}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{ns}\left(x,k\right)-\operatorname{ds}\left(x,k\right)\right)}}$	int(JacobiCS(x, k), x)= ln(JacobiNS(x, k)- JacobiDS(x, k))	Integrate[JacobiCS[x, (k)^2], x]= Log[JacobiNS[x, (k)^2]- JacobiDS[x, (k)^2]]	Successful	Successful	-	-
22.14.E13	${\displaystyle{\displaystyle\int\frac{\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\ln\left(\frac{\operatorname{sn}\left(x,k\right)}{\operatorname{cn}\left(x,k\right)+\operatorname{dn}\left(x,k\right)}\right)}}$	int((1)/(JacobiSN(x, k)), x)= ln((JacobiSN(x, k))/(JacobiCN(x, k)+ JacobiDN(x, k)))	Integrate[Divide[1,JacobiSN[x, (k)^2]], x]= Log[Divide[JacobiSN[x, (k)^2],JacobiCN[x, (k)^2]+ JacobiDN[x, (k)^2]]]	Successful	Successful	-	-
22.14.E14	${\displaystyle{\displaystyle\int\frac{\operatorname{cn}\left(x,k\right)\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\frac{1}{2}\ln\left(\frac{1-\operatorname{dn}\left(x,k\right)}{1+\operatorname{dn}\left(x,k\right)}\right)}}$	int((JacobiCN(x, k))/(JacobiSN(x, k)), x)=(1)/(2)*ln((1 - JacobiDN(x, k))/(1 + JacobiDN(x, k)))	Integrate[Divide[JacobiCN[x, (k)^2],JacobiSN[x, (k)^2]], x]=Divide[1,2]*Log[Divide[1 - JacobiDN[x, (k)^2],1 + JacobiDN[x, (k)^2]]]	Failure	Failure	Skip	Fail Complex[0.6931471805599447, 1.0816575901446905*^-16] <- {Rule[k, 2], Rule[x, 1]} Complex[0.6931471805599456, -3.1415926535897927] <- {Rule[k, 2], Rule[x, 2]} Complex[0.6931471805599457, -3.141592653589793] <- {Rule[k, 2], Rule[x, 3]} Complex[1.098612288668117, 4.781590265396833*^-15] <- {Rule[k, 3], Rule[x, 1]} ... skip entries to safe data
22.14.E15	${\displaystyle{\displaystyle\int\frac{\operatorname{cn}\left(x,k\right)\mathrm{d}x}{{\operatorname{sn}^{2}}\left(x,k\right)}=-\frac{\operatorname{dn}\left(x,k\right)}{\operatorname{sn}\left(x,k\right)}}}$	int((JacobiCN(x, k))/((JacobiSN(x, k))^(2)), x)= -(JacobiDN(x, k))/(JacobiSN(x, k))	Integrate[Divide[JacobiCN[x, (k)^2],(JacobiSN[x, (k)^2])^(2)], x]= -Divide[JacobiDN[x, (k)^2],JacobiSN[x, (k)^2]]	Successful	Successful	-	-
22.14.E16	${\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{sn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{1}{4}\!{K^{\prime}}\left(k\right)-\tfrac{1}{2}\!K\left(k\right)\ln k}}$	int(ln(JacobiSN(t, k)), t = 0..EllipticK(k))= -(1)/(4)*EllipticCK(k)-(1)/(2)*EllipticK(k)*ln(k)	Integrate[Log[JacobiSN[t, (k)^2]], {t, 0, EllipticK[(k)^2]}]= -Divide[1,4]*EllipticK[1-(k)^2]-Divide[1,2]*EllipticK[(k)^2]*Log[k]	Failure	Failure	Skip	Successful
22.14.E17	${\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{1}{4}\!{K^{\prime}}\left(k\right)+\tfrac{1}{2}\!K\left(k\right)\ln\left(k^{\prime}/k\right)}}$	int(ln(JacobiCN(t, k)), t = 0..EllipticK(k))= -(1)/(4)*EllipticCK(k)+(1)/(2)*EllipticK(k)*ln(sqrt(1 - (k)^(2))/ k)	Integrate[Log[JacobiCN[t, (k)^2]], {t, 0, EllipticK[(k)^2]}]= -Divide[1,4]*EllipticK[1-(k)^2]+Divide[1,2]*EllipticK[(k)^2]*Log[Sqrt[1 - (k)^(2)]/ k]	Failure	Failure	Skip	Successful
22.15.E1	${\displaystyle{\displaystyle\operatorname{sn}\left(\xi,k\right)=x}}$	JacobiSN(xi, k)= x	JacobiSN[\[Xi], (k)^2]= x	Failure	Failure	Skip	Fail Complex[0.11837413740083425, 0.04087130856331978] <- {Rule[k, 1], Rule[x, 1], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.16253964807069676, 0.7594854905054264] <- {Rule[k, 2], Rule[x, 1], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.3727323062986447, 0.1514985571826523] <- {Rule[k, 3], Rule[x, 1], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.11837413740083425, -0.04087130856331978] <- {Rule[k, 1], Rule[x, 1], Rule[ξ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.15.E2	${\displaystyle{\displaystyle\operatorname{cn}\left(\eta,k\right)=x}}$	JacobiCN(eta, k)= x	JacobiCN[\[Eta], (k)^2]= x	Failure	Failure	Skip	Fail Complex[-0.9098721588744132, -0.507161981115838] <- {Rule[k, 1], Rule[x, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.0999336041995775, 0.5782521633446407] <- {Rule[k, 2], Rule[x, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.9421331411733305, -0.059936758566078566] <- {Rule[k, 3], Rule[x, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.9098721588744132, 0.507161981115838] <- {Rule[k, 1], Rule[x, 1], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.15.E3	${\displaystyle{\displaystyle\operatorname{dn}\left(\zeta,k\right)=x}}$	JacobiDN(zeta, k)= x	JacobiDN[\[zeta], (k)^2]= x	Failure	Failure	Skip	Error
