22.11: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/22.11.E1 22.11.E1] || [[Item:Q7025|<math>\Jacobiellsnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSN(z, k) = (2*Pi)/(K*k)*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* sin((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSN[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Sin[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E1 22.11.E1] || <math qid="Q7025">\Jacobiellsnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSN(z, k) = (2*Pi)/(K*k)*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* sin((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSN[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Sin[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E2 22.11.E2] || [[Item:Q7026|<math>\Jacobiellcnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCN(z, k) = (2*Pi)/(K*k)*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* cos((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCN[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Cos[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E2 22.11.E2] || <math qid="Q7026">\Jacobiellcnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCN(z, k) = (2*Pi)/(K*k)*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* cos((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCN[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Cos[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E3 22.11.E3] || [[Item:Q7027|<math>\Jacobielldnk@{z}{k} = \frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{n}\cos@{2n\zeta}}{1+q^{2n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldnk@{z}{k} = \frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{n}\cos@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDN(z, k) = (Pi)/(2*EllipticK(k))+(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* cos(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDN[z, (k)^2] == Divide[Pi,2*EllipticK[(k)^2]]+Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Cos[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E3 22.11.E3] || <math qid="Q7027">\Jacobielldnk@{z}{k} = \frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{n}\cos@{2n\zeta}}{1+q^{2n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldnk@{z}{k} = \frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{n}\cos@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDN(z, k) = (Pi)/(2*EllipticK(k))+(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* cos(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDN[z, (k)^2] == Divide[Pi,2*EllipticK[(k)^2]]+Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Cos[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E4 22.11.E4] || [[Item:Q7028|<math>\Jacobiellcdk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcdk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCD(z, k) = (2*Pi)/(K*k)*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* cos((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCD[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Cos[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E4 22.11.E4] || <math qid="Q7028">\Jacobiellcdk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcdk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCD(z, k) = (2*Pi)/(K*k)*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* cos((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCD[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Cos[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E5 22.11.E5] || [[Item:Q7029|<math>\Jacobiellsdk@{z}{k} = \frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsdk@{z}{k} = \frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSD(z, k) = (2*Pi)/(K*k*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* sin((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSD[z, (k)^2] == Divide[2*Pi,K*k*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Sin[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E5 22.11.E5] || <math qid="Q7029">\Jacobiellsdk@{z}{k} = \frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsdk@{z}{k} = \frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSD(z, k) = (2*Pi)/(K*k*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* sin((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSD[z, (k)^2] == Divide[2*Pi,K*k*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Sin[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E6 22.11.E6] || [[Item:Q7030|<math>\Jacobiellndk@{z}{k} = \frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos@{2n\zeta}}{1+q^{2n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellndk@{z}{k} = \frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiND(z, k) = (Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))+(2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* cos(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiND[z, (k)^2] == Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]+Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Cos[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E6 22.11.E6] || <math qid="Q7030">\Jacobiellndk@{z}{k} = \frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos@{2n\zeta}}{1+q^{2n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellndk@{z}{k} = \frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiND(z, k) = (Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))+(2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* cos(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiND[z, (k)^2] == Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]+Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Cos[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E7 22.11.E7] || [[Item:Q7031|<math>\Jacobiellnsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellnsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiNS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta) = (2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* sin((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiNS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Csc[\[Zeta]] == Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Sin[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E7 22.11.E7] || <math qid="Q7031">\Jacobiellnsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellnsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiNS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta) = (2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* sin((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiNS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Csc[\[Zeta]] == Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Sin[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E8 22.11.E8] || [[Item:Q7032|<math>\Jacobielldsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = -\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = -\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta) = -(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* sin((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Csc[\[Zeta]] == -Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Sin[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E8 22.11.E8] || <math