Jacobian Elliptic Functions - 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.12.E1 τ = i ⁢ K ′ ⁡ ( k ) / K ⁡ ( k ) 𝜏 𝑖 complementary-complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\tau=i{K^{\prime}}\left(k\right)/K\left(k\right)}}
\tau = i\ccompellintKk@{k}/\compellintKk@{k}		 

tau = I*EllipticCK(k)/EllipticK(k)

\[Tau] == I*EllipticK[1-(k)^2]/EllipticK[(k)^2]
Failure Failure Error
Failed [30 / 30]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.4867361401447923, 0.0147898206680519]
Test Values: {Rule[k, 2], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.12.E2 2 ⁢ K ⁢ k ⁢ sn ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) 2 𝐾 𝑘 Jacobi-elliptic-sn 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}}}
2Kk\Jacobiellsnk@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t-(n+\frac{1}{2})\tau)}}		 

2*K*k*JacobiSN(2*K*t, k) = sum((Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity)

2*K*k*JacobiSN[2*K*t, (k)^2] == Sum[Divide[Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E2 ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t - m - ( n + 1 2 ) ⁢ τ ) superscript subscript 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(% t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}% ^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right)}}
\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t-(n+\frac{1}{2})\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right)		 

sum((Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m))/(t - m -(n +(1)/(2))*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m),t - m -(n +Divide[1,2])*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Successful Aborted Skip - symbolical successful subtest Skipped - Because timed out
22.12.E3 2 ⁢ i ⁢ K ⁢ k ⁢ cn ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) 2 𝑖 𝐾 𝑘 Jacobi-elliptic-cn 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2iKk\operatorname{cn}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)% }}}
2iKk\Jacobiellcnk@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t-(n+\frac{1}{2})\tau)}}		 

2*I*K*k*JacobiCN(2*K*t, k) = sum(((- 1)^(n)* Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity)

2*I*K*k*JacobiCN[2*K*t, (k)^2] == Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
22.12.E3 ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m + n t - m - ( n + 1 2 ) ⁢ τ ) superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑛 𝑡 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin% \left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{% m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-(n+\frac{1}{2})\tau}\right)}}
\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t-(n+\frac{1}{2})\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-(n+\frac{1}{2})\tau}\right)		 

sum(((- 1)^(n)* Pi)/(sin(Pi*(t -(n +(1)/(2))*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m + n))/(t - m -(n +(1)/(2))*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m + n),t - m -(n +Divide[1,2])*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E4 2 ⁢ i ⁢ K ⁢ dn ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) 2 𝑖 𝐾 Jacobi-elliptic-dn 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2iK\operatorname{dn}\left(2Kt,k\right)=\lim_{N\to% \infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t-(n+\frac{1}{2})\tau)% \right)}}}
2iK\Jacobielldnk@{2Kt}{k} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t-(n+\frac{1}{2})\tau)}}		 

2*I*EllipticK(k)*JacobiDN(2*K*t, k) = limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t -(n +(1)/(2))*tau))), n = - N..N), N = infinity)

2*I*EllipticK[(k)^2]*JacobiDN[2*K*t, (k)^2] == Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E4 lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ ( lim M → ∞ ⁡ ∑ m = - M M 1 t - m - ( n + 1 2 ) ⁢ τ ) subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}% {\tan\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}% (-1)^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-(n+\frac{1}{2})\tau% }\right)}}
\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t-(n+\frac{1}{2})\tau)}} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-(n+\frac{1}{2})\tau}\right)		 

limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t -(n +(1)/(2))*tau))), n = - N..N), N = infinity) = limit(sum((- 1)^(n)*(limit(sum((1)/(t - m -(n +(1)/(2))*tau), m = - M..M), M = infinity)), n = - N..N), N = infinity)

Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t -(n +Divide[1,2])*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None] == Limit[Sum[(- 1)^(n)*(Limit[Sum[Divide[1,t - m -(n +Divide[1,2])*\[Tau]], {m, - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None]), {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E5 2 ⁢ K ⁢ k ⁢ cd ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) 2 𝐾 𝑘 Jacobi-elliptic-cd 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2Kk\operatorname{cd}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)% \right)}}}
2Kk\Jacobiellcdk@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)}}		 

2*K*k*JacobiCD(2*K*t, k) = sum((Pi)/(sin(Pi*(t +(1)/(2)-(n +(1)/(2))*tau))), n = - infinity..infinity)

2*K*k*JacobiCD[2*K*t, (k)^2] == Sum[Divide[Pi,Sin[Pi*(t +Divide[1,2]-(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E5 ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t + 1 2 - m - ( n + 1 2 ) ⁢ τ ) superscript subscript 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 1 2 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(% t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum% _{m=-\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}% \right)}}
\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right)		 

sum((Pi)/(sin(Pi*(t +(1)/(2)-(n +(1)/(2))*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m))/(t +(1)/(2)- m -(n +(1)/(2))*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[Pi,Sin[Pi*(t +Divide[1,2]-(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m),t +Divide[1,2]- m -(n +Divide[1,2])*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E6 - 2 ⁢ i ⁢ K ⁢ k ⁢ k ′ ⁢ sd ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) 2 𝑖 𝐾 𝑘 superscript 𝑘 ′ Jacobi-elliptic-sd 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 {\displaystyle{\displaystyle-2iKkk^{\prime}\operatorname{sd}\left(2Kt,k\right)% =\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-(n+% \frac{1}{2})\tau)\right)}}}
-2iKkk^{\prime}\Jacobiellsdk@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)}}		 

- 2*I*K*k*sqrt(1 - (k)^(2))*JacobiSD(2*K*t, k) = sum(((- 1)^(n)* Pi)/(sin(Pi*(t +(1)/(2)-(n +(1)/(2))*tau))), n = - infinity..infinity)

- 2*I*K*k*Sqrt[1 - (k)^(2)]*JacobiSD[2*K*t, (k)^2] == Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t +Divide[1,2]-(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E6 ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m + n t + 1 2 - m - ( n + 1 2 ) ⁢ τ ) superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑛 𝑡 1 2 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin% \left(\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}% \left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-(n+\frac{1}{2% })\tau}\right)}}
\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right)		 

sum(((- 1)^(n)* Pi)/(sin(Pi*(t +(1)/(2)-(n +(1)/(2))*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m + n))/(t +(1)/(2)- m -(n +(1)/(2))*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t +Divide[1,2]-(n +Divide[1,2])*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m + n),t +Divide[1,2]- m -(n +Divide[1,2])*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E7 2 ⁢ i ⁢ K ⁢ k ′ ⁢ nd ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) 2 𝑖 𝐾 superscript 𝑘 ′ Jacobi-elliptic-nd 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2iKk^{\prime}\operatorname{nd}\left(2Kt,k\right)=% \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t+\frac{1}{2}% -(n+\frac{1}{2})\tau)\right)}}}
2iKk^{\prime}\Jacobiellndk@{2Kt}{k} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)}}		 

2*I*EllipticK(k)*sqrt(1 - (k)^(2))*JacobiND(2*K*t, k) = limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t +(1)/(2)-(n +(1)/(2))*tau))), n = - N..N), N = infinity)

2*I*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]*JacobiND[2*K*t, (k)^2] == Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t +Divide[1,2]-(n +Divide[1,2])*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E7 lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ lim M → ∞ ⁡ ( ∑ m = - M M 1 t + 1 2 - m - ( n + 1 2 ) ⁢ τ ) subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 1 2 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}% {\tan\left(\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)\right)}=\lim_{N\to\infty}% \sum_{n=-N}^{N}(-1)^{n}\lim_{M\to\infty}\left(\sum_{m=-M}^{M}\frac{1}{t+\frac{% 1}{2}-m-(n+\frac{1}{2})\tau}\right)}}
\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t+\frac{1}{2}-(n+\frac{1}{2})\tau)}} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\lim_{M\to\infty}\left(\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right)		 

limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t +(1)/(2)-(n +(1)/(2))*tau))), n = - N..N), N = infinity) = limit(sum((- 1)^(n)* limit(sum((1)/(t +(1)/(2)- m -(n +(1)/(2))*tau), m = - M..M), M = infinity), n = - N..N), N = infinity)

Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t +Divide[1,2]-(n +Divide[1,2])*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None] == Limit[Sum[(- 1)^(n)* Limit[Sum[Divide[1,t +Divide[1,2]- m -(n +Divide[1,2])*\[Tau]], {m, - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E8 2 ⁢ K ⁢ dc ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) 2 𝐾 Jacobi-elliptic-dc 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{dc}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)\right)}}}
2K\Jacobielldck@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t+\frac{1}{2}-n\tau)}}		 

2*EllipticK(k)*JacobiDC(2*K*t, k) = sum((Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity)

2*EllipticK[(k)^2]*JacobiDC[2*K*t, (k)^2] == Sum[Divide[Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E8 ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t + 1 2 - m - n ⁢ τ ) superscript subscript 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 1 2 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(% t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right)}}
\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t+\frac{1}{2}-n\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right)		 

sum((Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m))/(t +(1)/(2)- m - n*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m),t +Divide[1,2]- m - n*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Successful Aborted Skip - symbolical successful subtest Skipped - Because timed out
22.12.E9 2 ⁢ K ⁢ k ′ ⁢ nc ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) 2 𝐾 superscript 𝑘 ′ Jacobi-elliptic-nc 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 {\displaystyle{\displaystyle 2Kk^{\prime}\operatorname{nc}\left(2Kt,k\right)=% \sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)% \right)}}}
2Kk^{\prime}\Jacobiellnck@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t+\frac{1}{2}-n\tau)}}		 

2*EllipticK(k)*sqrt(1 - (k)^(2))*JacobiNC(2*K*t, k) = sum(((- 1)^(n)* Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity)

2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]*JacobiNC[2*K*t, (k)^2] == Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
22.12.E9 ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m + n t + 1 2 - m - n ⁢ τ ) superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑛 𝑡 1 2 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin% \left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=% -\infty}^{\infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-n\tau}\right)}}
\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t+\frac{1}{2}-n\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-n\tau}\right)		 

sum(((- 1)^(n)* Pi)/(sin(Pi*(t +(1)/(2)- n*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m + n))/(t +(1)/(2)- m - n*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m + n),t +Divide[1,2]- m - n*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Successful Aborted Skip - symbolical successful subtest Skipped - Because timed out
22.12.E10 - 2 ⁢ K ⁢ k ′ ⁢ sc ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) 2 𝐾 superscript 𝑘 ′ Jacobi-elliptic-sc 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 {\displaystyle{\displaystyle-2Kk^{\prime}\operatorname{sc}\left(2Kt,k\right)=% \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t+\frac{1}{2}% -n\tau)\right)}}}
-2Kk^{\prime}\Jacobiellsck@{2Kt}{k} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t+\frac{1}{2}-n\tau)}}		 

- 2*EllipticK(k)*sqrt(1 - (k)^(2))*JacobiSC(2*K*t, k) = limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t +(1)/(2)- n*tau))), n = - N..N), N = infinity)

- 2*EllipticK[(k)^2]*Sqrt[1 - (k)^(2)]*JacobiSC[2*K*t, (k)^2] == Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E10 lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ ( lim M → ∞ ⁡ ∑ m = - M M 1 t + 1 2 - m - n ⁢ τ ) subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 1 2 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}% {\tan\left(\pi(t+\frac{1}{2}-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-% 1)^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-n\tau}% \right)}}
\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t+\frac{1}{2}-n\tau)}} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-n\tau}\right)		 

limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t +(1)/(2)- n*tau))), n = - N..N), N = infinity) = limit(sum((- 1)^(n)*(limit(sum((1)/(t +(1)/(2)- m - n*tau), m = - M..M), M = infinity)), n = - N..N), N = infinity)

Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t +Divide[1,2]- n*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None] == Limit[Sum[(- 1)^(n)*(Limit[Sum[Divide[1,t +Divide[1,2]- m - n*\[Tau]], {m, - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None]), {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E11 2 ⁢ K ⁢ ns ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t - n ⁢ τ ) ) 2 𝐾 Jacobi-elliptic-ns 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-n\tau)\right)}}}
2K\Jacobiellnsk@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t-n\tau)}}		 

2*EllipticK(k)*JacobiNS(2*K*t, k) = sum((Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity)

2*EllipticK[(k)^2]*JacobiNS[2*K*t, (k)^2] == Sum[Divide[Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E11 ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t - m - n ⁢ τ ) superscript subscript 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin\left(\pi(% t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac% {(-1)^{m}}{t-m-n\tau}\right)}}
\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin@{\pi(t-n\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-n\tau}\right)		 

sum((Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m))/(t - m - n*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m),t - m - n*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Successful Aborted Skip - symbolical successful subtest Skipped - Because timed out
22.12.E12 2 ⁢ K ⁢ ds ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t - n ⁢ τ ) ) 2 𝐾 Jacobi-elliptic-ds 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{ds}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-n\tau)\right)}}}
2K\Jacobielldsk@{2Kt}{k} = \sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t-n\tau)}}		 

2*EllipticK(k)*JacobiDS(2*K*t, k) = sum(((- 1)^(n)* Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity)

2*EllipticK[(k)^2]*JacobiDS[2*K*t, (k)^2] == Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
22.12.E12 ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m + n t - m - n ⁢ τ ) superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑛 𝑡 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin% \left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m+n}}{t-m-n\tau}\right)}}
\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin@{\pi(t-n\tau)}} = \sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-n\tau}\right)		 

sum(((- 1)^(n)* Pi)/(sin(Pi*(t - n*tau))), n = - infinity..infinity) = sum(sum(((- 1)^(m + n))/(t - m - n*tau), m = - infinity..infinity), n = - infinity..infinity)

Sum[Divide[(- 1)^(n)* Pi,Sin[Pi*(t - n*\[Tau])]], {n, - Infinity, Infinity}, GenerateConditions->None] == Sum[Sum[Divide[(- 1)^(m + n),t - m - n*\[Tau]], {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E13 2 ⁢ K ⁢ cs ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t - n ⁢ τ ) ) 2 𝐾 Jacobi-elliptic-cs 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to% \infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t-n\tau)\right)}}}
2K\Jacobiellcsk@{2Kt}{k} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t-n\tau)}}		 

2*EllipticK(k)*JacobiCS(2*K*t, k) = limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t - n*tau))), n = - N..N), N = infinity)

2*EllipticK[(k)^2]*JacobiCS[2*K*t, (k)^2] == Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t - n*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.12.E13 lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t - n ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ ( lim M → ∞ ⁡ ∑ m = - M M 1 t - m - n ⁢ τ ) subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}% {\tan\left(\pi(t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(% \lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right)}}
\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan@{\pi(t-n\tau)}} = \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right)		 

limit(sum((- 1)^(n)*(Pi)/(tan(Pi*(t - n*tau))), n = - N..N), N = infinity) = limit(sum((- 1)^(n)*(limit(sum((1)/(t - m - n*tau), m = - M..M), M = infinity)), n = - N..N), N = infinity)

Limit[Sum[(- 1)^(n)*Divide[Pi,Tan[Pi*(t - n*\[Tau])]], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None] == Limit[Sum[(- 1)^(n)*(Limit[Sum[Divide[1,t - m - n*\[Tau]], {m, - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None]), {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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