# Results of Jacobian Elliptic Functions

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DLMF Formula Maple Mathematica Symbolic
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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