qid="Q7032">\Jacobielldsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = -\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = -\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta) = -(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* sin((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Csc[\[Zeta]] == -Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Sin[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E9 22.11.E9] || [[Item:Q7033|<math>\Jacobiellcsk@{z}{k}-\frac{\pi}{2K}\cot@@{\zeta} = -\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcsk@{z}{k}-\frac{\pi}{2K}\cot@@{\zeta} = -\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCS(z, k)-(Pi)/(2*EllipticK(k))*cot(zeta) = -(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)* sin(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Cot[\[Zeta]] == -Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)* Sin[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E9 22.11.E9] || <math qid="Q7033">\Jacobiellcsk@{z}{k}-\frac{\pi}{2K}\cot@@{\zeta} = -\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcsk@{z}{k}-\frac{\pi}{2K}\cot@@{\zeta} = -\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCS(z, k)-(Pi)/(2*EllipticK(k))*cot(zeta) = -(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)* sin(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCS[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Cot[\[Zeta]] == -Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)* Sin[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E10 22.11.E10] || [[Item:Q7034|<math>\Jacobielldck@{z}{k}-\frac{\pi}{2K}\sec@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldck@{z}{k}-\frac{\pi}{2K}\sec@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDC(z, k)-(Pi)/(2*EllipticK(k))*sec(zeta) = (2*Pi)/(EllipticK(k))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* cos((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Sec[\[Zeta]] == Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Cos[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E10 22.11.E10] || <math qid="Q7034">\Jacobielldck@{z}{k}-\frac{\pi}{2K}\sec@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldck@{z}{k}-\frac{\pi}{2K}\sec@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDC(z, k)-(Pi)/(2*EllipticK(k))*sec(zeta) = (2*Pi)/(EllipticK(k))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* cos((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]]*Sec[\[Zeta]] == Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Cos[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E11 22.11.E11] || [[Item:Q7035|<math>\Jacobiellnck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\sec@@{\zeta} = -\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellnck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\sec@@{\zeta} = -\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiNC(z, k)-(Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))*sec(zeta) = -(2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* cos((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiNC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sec[\[Zeta]] == -Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Cos[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E11 22.11.E11] || <math qid="Q7035">\Jacobiellnck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\sec@@{\zeta} = -\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellnck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\sec@@{\zeta} = -\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiNC(z, k)-(Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))*sec(zeta) = -(2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* cos((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiNC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sec[\[Zeta]] == -Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Cos[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E12 22.11.E12] || [[Item:Q7036|<math>\Jacobiellsck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\tan@@{\zeta} = \frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\tan@@{\zeta} = \frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSC(z, k)-(Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))*tan(zeta) = (2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)* sin(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Tan[\[Zeta]] == Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)* Sin[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E12 22.11.E12] || <math qid="Q7036">\Jacobiellsck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\tan@@{\zeta} = \frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\tan@@{\zeta} = \frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin@{2n\zeta}}{1+q^{2n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSC(z, k)-(Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))*tan(zeta) = (2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)* sin(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Tan[\[Zeta]] == Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)* Sin[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/22.11.E13 22.11.E13] || [[Item:Q7037|<math>\Jacobiellsnk^{2}@{z}{k} = \frac{1}{k^{2}}\left(1-\frac{\compellintEk@@{k}}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}\cos@{2n\zeta}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsnk^{2}@{z}{k} = \frac{1}{k^{2}}\left(1-\frac{\compellintEk@@{k}}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}\cos@{2n\zeta}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(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 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n))*cos(2*n*zeta), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(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 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)]*Cos[2*n*\[Zeta]], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/22.11.E13 22.11.E13] || <math qid="Q7037">\Jacobiellsnk^{2}@{z}{k} = \frac{1}{k^{2}}\left(1-\frac{\compellintEk@@{k}}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}\cos@{2n\zeta}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsnk^{2}@{z}{k} = \frac{1}{k^{2}}\left(1-\frac{\compellintEk@@{k}}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}\cos@{2n\zeta}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(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 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n))*cos(2*n*zeta), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(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 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)]*Cos[2*n*\[Zeta]], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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Latest revision as of 12:58, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.11.E1 sn ⁡ ( z , k ) = 2 ⁢ π K ⁢ k ⁢ ∑ n = 0 ∞ q n + 1 2 ⁢ sin ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 - q 2 ⁢ n + 1 Jacobi-elliptic-sn 𝑧 𝑘 2 𝜋 𝐾 𝑘 superscript subscript 𝑛 0 superscript 𝑞 𝑛 1 2 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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^{2% n+1}}}}
\Jacobiellsnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}		 