22.15#Ex1	${\displaystyle{\displaystyle\xi=\operatorname{arcsn}\left(x,k\right)}}$	xi = InverseJacobiSN(x, k)	\[Xi]= InverseJacobiSN[x, (k)^2]	Failure	Failure	Error	Fail DirectedInfinity[-1] <- {Rule[k, 1], Rule[x, 1], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.8649074180390403, 2.9850098891679915] <- {Rule[k, 1], Rule[x, 2], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.0676399720931227, 2.9850098891679915] <- {Rule[k, 1], Rule[x, 3], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.571338384966797, 2.492471386122917] <- {Rule[k, 2], Rule[x, 1], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.15#Ex2	${\displaystyle{\displaystyle\eta=\operatorname{arccn}\left(x,k\right)}}$	eta = InverseJacobiCN(x, k)	\[Eta]= InverseJacobiCN[x, (k)^2]	Failure	Failure	Fail 1.414213562+1.414213562*I <- {eta = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 1.414213562+.367016011*I <- {eta = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 1.414213562+.183254145*I <- {eta = 2^(1/2)+I*2^(1/2), k = 1, x = 3} 1.414213562+1.414213562*I <- {eta = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[k, 1], Rule[x, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, 0.3670160111764975] <- {Rule[k, 1], Rule[x, 2], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, 0.1832541450323204] <- {Rule[k, 1], Rule[x, 3], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[k, 2], Rule[x, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.15#Ex3	${\displaystyle{\displaystyle\zeta=\operatorname{arcdn}\left(x,k\right)}}$	zeta = InverseJacobiDN(x, k)	\[zeta]= InverseJacobiDN[x, (k)^2]	Failure	Failure	Fail 1.414213562+1.414213562*I <- {zeta = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 1.414213562+.367016011*I <- {zeta = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 1.414213562+.183254145*I <- {zeta = 2^(1/2)+I*2^(1/2), k = 1, x = 3} 1.414213562+1.414213562*I <- {zeta = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Error
22.15.E5	${\displaystyle{\displaystyle-K<=\operatorname{arcsn}\left(x,k\right)}}$	- EllipticK(k)< = InverseJacobiSN(x, k)	- EllipticK[(k)^2]< = InverseJacobiSN[x, (k)^2]	Failure	Failure	Error	Successful
22.15.E5	${\displaystyle{\displaystyle\operatorname{arcsn}\left(x,k\right)<=K}}$	InverseJacobiSN(x, k)< = EllipticK(k)	InverseJacobiSN[x, (k)^2]< = EllipticK[(k)^2]	Failure	Failure	Error	Successful
22.15.E6	${\displaystyle{\displaystyle 0<=\operatorname{arccn}\left(x,k\right)}}$	0 < = InverseJacobiCN(x, k)	0 < = InverseJacobiCN[x, (k)^2]	Failure	Failure	Successful	Successful
22.15.E6	${\displaystyle{\displaystyle\operatorname{arccn}\left(x,k\right)<=2K}}$	InverseJacobiCN(x, k)< = 2*EllipticK(k)	InverseJacobiCN[x, (k)^2]< = 2*EllipticK[(k)^2]	Failure	Failure	Error	Successful
22.15.E7	${\displaystyle{\displaystyle 0<=\operatorname{arcdn}\left(x,k\right)}}$	0 < = InverseJacobiDN(x, k)	0 < = InverseJacobiDN[x, (k)^2]	Failure	Failure	Successful	Successful
22.15.E7	${\displaystyle{\displaystyle\operatorname{arcdn}\left(x,k\right)<=K}}$	InverseJacobiDN(x, k)< = EllipticK(k)	InverseJacobiDN[x, (k)^2]< = EllipticK[(k)^2]	Failure	Failure	Error	Successful
22.15.E8	${\displaystyle{\displaystyle\xi=(-1)^{m}\operatorname{arcsn}\left(x,k\right)+2mK}}$	xi =(- 1)^(m)* InverseJacobiSN(x, k)+ 2*m*K	\[Xi]=(- 1)^(m)* InverseJacobiSN[x, (k)^2]+ 2*m*K	Failure	Failure	Error	Skip
22.15.E9	${\displaystyle{\displaystyle\eta=+\operatorname{arccn}\left(x,k\right)+4mK}}$	eta = + InverseJacobiCN(x, k)+ 4*m*K	\[Eta]= + InverseJacobiCN[x, (k)^2]+ 4*m*K	Failure	Failure	Fail -4.242640686-4.242640686*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 1} -4.242640686-5.289838237*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 2} -4.242640686-5.473600103*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 3} -9.899494934-9.899494934*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 2, x = 1} ... skip entries to safe data	Skip
22.15.E9	${\displaystyle{\displaystyle\eta=-\operatorname{arccn}\left(x,k\right)+4mK}}$	eta = - InverseJacobiCN(x, k)+ 4*m*K	\[Eta]= - InverseJacobiCN[x, (k)^2]+ 4*m*K	Failure	Failure	Fail -4.242640686-4.242640686*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 1} -4.242640686-3.195443135*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 2} -4.242640686-3.011681269*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 3} -9.899494934-9.899494934*I <- {K = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), k = 1, m = 2, x = 1} ... skip entries to safe data	Skip
