Jacobian Elliptic Functions - 22.13 Derivatives and Differential Equations

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
22.13.E1	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k\right)\right)^{2}=\left(1-{\operatorname{sn}^{2}}\left(z,k\right)\right)\left(1-k^{2}{\operatorname{sn}^{2}}\left(z,k\right)\right)}}$ \left(\deriv{}{z}\Jacobiellsnk@{z}{k}\right)^{2} = \left(1-\Jacobiellsnk^{2}@{z}{k}\right)\left(1-k^{2}\Jacobiellsnk^{2}@{z}{k}\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	-	Successful [Tested: 21]
22.13.E2	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{2}={\left(1-{\operatorname{cn}^{2}}\left(z,k\right)\right)}{\left({k^{\prime}}^{2}+k^{2}{\operatorname{cn}^{2}}\left(z,k\right)\right)}}}$ \left(\deriv{}{z}\Jacobiellcnk@{z}{k}\right)^{2} = {\left(1-\Jacobiellcnk^{2}@{z}{k}\right)}{\left({k^{\prime}}^{2}+k^{2}\Jacobiellcnk^{2}@{z}{k}\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	-	Successful [Tested: 21]
22.13.E3	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dn}\left(z,k\right)\right)^{2}=\left(1-{\operatorname{dn}^{2}}\left(z,k\right)\right)\left({\operatorname{dn}^{2}}\left(z,k\right)-{k^{\prime}}^{2}\right)}}$ \left(\deriv{}{z}\Jacobielldnk@{z}{k}\right)^{2} = \left(1-\Jacobielldnk^{2}@{z}{k}\right)\left(\Jacobielldnk^{2}@{z}{k}-{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	Failed [21 / 21] Result: 1.137161176+.7719908960*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 14.77981366-.6810923425*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[1.1371611759337996, 0.7719908961474706] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[14.779813656775712, -0.6810923360985438] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E4	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cd}\left(z,k\right)\right)^{2}=\left(1-{\operatorname{cd}^{2}}\left(z,k\right)\right)\left(1-k^{2}{\operatorname{cd}^{2}}\left(z,k\right)\right)}}$ \left(\deriv{}{z}\Jacobiellcdk@{z}{k}\right)^{2} = \left(1-\Jacobiellcdk^{2}@{z}{k}\right)\left(1-k^{2}\Jacobiellcdk^{2}@{z}{k}\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	-	Successful [Tested: 21]
22.13.E5	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sd}\left(z,k\right)\right)^{2}={\left(1-{k^{\prime}}^{2}{\operatorname{sd}^{2}}\left(z,k\right)\right)}{\left(1+k^{2}{\operatorname{sd}^{2}}\left(z,k\right)\right)}}}$ \left(\deriv{}{z}\Jacobiellsdk@{z}{k}\right)^{2} = {\left(1-{k^{\prime}}^{2}\Jacobiellsdk^{2}@{z}{k}\right)}{\left(1+k^{2}\Jacobiellsdk^{2}@{z}{k}\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	Failed [21 / 21] Result: .3306277626+2.965675443*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 3.240181814+.5678364413*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[0.33062776288262774, 2.9656754410633357] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.24018181473062, 0.5678364360004244] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E6	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nd}\left(z,k\right)\right)^{2}=\left({\operatorname{nd}^{2}}\left(z,k\right)-1\right)\left(1-{k^{\prime}}^{2}{\operatorname{nd}^{2}}\left(z,k\right)\right)}}$ \left(\deriv{}{z}\Jacobiellndk@{z}{k}\right)^{2} = \left(\Jacobiellndk^{2}@{z}{k}-1\right)\left(1-{k^{\prime}}^{2}\Jacobiellndk^{2}@{z}{k}\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	Failed [21 / 21] Result: -.6693722376+2.965675443*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 15.46527968+2.623409101*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[-0.6693722371173725, 2.965675441063337] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[15.465279679493392, 2.6234090772942062] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E7	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{2}=\left({\operatorname{dc}^{2}}\left(z,k\right)-1\right)\left({\operatorname{dc}^{2}}\left(z,k\right)-k^{2}\right)}}$ \left(\deriv{}{z}\Jacobielldck@{z}{k}\right)^{2} = \left(\Jacobielldck^{2}@{z}{k}-1\right)\left(\Jacobielldck^{2}@{z}{k}-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	-	Successful [Tested: 21]