JacobiSN(z, k) = (2*Pi)/(K*k)*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* sin((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)

JacobiSN[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Sin[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E2 cn ⁡ ( z , k ) = 2 ⁢ π K ⁢ k ⁢ ∑ n = 0 ∞ q n + 1 2 ⁢ cos ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 Jacobi-elliptic-cn 𝑧 𝑘 2 𝜋 𝐾 𝑘 superscript subscript 𝑛 0 superscript 𝑞 𝑛 1 2 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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^{2% n+1}}}}
\Jacobiellcnk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}		 

JacobiCN(z, k) = (2*Pi)/(K*k)*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* cos((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)

JacobiCN[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Cos[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E3 dn ⁡ ( z , k ) = π 2 ⁢ K + 2 ⁢ π K ⁢ ∑ n = 1 ∞ q n ⁢ cos ⁡ ( 2 ⁢ n ⁢ ζ ) 1 + q 2 ⁢ n Jacobi-elliptic-dn 𝑧 𝑘 𝜋 2 𝐾 2 𝜋 𝐾 superscript subscript 𝑛 1 superscript 𝑞 𝑛 2 𝑛 𝜁 1 superscript 𝑞 2 𝑛 {\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}% }}}
\Jacobielldnk@{z}{k} = \frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{n}\cos@{2n\zeta}}{1+q^{2n}}		 

JacobiDN(z, k) = (Pi)/(2*EllipticK(k))+(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* cos(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)

JacobiDN[z, (k)^2] == Divide[Pi,2*EllipticK[(k)^2]]+Divide[2*Pi,EllipticK[(k)^2]]*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Cos[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E4 cd ⁡ ( z , k ) = 2 ⁢ π K ⁢ k ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ q n + 1 2 ⁢ cos ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 - q 2 ⁢ n + 1 Jacobi-elliptic-cd 𝑧 𝑘 2 𝜋 𝐾 𝑘 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 𝑛 1 2 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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}}}}
\Jacobiellcdk@{z}{k} = \frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}		 

JacobiCD(z, k) = (2*Pi)/(K*k)*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* cos((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)

JacobiCD[z, (k)^2] == Divide[2*Pi,K*k]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Cos[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E5 sd ⁡ ( z , k ) = 2 ⁢ π K ⁢ k ⁢ k ′ ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ q n + 1 2 ⁢ sin ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 Jacobi-elliptic-sd 𝑧 𝑘 2 𝜋 𝐾 𝑘 superscript 𝑘 ′ superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 𝑛 1 2 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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}}}}
\Jacobiellsdk@{z}{k} = \frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}		 

JacobiSD(z, k) = (2*Pi)/(K*k*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n +(1)/(2))* sin((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)

JacobiSD[z, (k)^2] == Divide[2*Pi,K*k*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n +Divide[1,2])* Sin[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E6 nd ⁡ ( z , k ) = π 2 ⁢ K ⁢ k ′ + 2 ⁢ π K ⁢ k ′ ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q n ⁢ cos ⁡ ( 2 ⁢ n ⁢ ζ ) 1 + q 2 ⁢ n Jacobi-elliptic-nd 𝑧 𝑘 𝜋 2 𝐾 superscript 𝑘 ′ 2 𝜋 𝐾 superscript 𝑘 ′ superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 𝑛 2 𝑛 𝜁 1 superscript 𝑞 2 𝑛 {\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}}}}
\Jacobiellndk@{z}{k} = \frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos@{2n\zeta}}{1+q^{2n}}		 

JacobiND(z, k) = (Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))+(2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* cos(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)

JacobiND[z, (k)^2] == Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]+Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Cos[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E7 ns ⁡ ( z , k ) - π 2 ⁢ K ⁢ csc ⁡ ζ = 2 ⁢ π K ⁢ ∑ n = 0 ∞ q 2 ⁢ n + 1 ⁢ sin ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 - q 2 ⁢ n + 1 Jacobi-elliptic-ns 𝑧 𝑘 𝜋 2 𝐾 𝜁 2 𝜋 𝐾 superscript subscript 𝑛 0 superscript 𝑞 2 𝑛 1 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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}}}}
\Jacobiellnsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1-q^{2n+1}}		 

JacobiNS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta) = (2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* sin((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(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[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Sin[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E8 ds ⁡ ( z , k ) - π 2 ⁢ K ⁢ csc ⁡ ζ = - 2 ⁢ π K ⁢ ∑ n = 0 ∞ q 2 ⁢ n + 1 ⁢ sin ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 Jacobi-elliptic-ds 𝑧 𝑘 𝜋 2 𝐾 𝜁 2 𝜋 𝐾 superscript subscript 𝑛 0 superscript 𝑞 2 𝑛 1 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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}}}}
\Jacobielldsk@{z}{k}-\frac{\pi}{2K}\csc@@{\zeta} = -\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin@{(2n+1)\zeta}}{1+q^{2n+1}}		 