22.15.E10	${\displaystyle{\displaystyle\zeta=+\operatorname{arcdn}\left(x,k\right)+2mK}}$	zeta = + InverseJacobiDN(x, k)+ 2*m*K	\[zeta]= + InverseJacobiDN[x, (k)^2]+ 2*m*K	Failure	Failure	Fail -1.414213562-1.414213562*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 1} -1.414213562-2.461411113*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 2} -1.414213562-2.645172979*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 3} -4.242640686-4.242640686*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 2, x = 1} ... skip entries to safe data	Error
22.15.E10	${\displaystyle{\displaystyle\zeta=-\operatorname{arcdn}\left(x,k\right)+2mK}}$	zeta = - InverseJacobiDN(x, k)+ 2*m*K	\[zeta]= - InverseJacobiDN[x, (k)^2]+ 2*m*K	Failure	Failure	Fail -1.414213562-1.414213562*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 1} -1.414213562-.367016011*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 2} -1.414213562-.183254145*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 1, x = 3} -4.242640686-4.242640686*I <- {K = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2), k = 1, m = 2, x = 1} ... skip entries to safe data	Error
22.15.E12	${\displaystyle{\displaystyle\operatorname{arcsn}\left(x,k\right)=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}}}$	InverseJacobiSN(x, k)= int((1)/(sqrt((1 - (t)^(2))*(1 - (k)^(2)* (t)^(2)))), t = 0..x)	InverseJacobiSN[x, (k)^2]= Integrate[Divide[1,Sqrt[(1 - (t)^(2))*(1 - (k)^(2)* (t)^(2))]], {t, 0, x}]	Failure	Failure	Skip	Error
22.15.E13	${\displaystyle{\displaystyle\operatorname{arccn}\left(x,k\right)=\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})({k^{\prime}}^{2}+k^{2}t^{2})}}}}$	InverseJacobiCN(x, k)= int((1)/(sqrt((1 - (t)^(2))*(1 - (k)^(2)+ (k)^(2)* (t)^(2)))), t = x..1)	InverseJacobiCN[x, (k)^2]= Integrate[Divide[1,Sqrt[(1 - (t)^(2))*(1 - (k)^(2)+ (k)^(2)* (t)^(2))]], {t, x, 1}]	Failure	Failure	Skip	Error
22.15.E14	${\displaystyle{\displaystyle\operatorname{arcdn}\left(x,k\right)=\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(t^{2}-{k^{\prime}}^{2})}}}}$	InverseJacobiDN(x, k)= int((1)/(sqrt((1 - (t)^(2))*((t)^(2)- 1 - (k)^(2)))), t = x..1)	InverseJacobiDN[x, (k)^2]= Integrate[Divide[1,Sqrt[(1 - (t)^(2))*((t)^(2)- 1 - (k)^(2))]], {t, x, 1}]	Error	Failure	-	Error
22.15.E16	${\displaystyle{\displaystyle\operatorname{arcsd}\left(x,k\right)=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-{k^{\prime}}^{2}t^{2})(1+k^{2}t^{2})}}}}$	InverseJacobiSD(x, k)= int((1)/(sqrt((1 - 1 - (k)^(2)* (t)^(2))*(1 + (k)^(2)* (t)^(2)))), t = 0..x)	InverseJacobiSD[x, (k)^2]= Integrate[Divide[1,Sqrt[(1 - 1 - (k)^(2)* (t)^(2))*(1 + (k)^(2)* (t)^(2))]], {t, 0, x}]	Error	Failure	-	Successful
22.15.E17	${\displaystyle{\displaystyle\operatorname{arcnd}\left(x,k\right)=\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{(t^{2}-1)(1-{k^{\prime}}^{2}t^{2})}}}}$	InverseJacobiND(x, k)= int((1)/(sqrt(((t)^(2)- 1)*(1 - 1 - (k)^(2)* (t)^(2)))), t = 1..x)	InverseJacobiND[x, (k)^2]= Integrate[Divide[1,Sqrt[((t)^(2)- 1)*(1 - 1 - (k)^(2)* (t)^(2))]], {t, 1, x}]	Failure	Failure	Skip	Successful
22.15.E19	${\displaystyle{\displaystyle\operatorname{arcnc}\left(x,k\right)=\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{(t^{2}-1)(k^{2}+{k^{\prime}}^{2}t^{2})}}}}$	InverseJacobiNC(x, k)= int((1)/(sqrt(((t)^(2)- 1)*((k)^(2)+ 1 - (k)^(2)* (t)^(2)))), t = 1..x)	InverseJacobiNC[x, (k)^2]= Integrate[Divide[1,Sqrt[((t)^(2)- 1)*((k)^(2)+ 1 - (k)^(2)* (t)^(2))]], {t, 1, x}]	Failure	Failure	Skip	Error
22.15.E20	${\displaystyle{\displaystyle\operatorname{arcsc}\left(x,k\right)=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1+t^{2})(1+{k^{\prime}}^{2}t^{2})}}}}$	InverseJacobiSC(x, k)= int((1)/(sqrt((1 + (t)^(2))*(1 + 1 - (k)^(2)* (t)^(2)))), t = 0..x)	InverseJacobiSC[x, (k)^2]= Integrate[Divide[1,Sqrt[(1 + (t)^(2))*(1 + 1 - (k)^(2)* (t)^(2))]], {t, 0, x}]	Failure	Failure	Skip	Error
22.15.E21	${\displaystyle{\displaystyle\operatorname{arcns}\left(x,k\right)=\int_{x}^{\infty}\frac{\mathrm{d}t}{\sqrt{(t^{2}-1)(t^{2}-k^{2})}}}}$	InverseJacobiNS(x, k)= int((1)/(sqrt(((t)^(2)- 1)*((t)^(2)- (k)^(2)))), t = x..infinity)	InverseJacobiNS[x, (k)^2]= Integrate[Divide[1,Sqrt[((t)^(2)- 1)*((t)^(2)- (k)^(2))]], {t, x, Infinity}]	Failure	Failure	Skip	Error