22.13.E8	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nc}\left(z,k\right)\right)^{2}={\left(k^{2}+{k^{\prime}}^{2}{\operatorname{nc}^{2}}\left(z,k\right)\right)}{\left({\operatorname{nc}^{2}}\left(z,k\right)-1\right)}}}$ \left(\deriv{}{z}\Jacobiellnck@{z}{k}\right)^{2} = {\left(k^{2}+{k^{\prime}}^{2}\Jacobiellnck^{2}@{z}{k}\right)}{\left(\Jacobiellnck^{2}@{z}{k}-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	Failed [20 / 21] Result: -1.244125150+.6620171546*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: .726292651-.1255426739*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [20 / 21] Result: Complex[-1.2441251486756877, 0.66201715389323] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.726292650669289, -0.12554267275387493] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E9	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sc}\left(z,k\right)\right)^{2}=\left(1+{\operatorname{sc}^{2}}\left(z,k\right)\right)\left(1+{k^{\prime}}^{2}{\operatorname{sc}^{2}}\left(z,k\right)\right)}}$ \left(\deriv{}{z}\Jacobiellsck@{z}{k}\right)^{2} = \left(1+\Jacobiellsck^{2}@{z}{k}\right)\left(1+{k^{\prime}}^{2}\Jacobiellsck^{2}@{z}{k}\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	Failed [21 / 21] Result: -2.244125150+.6620171546*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.273707349-.1255426740*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[-2.244125148675687, 0.6620171538932291] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.27370734933071006, -0.12554267275387854] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E10	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ns}\left(z,k\right)\right)^{2}=\left({\operatorname{ns}^{2}}\left(z,k\right)-k^{2}\right)\left({\operatorname{ns}^{2}}\left(z,k\right)-1\right)}}$ \left(\deriv{}{z}\Jacobiellnsk@{z}{k}\right)^{2} = \left(\Jacobiellnsk^{2}@{z}{k}-k^{2}\right)\left(\Jacobiellnsk^{2}@{z}{k}-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	-	Successful [Tested: 21]
22.13.E11	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ds}\left(z,k\right)\right)^{2}=\left({\operatorname{ds}^{2}}\left(z,k\right)-{k^{\prime}}^{2}\right)\left(k^{2}+{\operatorname{ds}^{2}}\left(z,k\right)\right)}}$ \left(\deriv{}{z}\Jacobielldsk@{z}{k}\right)^{2} = \left(\Jacobielldsk^{2}@{z}{k}-{k^{\prime}}^{2}\right)\left(k^{2}+\Jacobielldsk^{2}@{z}{k}\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	Failed [21 / 21] Result: 2.407829919-1.634616811*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 17.28421715+.7965017848*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[2.4078299188565357, -1.6346168126100018] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[17.284217154319762, 0.7965017768592271] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E12	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cs}\left(z,k\right)\right)^{2}=\left(1+{\operatorname{cs}^{2}}\left(z,k\right)\right)\left({k^{\prime}}^{2}+{\operatorname{cs}^{2}}\left(z,k\right)\right)}}$ \left(\deriv{}{z}\Jacobiellcsk@{z}{k}\right)^{2} = \left(1+\Jacobiellcsk^{2}@{z}{k}\right)\left({k^{\prime}}^{2}+\Jacobiellcsk^{2}@{z}{k}\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	-	Successful [Tested: 21]
22.13.E13	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sn}\left(z,k\right)=-(1+k^{2})\operatorname{sn}\left(z,k\right)+2k^{2}{\operatorname{sn}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellsnk@{z}{k} = -(1+k^{2})\Jacobiellsnk@{z}{k}+2k^{2}\Jacobiellsnk^{3}@{z}{k}		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	-	Successful [Tested: 21]
22.13.E14	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cn}\left(z,k\right)=-({k^{\prime}}^{2}-k^{2})\operatorname{cn}\left(z,k\right)-2k^{2}{\operatorname{cn}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellcnk@{z}{k} = -({k^{\prime}}^{2}-k^{2})\Jacobiellcnk@{z}{k}-2k^{2}\Jacobiellcnk^{3}@{z}{k}		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	-	Successful [Tested: 21]
22.13.E15	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dn}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{dn}\left(z,k\right)-2{\operatorname{dn}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobielldnk@{z}{k} = (1+{k^{\prime}}^{2})\Jacobielldnk@{z}{k}-2\Jacobielldnk^{3}@{z}{k}		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	-	Successful [Tested: 21]
22.13.E16	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cd}\left(z,k\right)=-(1+k^{2})\operatorname{cd}\left(z,k\right)+2k^{2}{\operatorname{cd}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellcdk@{z}{k} = -(1+k^{2})\Jacobiellcdk@{z}{k}+2k^{2}\Jacobiellcdk^{3}@{z}{k}		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	-	Successful [Tested: 21]