JacobiDS(z, k)-(Pi)/(2*EllipticK(k))*csc(zeta) = -(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* sin((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(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[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Sin[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E9 cs ⁡ ( z , k ) - π 2 ⁢ K ⁢ cot ⁡ ζ = - 2 ⁢ π K ⁢ ∑ n = 1 ∞ q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ ζ ) 1 + q 2 ⁢ n Jacobi-elliptic-cs 𝑧 𝑘 𝜋 2 𝐾 𝜁 2 𝜋 𝐾 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝜁 1 superscript 𝑞 2 𝑛 {\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}}}}
\Jacobiellcsk@{z}{k}-\frac{\pi}{2K}\cot@@{\zeta} = -\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{2n}\sin@{2n\zeta}}{1+q^{2n}}		 

JacobiCS(z, k)-(Pi)/(2*EllipticK(k))*cot(zeta) = -(2*Pi)/(EllipticK(k))*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)* sin(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(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[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)* Sin[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E10 dc ⁡ ( z , k ) - π 2 ⁢ K ⁢ sec ⁡ ζ = 2 ⁢ π K ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ q 2 ⁢ n + 1 ⁢ cos ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 - q 2 ⁢ n + 1 Jacobi-elliptic-dc 𝑧 𝑘 𝜋 2 𝐾 𝜁 2 𝜋 𝐾 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 2 𝑛 1 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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}}}}
\Jacobielldck@{z}{k}-\frac{\pi}{2K}\sec@@{\zeta} = \frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1-q^{2n+1}}		 

JacobiDC(z, k)-(Pi)/(2*EllipticK(k))*sec(zeta) = (2*Pi)/(EllipticK(k))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* cos((2*n + 1)*zeta))/(1 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(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)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Cos[(2*n + 1)*\[Zeta]],1 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E11 nc ⁡ ( z , k ) - π 2 ⁢ K ⁢ k ′ ⁢ sec ⁡ ζ = - 2 ⁢ π K ⁢ k ′ ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ q 2 ⁢ n + 1 ⁢ cos ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 Jacobi-elliptic-nc 𝑧 𝑘 𝜋 2 𝐾 superscript 𝑘 ′ 𝜁 2 𝜋 𝐾 superscript 𝑘 ′ superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 2 𝑛 1 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\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}}}}
\Jacobiellnck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\sec@@{\zeta} = -\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos@{(2n+1)\zeta}}{1+q^{2n+1}}		 

JacobiNC(z, k)-(Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))*sec(zeta) = -(2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)* cos((2*n + 1)*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n + 1)), n = 0..infinity)

JacobiNC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sec[\[Zeta]] == -Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)* Cos[(2*n + 1)*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E12 sc ⁡ ( z , k ) - π 2 ⁢ K ⁢ k ′ ⁢ tan ⁡ ζ = 2 ⁢ π K ⁢ k ′ ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ ζ ) 1 + q 2 ⁢ n Jacobi-elliptic-sc 𝑧 𝑘 𝜋 2 𝐾 superscript 𝑘 ′ 𝜁 2 𝜋 𝐾 superscript 𝑘 ′ superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 2 𝑛 2 𝑛 𝜁 1 superscript 𝑞 2 𝑛 {\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}}}}
\Jacobiellsck@{z}{k}-\frac{\pi}{2Kk^{\prime}}\tan@@{\zeta} = \frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin@{2n\zeta}}{1+q^{2n}}		 

JacobiSC(z, k)-(Pi)/(2*EllipticK(k)*sqrt(1 - (k)^(2)))*tan(zeta) = (2*Pi)/(EllipticK(k)*sqrt(1 - (k)^(2)))*sum(((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)* sin(2*n*zeta))/(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n)), n = 1..infinity)

JacobiSC[z, (k)^2]-Divide[Pi,2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Tan[\[Zeta]] == Divide[2*Pi,EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]]*Sum[Divide[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)* Sin[2*n*\[Zeta]],1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.11.E13 sn 2 ⁡ ( z , k ) = 1 k 2 ⁢ ( 1 - E ⁡ K ) - 2 ⁢ π 2 k 2 ⁢ K 2 ⁢ ∑ n = 1 ∞ n ⁢ q n 1 - q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ n ⁢ ζ ) Jacobi-elliptic-sn 2 𝑧 𝑘 1 superscript 𝑘 2 1 complete-elliptic-integral-second-kind-E 𝑘 𝐾 2 superscript 𝜋 2 superscript 𝑘 2 superscript 𝐾 2 superscript subscript 𝑛 1 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝜁 {\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)}}
\Jacobiellsnk^{2}@{z}{k} = \frac{1}{k^{2}}\left(1-\frac{\compellintEk@@{k}}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}\cos@{2n\zeta}		 

(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 -(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(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 -(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n)]*Cos[2*n*\[Zeta]], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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