22.15.E22	${\displaystyle{\displaystyle\operatorname{arcds}\left(x,k\right)=\int_{x}^{\infty}\frac{\mathrm{d}t}{\sqrt{(t^{2}+k^{2})(t^{2}-{k^{\prime}}^{2})}}}}$	InverseJacobiDS(x, k)= int((1)/(sqrt(((t)^(2)+ (k)^(2))*((t)^(2)- 1 - (k)^(2)))), t = x..infinity)	InverseJacobiDS[x, (k)^2]= Integrate[Divide[1,Sqrt[((t)^(2)+ (k)^(2))*((t)^(2)- 1 - (k)^(2))]], {t, x, Infinity}]	Error	Failure	-	Error
22.15.E23	${\displaystyle{\displaystyle\operatorname{arccs}\left(x,k\right)=\int_{x}^{\infty}\frac{\mathrm{d}t}{\sqrt{(1+t^{2})(t^{2}+{k^{\prime}}^{2})}}}}$	InverseJacobiCS(x, k)= int((1)/(sqrt((1 + (t)^(2))*((t)^(2)+ 1 - (k)^(2)))), t = x..infinity)	InverseJacobiCS[x, (k)^2]= Integrate[Divide[1,Sqrt[(1 + (t)^(2))*((t)^(2)+ 1 - (k)^(2))]], {t, x, Infinity}]	Failure	Failure	Skip	Error
22.16.E2	${\displaystyle{\displaystyle\operatorname{am}\left(x+2K,k\right)=\operatorname{am}\left(x,k\right)+\pi}}$	JacobiAM(x + 2*EllipticK(k), k)= JacobiAM(x, k)+ Pi	Error	Failure	Error	Error	-
22.16.E3	${\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\int_{0}^{x}\operatorname{dn}\left(t,k\right)\mathrm{d}t}}$	JacobiAM(x, k)= int(JacobiDN(t, k), t = 0..x)	Error	Failure	Error	Skip	-
22.16.E4	${\displaystyle{\displaystyle\operatorname{am}\left(x,0\right)=x}}$	JacobiAM(x, 0)= x	Error	Successful	Error	-	-
22.16.E5	${\displaystyle{\displaystyle\operatorname{am}\left(x,1\right)=\operatorname{gd}\left(x\right)}}$	JacobiAM(x, 1)= arctan(sinh(x))	Error	Failure	Error	Successful	-
22.16.E9	${\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^{n}\sin\left(2n\zeta\right)}{n(1+q^{2n})}}}$	JacobiAM(x, k)=(Pi)/(2*EllipticK(k))*x + 2*sum(((q)^(n)* sin(2*n*zeta))/(n*(1 + (q)^(2*n))), n = 1..infinity)	Error	Failure	Error	Skip	-
22.16.E10	${\displaystyle{\displaystyle x=F\left(\phi,k\right)}}$	x = EllipticF(sin(phi), k)	x = EllipticF[\[Phi], (k)^2]	Failure	Failure	Fail .5109255834-1.490381835*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 1.510925583-1.490381835*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 2.510925583-1.490381835*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 3} .7620067201-1.041441678*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Fail Complex[0.5109255837122914, -1.4903818357543746] <- {Rule[k, 1], Rule[x, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.5109255837122912, -1.4903818357543746] <- {Rule[k, 1], Rule[x, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.5109255837122912, -1.4903818357543746] <- {Rule[k, 1], Rule[x, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.762006720261616, -1.0414416780368256] <- {Rule[k, 2], Rule[x, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.16.E11	${\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\phi}}$	JacobiAM(x, k)= phi	Error	Failure	Error	Fail -.548444079-1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 1} -.112453227-1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 2} .57090779e-1-1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 3} -.9119062827-1.414213562*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	-
22.16.E12	${\displaystyle{\displaystyle\operatorname{sn}\left(x,k\right)=\sin\phi}}$	JacobiSN(x, k)= sin(phi)	JacobiSN[x, (k)^2]= Sin[\[Phi]]	Failure	Failure	Fail -1.389941384-.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 1} -1.187507960-.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 2} -1.156480786-.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 3} -1.670086451-.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Fail Complex[-1.3899413853835214, -0.30176146986776087] <- {Rule[k, 1], Rule[x, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.1875079612634694, -0.30176146986776087] <- {Rule[k, 1], Rule[x, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.1564807876525558, -0.30176146986776087] <- {Rule[k, 1], Rule[x, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.6700864525401473, -0.3017614698677609] <- {Rule[k, 2], Rule[x, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.16.E12	${\displaystyle{\displaystyle\sin\phi=\sin\left(\operatorname{am}\left(x,k\right)\right)}}$	sin(phi)= sin(JacobiAM(x, k))	Error	Failure	Error	Fail 1.389941384+.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 1.187507960+.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 1.156480786+.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 3} 1.670086451+.3017614705*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	-