22.13.E17	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sd}\left(z,k\right)=(k^{2}-{k^{\prime}}^{2})\operatorname{sd}\left(z,k\right)-2k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellsdk@{z}{k} = (k^{2}-{k^{\prime}}^{2})\Jacobiellsdk@{z}{k}-2k^{2}{k^{\prime}}^{2}\Jacobiellsdk^{3}@{z}{k}		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	Failed [21 / 21] Result: 3.191457484+2.523217914*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 8.747979617-5.269762671*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[3.1914574835245033, 2.523217912470552] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[8.747979609525483, -5.269762670615425] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E18	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{nd}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{nd}\left(z,k\right)-2{k^{\prime}}^{2}{\operatorname{nd}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellndk@{z}{k} = (1+{k^{\prime}}^{2})\Jacobiellndk@{z}{k}-2{k^{\prime}}^{2}\Jacobiellndk^{3}@{z}{k}		diff(JacobiND(z, k), [z$(2)]) = (1 +1 - (k)^(2))*JacobiND(z, k)- 2*1 - (k)^(2)*(JacobiND(z, k))^(3)	D[JacobiND[z, (k)^2], {z, 2}] == (1 +1 - (k)^(2))*JacobiND[z, (k)^2]- 2*1 - (k)^(2)*(JacobiND[z, (k)^2])^(3)	Failure	Failure	Failed [21 / 21] Result: 3.040301731+2.018052700*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 3.903394000-12.57828103*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[3.0403017307041966, 2.01805269920667] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.903393981406644, -12.578281030301023] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E19	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dc}\left(z,k\right)=-(1+k^{2})\operatorname{dc}\left(z,k\right)+2{\operatorname{dc}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobielldck@{z}{k} = -(1+k^{2})\Jacobielldck@{z}{k}+2\Jacobielldck^{3}@{z}{k}		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	-	Successful [Tested: 21]
22.13.E20	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{nc}\left(z,k\right)=(k^{2}-{k^{\prime}}^{2})\operatorname{nc}\left(z,k\right)+2{k^{\prime}}^{2}{\operatorname{nc}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellnck@{z}{k} = (k^{2}-{k^{\prime}}^{2})\Jacobiellnck@{z}{k}+2{k^{\prime}}^{2}\Jacobiellnck^{3}@{z}{k}		diff(JacobiNC(z, k), [z$(2)]) = ((k)^(2)-1 - (k)^(2))*JacobiNC(z, k)+ 2*1 - (k)^(2)*(JacobiNC(z, k))^(3)	D[JacobiNC[z, (k)^2], {z, 2}] == ((k)^(2)-1 - (k)^(2))*JacobiNC[z, (k)^2]+ 2*1 - (k)^(2)*(JacobiNC[z, (k)^2])^(3)	Failure	Failure	Failed [21 / 21] Result: 1.495832765+2.956203453*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 3.847566639+.844372345e-1*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[1.4958327644324174, 2.9562034517436775] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.8475666387741003, 0.08443723368166078] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E21	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sc}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{sc}\left(z,k\right)+2{k^{\prime}}^{2}{\operatorname{sc}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellsck@{z}{k} = (1+{k^{\prime}}^{2})\Jacobiellsck@{z}{k}+2{k^{\prime}}^{2}\Jacobiellsck^{3}@{z}{k}		diff(JacobiSC(z, k), [z$(2)]) = (1 +1 - (k)^(2))*JacobiSC(z, k)+ 2*1 - (k)^(2)*(JacobiSC(z, k))^(3)	D[JacobiSC[z, (k)^2], {z, 2}] == (1 +1 - (k)^(2))*JacobiSC[z, (k)^2]+ 2*1 - (k)^(2)*(JacobiSC[z, (k)^2])^(3)	Failure	Failure	Failed [21 / 21] Result: -2.525815950+1.181755196*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: -3.577866152+.2036740201*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[-2.5258159501097865, 1.1817551948561285] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-3.5778661524913966, 0.20367401847233424] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E22	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ns}\left(z,k\right)=-(1+k^{2})\operatorname{ns}\left(z,k\right)+2{\operatorname{ns}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellnsk@{z}{k} = -(1+k^{2})\Jacobiellnsk@{z}{k}+2\Jacobiellnsk^{3}@{z}{k}		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	-	Successful [Tested: 21]
22.13.E23	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ds}\left(z,k\right)=(k^{2}-{k^{\prime}}^{2})\operatorname{ds}\left(z,k\right)+2{\operatorname{ds}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobielldsk@{z}{k} = (k^{2}-{k^{\prime}}^{2})\Jacobielldsk@{z}{k}+2\Jacobielldsk^{3}@{z}{k}		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	Failed [21 / 21] Result: 1.446566498-1.129997698*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.2935291263-10.85414309*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [21 / 21] Result: Complex[1.4465664983977982, -1.1299976975966786] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.293529123621927, -10.854143085101464] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.13.E24	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cs}\left(z,k\right)=(1+{k^{\prime}}^{2})\operatorname{cs}\left(z,k\right)+2{\operatorname{cs}^{3}}\left(z,k\right)}}$ \deriv[2]{}{z}\Jacobiellcsk@{z}{k} = (1+{k^{\prime}}^{2})\Jacobiellcsk@{z}{k}+2\Jacobiellcsk^{3}@{z}{k}		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	-	Successful [Tested: 21]