22.16.E13	${\displaystyle{\displaystyle\operatorname{cn}\left(x,k\right)=\cos\phi}}$	JacobiCN(x, k)= cos(phi)	JacobiCN[x, (k)^2]= Cos[\[Phi]]	Failure	Failure	Fail .3083802813+1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 1} -.738717636e-1+1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 2} -.2403460650+1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 3} .5368000659+1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Fail Complex[0.3083802819691609, 1.9113931101642103] <- {Rule[k, 1], Rule[x, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.07387176286064484, 1.9113931101642103] <- {Rule[k, 1], Rule[x, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.24034606427529137, 1.9113931101642103] <- {Rule[k, 1], Rule[x, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.5368000666176019, 1.9113931101642103] <- {Rule[k, 2], Rule[x, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.16.E13	${\displaystyle{\displaystyle\cos\phi=\cos\left(\operatorname{am}\left(x,k\right)\right)}}$	cos(phi)= cos(JacobiAM(x, k))	Error	Failure	Error	Fail -.3083802813-1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 1} .738717636e-1-1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 2} .2403460650-1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, x = 3} -.5368000659-1.911393109*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	-
22.16.E33	${\displaystyle{\displaystyle\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k\right)-k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right)}}$	JacobiZeta(x + EllipticK(k), k)= JacobiZeta(x, k)- (k)^(2)* JacobiSN(x, k)*JacobiCD(x, k)	JacobiZeta[x + EllipticK[(k)^2], k]= JacobiZeta[x, k]- (k)^(2)* JacobiSN[x, (k)^2]*JacobiCD[x, (k)^2]	Failure	Failure	Error	Fail Complex[-9.092646035261055, 0.7547542476789773] <- {Rule[k, 2], Rule[x, 1]} Complex[0.5691890271397613, -1.7224198606648595] <- {Rule[k, 2], Rule[x, 2]} Complex[-0.6254417826653399, -1.1984364919906034] <- {Rule[k, 2], Rule[x, 3]} Complex[0.24923293591578033, 0.8848347496931445] <- {Rule[k, 3], Rule[x, 1]} ... skip entries to safe data
22.16.E34	${\displaystyle{\displaystyle\mathrm{Z}\left(x+2K|k\right)=\mathrm{Z}\left(x|k\right)}}$	JacobiZeta(x + 2*EllipticK(k), k)= JacobiZeta(x, k)	JacobiZeta[x + 2*EllipticK[(k)^2], k]= JacobiZeta[x, k]	Successful	Failure	-	Fail Complex[-3.520658209011301, -4.348209951262927] <- {Rule[k, 2], Rule[x, 1]} Complex[2.8383377769721507, -5.069497738677276] <- {Rule[k, 2], Rule[x, 2]} Complex[5.651309663980639, 0.2964137811420908] <- {Rule[k, 2], Rule[x, 3]} Complex[-5.093513410963294, -0.873054809617268] <- {Rule[k, 3], Rule[x, 1]} ... skip entries to safe data
22.17.E1	${\displaystyle{\displaystyle\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right)}}$	genJacobiellk(p)*q* z*k = genJacobiellk(p)*q* z- k	genJacobiellk(p)*q* z*k = genJacobiellk(p)*q* z- k	Failure	Failure	Error	Skip
22.17.E2	${\displaystyle{\displaystyle\operatorname{sn}\left(z,1/k\right)=k\operatorname{sn}\left(z/k,k\right)}}$	JacobiSN(z, 1/ k)= k*JacobiSN(z/ k, k)	JacobiSN[z, (1/ k)^2]= k*JacobiSN[z/ k, (k)^2]	Failure	Failure	Successful	Successful
22.17.E3	${\displaystyle{\displaystyle\operatorname{cn}\left(z,1/k\right)=\operatorname{dn}\left(z/k,k\right)}}$	JacobiCN(z, 1/ k)= JacobiDN(z/ k, k)	JacobiCN[z, (1/ k)^2]= JacobiDN[z/ k, (k)^2]	Failure	Failure	Successful	Successful
22.17.E4	${\displaystyle{\displaystyle\operatorname{dn}\left(z,1/k\right)=\operatorname{cn}\left(z/k,k\right)}}$	JacobiDN(z, 1/ k)= JacobiCN(z/ k, k)	JacobiDN[z, (1/ k)^2]= JacobiCN[z/ k, (k)^2]	Failure	Failure	Successful	Successful
22.17.E6	${\displaystyle{\displaystyle\operatorname{sn}\left(z,ik\right)=k^{\prime}_{1}\operatorname{sd}\left(z/k^{\prime}_{1},k_{1}\right)}}$	JacobiSN(z, I*k)=sqrt(1 - (k)^(2))[1]*JacobiSD(z/sqrt(1 - (k)^(2))[1], k[1])	JacobiSN[z, (I*k)^2]=Subscript[Sqrt[1 - (k)^(2)], 1]*JacobiSD[z/Subscript[Sqrt[1 - (k)^(2)], 1], (Subscript[k, 1])^2]	Failure	Failure	Error	Successful
22.17.E7	${\displaystyle{\displaystyle\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k^{\prime}_{1},k_{1}\right)}}$	JacobiCN(z, I*k)= JacobiCD(z/sqrt(1 - (k)^(2))[1], k[1])	JacobiCN[z, (I*k)^2]= JacobiCD[z/Subscript[Sqrt[1 - (k)^(2)], 1], (Subscript[k, 1])^2]	Failure	Failure	Error	Successful
22.17.E8	${\displaystyle{\displaystyle\operatorname{dn}\left(z,ik\right)=\operatorname{nd}\left(z/k^{\prime}_{1},k_{1}\right)}}$	JacobiDN(z, I*k)= JacobiND(z/sqrt(1 - (k)^(2))[1], k[1])	JacobiDN[z, (I*k)^2]= JacobiND[z/Subscript[Sqrt[1 - (k)^(2)], 1], (Subscript[k, 1])^2]	Failure	Failure	Error	Successful
22.18#Ex1	${\displaystyle{\displaystyle x=a\operatorname{sn}\left(u,k\right)}}$	x = a*JacobiSN(u, k)	x = a*JacobiSN[u, (k)^2]	Failure	Failure	Fail -.523819113-1.639420631*I <- {a = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 1, x = 1} .476180887-1.639420631*I <- {a = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 1.476180887-1.639420631*I <- {a = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 1, x = 3} .8897268907-2.258422469*I <- {a = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Skip
22.18#Ex2	${\displaystyle{\displaystyle y=b\operatorname{cn}\left(u,k\right)}}$	y = b*JacobiCN(u, k)	y = b*JacobiCN[u, (k)^2]	Failure	Failure	Fail .1553046323+.5897753365*I <- {b = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 1, y = 1} 1.155304632+.5897753365*I <- {b = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 1, y = 2} 2.155304632+.5897753365*I <- {b = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 1, y = 3} 3.373313072+.7377689657*I <- {b = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), k = 2, y = 1} ... skip entries to safe data	Skip
22.18.E4	${\displaystyle{\displaystyle l(r)=(1/\sqrt{2})\operatorname{arccn}\left(r,1/\sqrt{2}\right)}}$	l*(r)=(1/sqrt(2))* InverseJacobiCN(r, 1/sqrt(2))	l*(r)=(1/Sqrt[2])* InverseJacobiCN[r, (1/Sqrt[2])^2]	Failure	Failure	Fail 1.062814328+2.373843104*I <- {r = 2^(1/2)+I*2^(1/2), l = 1} 2.477027890+3.788056666*I <- {r = 2^(1/2)+I*2^(1/2), l = 2} 3.891241452+5.202270228*I <- {r = 2^(1/2)+I*2^(1/2), l = 3} 1.062814328-2.373843104*I <- {r = 2^(1/2)-I*2^(1/2), l = 1} ... skip entries to safe data	Fail Complex[1.0628143278928741, 2.3738431050389313] <- {Rule[l, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.4770278902659695, 3.788056667412026] <- {Rule[l, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.891241452639065, 5.202270229785122] <- {Rule[l, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.0628143278928741, -2.3738431050389313] <- {Rule[l, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.18.E5	${\displaystyle{\displaystyle r=\operatorname{cn}\left(\sqrt{2}l,1/\sqrt{2}\right)}}$	r = JacobiCN(sqrt(2)*l, 1/sqrt(2))	r = JacobiCN[Sqrt[2]*l, (1/Sqrt[2])^2]	Failure	Failure	Fail 1.103475632+1.414213562*I <- {r = 2^(1/2)+I*2^(1/2), l = 1} 2.087946761+1.414213562*I <- {r = 2^(1/2)+I*2^(1/2), l = 2} 2.280767714+1.414213562*I <- {r = 2^(1/2)+I*2^(1/2), l = 3} 1.103475632-1.414213562*I <- {r = 2^(1/2)-I*2^(1/2), l = 1} ... skip entries to safe data	Fail Complex[1.103475632039239, 1.4142135623730951] <- {Rule[l, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.087946761347629, 1.4142135623730951] <- {Rule[l, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.280767714220744, 1.4142135623730951] <- {Rule[l, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.103475632039239, -1.4142135623730951] <- {Rule[l, 1], Rule[r, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.18#Ex3	${\displaystyle{\displaystyle x=\operatorname{cn}\left(\sqrt{2}l,1/\sqrt{2}\right)\operatorname{dn}\left(\sqrt{2}l,1/\sqrt{2}\right)}}$	x = JacobiCN(sqrt(2)*l, 1/sqrt(2))*JacobiDN(sqrt(2)*l, 1/sqrt(2))	x = JacobiCN[Sqrt[2]*l, (1/Sqrt[2])^2]*JacobiDN[Sqrt[2]*l, (1/Sqrt[2])^2]	Failure	Failure	Fail .7699114076 <- {l = 1, x = 1} 1.769911408 <- {l = 1, x = 2} 2.769911408 <- {l = 1, x = 3} 1.574437352 <- {l = 2, x = 1} ... skip entries to safe data	Fail Complex[0.7699114077583536, 0.0] <- {Rule[l, 1], Rule[x, 1]} Complex[1.7699114077583538, 0.0] <- {Rule[l, 1], Rule[x, 2]} Complex[2.7699114077583538, 0.0] <- {Rule[l, 1], Rule[x, 3]} Complex[1.574437352038115, 0.0] <- {Rule[l, 2], Rule[x, 1]} ... skip entries to safe data
22.19.E1	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}\theta(t)}{{\mathrm{d}t}^{2}}=-\sin\theta(t)}}$	diff(theta*(t), [t$(2)])= - sin(theta*(t))	D[\[Theta]*(t), {t, 2}]= - Sin[\[Theta]*(t)]	Failure	Failure	Fail 0.+27.28991714*I <- {t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} -.7568024940+0.*I <- {t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} 0.-27.28991714*I <- {t = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .7568024940+0.*I <- {t = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.0, 27.28991719712775] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} -0.7568024953079282 <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.0, -27.28991719712775] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} 0.7568024953079282 <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.19.E2	${\displaystyle{\displaystyle\sin\left(\tfrac{1}{2}\theta(t)\right)=\sin\left(\frac{1}{2}\alpha\right)\operatorname{sn}\left(t+K,\sin\left(\tfrac{1}{2}\alpha\right)\right)}}$	sin((1)/(2)*theta*(t))= sin((1)/(2)*alpha)*JacobiSN(t + EllipticK(k), sin((1)/(2)*alpha))	Sin[Divide[1,2]*\[Theta]*(t)]= Sin[Divide[1,2]*\[Alpha]]*JacobiSN[t + EllipticK[(k)^2], (Sin[Divide[1,2]*\[Alpha]])^2]	Failure	Failure	Error	Skip
22.19.E3	${\displaystyle{\displaystyle\theta(t)=2\operatorname{am}\left(t\sqrt{E/2},\sqrt{2/E}\right)}}$	theta*(t)= 2*JacobiAM(t*sqrt(E/ 2), sqrt(2/ E))	Error	Failure	Error	Fail -2.607335908+2.693048555*I <- {E = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} 1.392664090-1.306951443*I <- {E = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} -2.607335908-5.306951441*I <- {E = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -6.607335906-1.306951443*I <- {E = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	-
22.19.E4	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}x(t)}{{\mathrm{d}t}^{2}}=-\frac{\mathrm{d}V(x)}{\mathrm{d}x}}}$	diff(x*(t), [t$(2)])= - diff(V*(x), x)	D[x*(t), {t, 2}]= - D[V*(x), x]	Failure	Failure	Fail 1.414213562+1.414213562*I <- {V = 2^(1/2)+I*2^(1/2)} 1.414213562-1.414213562*I <- {V = 2^(1/2)-I*2^(1/2)} -1.414213562-1.414213562*I <- {V = -2^(1/2)-I*2^(1/2)} -1.414213562+1.414213562*I <- {V = -2^(1/2)+I*2^(1/2)}	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[V, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[V, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
22.19.E6	${\displaystyle{\displaystyle x(t)=a\operatorname{cn}\left(t\sqrt{1+2\eta},k\right)}}$	x*(t)= a*JacobiCN(t*sqrt(1 + 2*eta), k)	x*(t)= a*JacobiCN[t*Sqrt[1 + 2*\[Eta]], (k)^2]	Failure	Failure	Fail 1.973069551+1.442641289*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 3.387283113+2.856854851*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 4.801496675+4.271068413*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 3} .191022968-.212139835*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Skip
22.19.E7	${\displaystyle{\displaystyle x(t)=a\operatorname{sn}\left(t\sqrt{1-\eta},k\right)}}$	x*(t)= a*JacobiSN(t*sqrt(1 - eta), k)	x*(t)= a*JacobiSN[t*Sqrt[1 - \[Eta]], (k)^2]	Failure	Failure	Fail .3281638e-2+.32333204e-1*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 1.417495200+1.446546766*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 2.831708762+2.860760328*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 3} 2.320220885+2.141378505*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Skip
22.19.E8	${\displaystyle{\displaystyle x(t)=a\operatorname{dn}\left(t\sqrt{\eta},k\right)}}$	x*(t)= a*JacobiDN(t*sqrt(eta), k)	x*(t)= a*JacobiDN[t*Sqrt[\[Eta]], (k)^2]	Failure	Failure	Fail 1.853632511+2.613297406*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 3.267846073+4.027510968*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 4.682059635+5.441724530*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 3} 1.885012698-.421765357*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Skip
22.19.E9	${\displaystyle{\displaystyle x(t)=a\operatorname{cn}\left(t\sqrt{2\eta-1},k\right)}}$	x*(t)= a*JacobiCN(t*sqrt(2*eta - 1), k)	x*(t)= a*JacobiCN[t*Sqrt[2*\[Eta]- 1], (k)^2]	Failure	Failure	Fail 2.558292975+2.017654962*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 1} 3.972506537+3.431868524*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 2} 5.386720099+4.846082086*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 1, x = 3} .335387665+1.074384225*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Skip
22.20.E4	${\displaystyle{\displaystyle\phi_{n-1}=\frac{1}{2}\left(\phi_{n}+\operatorname{arcsin}\left(\frac{c_{n}}{a_{n}}\sin\phi_{n}\right)\right)}}$	phi[n - 1]=(1)/(2)*(phi[n]+ arcsin((c[n])/(a[n])*sin(phi[n])))	Subscript[\[Phi], n - 1]=Divide[1,2]*(Subscript[\[Phi], n]+ ArcSin[Divide[Subscript[c, n],Subscript[a, n]]*Sin[Subscript[\[Phi], n]]])	Failure	Failure	Fail -2.828427124*I <- {a[n] = 2^(1/2)+I*2^(1/2), c[n] = 2^(1/2)+I*2^(1/2), phi[n] = 2^(1/2)+I*2^(1/2), phi[n-1] = 2^(1/2)-I*2^(1/2)} -2.828427124-2.828427124*I <- {a[n] = 2^(1/2)+I*2^(1/2), c[n] = 2^(1/2)+I*2^(1/2), phi[n] = 2^(1/2)+I*2^(1/2), phi[n-1] = -2^(1/2)-I*2^(1/2)} -2.828427124 <- {a[n] = 2^(1/2)+I*2^(1/2), c[n] = 2^(1/2)+I*2^(1/2), phi[n] = 2^(1/2)+I*2^(1/2), phi[n-1] = -2^(1/2)+I*2^(1/2)} 2.828427124*I <- {a[n] = 2^(1/2)+I*2^(1/2), c[n] = 2^(1/2)+I*2^(1/2), phi[n] = 2^(1/2)-I*2^(1/2), phi[n-1] = 2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.0, 2.8284271247461903] <- {Rule[Subscript[a, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[c, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, n], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Subscript[a, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[c, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, n], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} 2.8284271247461903 <- {Rule[Subscript[a, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[c, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, n], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[0.0, -2.8284271247461903] <- {Rule[Subscript[a, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[c, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, Plus[-1, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.20#Ex4	${\displaystyle{\displaystyle\operatorname{sn}\left(x,k\right)=\sin\phi_{0}}}$	JacobiSN(x, k)= sin(phi[0])	JacobiSN[x, (k)^2]= Sin[Subscript[\[Phi], 0]]	Failure	Failure	Fail -1.389941384-.3017614705*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 1, x = 1} -1.187507960-.3017614705*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 1, x = 2} -1.156480786-.3017614705*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 1, x = 3} -1.670086451-.3017614705*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Fail Complex[-1.3899413853835214, -0.30176146986776087] <- {Rule[k, 1], Rule[x, 1], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.1875079612634694, -0.30176146986776087] <- {Rule[k, 1], Rule[x, 2], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.1564807876525558, -0.30176146986776087] <- {Rule[k, 1], Rule[x, 3], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.6700864525401473, -0.3017614698677609] <- {Rule[k, 2], Rule[x, 1], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.20#Ex5	${\displaystyle{\displaystyle\operatorname{cn}\left(x,k\right)=\cos\phi_{0}}}$	JacobiCN(x, k)= cos(phi[0])	JacobiCN[x, (k)^2]= Cos[Subscript[\[Phi], 0]]	Failure	Failure	Fail .3083802813+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 1, x = 1} -.738717636e-1+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 1, x = 2} -.2403460650+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 1, x = 3} .5368000659+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Fail Complex[0.3083802819691609, 1.9113931101642103] <- {Rule[k, 1], Rule[x, 1], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.07387176286064484, 1.9113931101642103] <- {Rule[k, 1], Rule[x, 2], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.24034606427529137, 1.9113931101642103] <- {Rule[k, 1], Rule[x, 3], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.5368000666176019, 1.9113931101642103] <- {Rule[k, 2], Rule[x, 1], Rule[Subscript[ϕ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
22.20#Ex6	${\displaystyle{\displaystyle\operatorname{dn}\left(x,k\right)=\frac{\cos\phi_{0}}{\cos\left(\phi_{1}-\phi_{0}\right)}}}$	JacobiDN(x, k)=(cos(phi[0]))/(cos(phi[1]- phi[0]))	JacobiDN[x, (k)^2]=Divide[Cos[Subscript[\[Phi], 0]],Cos[Subscript[\[Phi], 1]- Subscript[\[Phi], 0]]]	Failure	Failure	Fail .3083802813+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 1, x = 1} -.738717636e-1+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 1, x = 2} -.2403460650+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 1, x = 3} -.6095389579+1.911393109*I <- {phi[0] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 2, x = 1} ... skip entries to safe data	Skip