Theta Functions - 20.5 Infinite Products and Related Results

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20.5.E5 θ 1 ⁡ ( z | τ ) = θ 1 ′ ⁡ ( 0 | τ ) ⁢ sin ⁡ z ⁢ ∏ n = 1 ∞ sin ⁡ ( n ⁢ π ⁢ τ + z ) ⁢ sin ⁡ ( n ⁢ π ⁢ τ - z ) sin 2 ⁡ ( n ⁢ π ⁢ τ ) Jacobi-theta-tau 1 𝑧 𝜏 diffop Jacobi-theta-tau 1 1 0 𝜏 𝑧 superscript subscript product 𝑛 1 𝑛 𝜋 𝜏 𝑧 𝑛 𝜋 𝜏 𝑧 2 𝑛 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=\theta_{1}'% \left(0\middle|\tau\right)\sin z\prod_{n=1}^{\infty}\frac{\sin\left(n\pi\tau+z% \right)\sin\left(n\pi\tau-z\right)}{{\sin^{2}}\left(n\pi\tau\right)}}}
\Jacobithetatau{1}@{z}{\tau} = \Jacobithetatau{1}'@{0}{\tau}\sin@@{z}\prod_{n=1}^{\infty}\frac{\sin@{n\pi\tau+z}\sin@{n\pi\tau-z}}{\sin^{2}@{n\pi\tau}}		 

JacobiTheta1(z,exp(I*Pi*tau)) = diff( JacobiTheta1(0,exp(I*Pi*tau)), 0$(1) )*sin(z)*product((sin(n*Pi*tau + z)*sin(n*Pi*tau - z))/((sin(n*Pi*tau))^(2)), n = 1..infinity)

EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == D[EllipticTheta[1, 0, Exp[I*Pi*(\[Tau])]], {0, 1}]*Sin[z]*Product[Divide[Sin[n*Pi*\[Tau]+ z]*Sin[n*Pi*\[Tau]- z],(Sin[n*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
20.5.E6 θ 2 ⁡ ( z | τ ) = θ 2 ⁡ ( 0 | τ ) ⁢ cos ⁡ z ⁢ ∏ n = 1 ∞ cos ⁡ ( n ⁢ π ⁢ τ + z ) ⁢ cos ⁡ ( n ⁢ π ⁢ τ - z ) cos 2 ⁡ ( n ⁢ π ⁢ τ ) Jacobi-theta-tau 2 𝑧 𝜏 Jacobi-theta-tau 2 0 𝜏 𝑧 superscript subscript product 𝑛 1 𝑛 𝜋 𝜏 𝑧 𝑛 𝜋 𝜏 𝑧 2 𝑛 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{2}% \left(0\middle|\tau\right)\cos z\prod_{n=1}^{\infty}\frac{\cos\left(n\pi\tau+z% \right)\cos\left(n\pi\tau-z\right)}{{\cos^{2}}\left(n\pi\tau\right)}}}
\Jacobithetatau{2}@{z}{\tau} = \Jacobithetatau{2}@{0}{\tau}\cos@@{z}\prod_{n=1}^{\infty}\frac{\cos@{n\pi\tau+z}\cos@{n\pi\tau-z}}{\cos^{2}@{n\pi\tau}}		 

JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta2(0,exp(I*Pi*tau))*cos(z)*product((cos(n*Pi*tau + z)*cos(n*Pi*tau - z))/((cos(n*Pi*tau))^(2)), n = 1..infinity)

EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]]*Cos[z]*Product[Divide[Cos[n*Pi*\[Tau]+ z]*Cos[n*Pi*\[Tau]- z],(Cos[n*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Error Skipped - Because timed out
20.5.E7 θ 3 ⁡ ( z | τ ) = θ 3 ⁡ ( 0 | τ ) ⁢ ∏ n = 1 ∞ cos ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ + z ) ⁢ cos ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ - z ) cos 2 ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ ) Jacobi-theta-tau 3 𝑧 𝜏 Jacobi-theta-tau 3 0 𝜏 superscript subscript product 𝑛 1 𝑛 1 2 𝜋 𝜏 𝑧 𝑛 1 2 𝜋 𝜏 𝑧 2 𝑛 1 2 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{3}% \left(0\middle|\tau\right)\prod_{n=1}^{\infty}\frac{\cos\left((n-\tfrac{1}{2})% \pi\tau+z\right)\cos\left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\cos^{2}}\left((n% -\tfrac{1}{2})\pi\tau\right)}}}
\Jacobithetatau{3}@{z}{\tau} = \Jacobithetatau{3}@{0}{\tau}\prod_{n=1}^{\infty}\frac{\cos@{(n-\tfrac{1}{2})\pi\tau+z}\cos@{(n-\tfrac{1}{2})\pi\tau-z}}{\cos^{2}@{(n-\tfrac{1}{2})\pi\tau}}		 

JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta3(0,exp(I*Pi*tau))*product((cos((n -(1)/(2))*Pi*tau + z)*cos((n -(1)/(2))*Pi*tau - z))/((cos((n -(1)/(2))*Pi*tau))^(2)), n = 1..infinity)

EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]]*Product[Divide[Cos[(n -Divide[1,2])*Pi*\[Tau]+ z]*Cos[(n -Divide[1,2])*Pi*\[Tau]- z],(Cos[(n -Divide[1,2])*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Error Skipped - Because timed out
20.5.E8 θ 4 ⁡ ( z | τ ) = θ 4 ⁡ ( 0 | τ ) ⁢ ∏ n = 1 ∞ sin ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ + z ) ⁢ sin ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ - z ) sin 2 ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ ) Jacobi-theta-tau 4 𝑧 𝜏 Jacobi-theta-tau 4 0 𝜏 superscript subscript product 𝑛 1 𝑛 1 2 𝜋 𝜏 𝑧 𝑛 1 2 𝜋 𝜏 𝑧 2 𝑛 1 2 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{4}% \left(0\middle|\tau\right)\prod_{n=1}^{\infty}\frac{\sin\left((n-\tfrac{1}{2})% \pi\tau+z\right)\sin\left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\sin^{2}}\left((n% -\tfrac{1}{2})\pi\tau\right)}}}
\Jacobithetatau{4}@{z}{\tau} = \Jacobithetatau{4}@{0}{\tau}\prod_{n=1}^{\infty}\frac{\sin@{(n-\tfrac{1}{2})\pi\tau+z}\sin@{(n-\tfrac{1}{2})\pi\tau-z}}{\sin^{2}@{(n-\tfrac{1}{2})\pi\tau}}		 

JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta4(0,exp(I*Pi*tau))*product((sin((n -(1)/(2))*Pi*tau + z)*sin((n -(1)/(2))*Pi*tau - z))/((sin((n -(1)/(2))*Pi*tau))^(2)), n = 1..infinity)

EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, 0, Exp[I*Pi*(\[Tau])]]*Product[Divide[Sin[(n -Divide[1,2])*Pi*\[Tau]+ z]*Sin[(n -Divide[1,2])*Pi*\[Tau]- z],(Sin[(n -Divide[1,2])*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Error Skipped - Because timed out
20.5.E9 θ 3 ⁡ ( π ⁢ z | τ ) = ∑ n = - ∞ ∞ p 2 ⁢ n ⁢ q n 2 Jacobi-theta-tau 3 𝜋 𝑧 𝜏 superscript subscript 𝑛 superscript 𝑝 2 𝑛 superscript 𝑞 superscript 𝑛 2 {\displaystyle{\displaystyle\theta_{3}\left(\pi z\middle|\tau\right)=\sum_{n=-% \infty}^{\infty}p^{2n}q^{n^{2}}\\ }}
\Jacobithetatau{3}@{\pi z}{\tau} = \sum_{n=-\infty}^{\infty}p^{2n}q^{n^{2}}\\		 

JacobiTheta3(Pi*z,exp(I*Pi*tau)) = sum((p)^(2*n)* (q)^((n)^(2)), n = - infinity..infinity)

EllipticTheta[3, Pi*z, Exp[I*Pi*(\[Tau])]] == Sum[(p)^(2*n)* (q)^((n)^(2)), {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [140 / 300]
Result: Indeterminate
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.5.E9 ∑ n = - ∞ ∞ p 2 ⁢ n ⁢ q n 2 = ∏ n = 1 ∞ ( 1 - q 2 ⁢ n ) ⁢ ( 1 + q 2 ⁢ n - 1 ⁢ p 2 ) ⁢ ( 1 + q 2 ⁢ n - 1 ⁢ p - 2 ) superscript subscript 𝑛 superscript 𝑝 2 𝑛 superscript 𝑞 superscript 𝑛 2 superscript subscript product 𝑛 1 1 superscript 𝑞 2 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 𝑝 2 1 superscript 𝑞 2 𝑛 1 superscript 𝑝 2 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}p^{2n}q^{n^{2}}\\ =\prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+% q^{2n-1}p^{-2}\right)}}
\sum_{n=-\infty}^{\infty}p^{2n}q^{n^{2}}\\ = \prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+q^{2n-1}p^{-2}\right)		 

sum((p)^(2*n)* (q)^((n)^(2)), n = - infinity..infinity) = product((1 - (q)^(2*n))*(1 + (q)^(2*n - 1)* (p)^(2))*(1 + (q)^(2*n - 1)* (p)^(- 2)), n = 1..infinity)

Sum[(p)^(2*n)* (q)^((n)^(2)), {n, - Infinity, Infinity}, GenerateConditions->None] == Product[(1 - (q)^(2*n))*(1 + (q)^(2*n - 1)* (p)^(2))*(1 + (q)^(2*n - 1)* (p)^(- 2)), {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out
Failed [20 / 100]
Result: DirectedInfinity[]
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
20.5.E10 θ 1 ′ ⁡ ( z , q ) θ 1 ⁡ ( z , q ) - cot ⁡ z = 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - 2 ⁢ q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n diffop Jacobi-theta 1 1 𝑧 𝑞 Jacobi-theta 1 𝑧 𝑞 𝑧 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 2 superscript 𝑞 2 𝑛 2 𝑧 superscript 𝑞 4 𝑛 {\displaystyle{\displaystyle\frac{\theta_{1}'\left(z,q\right)}{\theta_{1}\left% (z,q\right)}-\cot z=4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^% {2n}\cos\left(2z\right)+q^{4n}}}}
\frac{\Jacobithetaq{1}'@{z}{q}}{\Jacobithetaq{1}@{z}{q}}-\cot@@{z} = 4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\cos@{2z}+q^{4n}}		 

(diff( JacobiTheta1(z, q), z$(1) ))/(JacobiTheta1(z, q))- cot(z) = 4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity)

Divide[D[EllipticTheta[1, z, q], {z, 1}],EllipticTheta[1, z, q]]- Cot[z] == 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
20.5.E10 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - 2 ⁢ q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n = 4 ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 2 superscript 𝑞 2 𝑛 2 𝑧 superscript 𝑞 4 𝑛 4 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle 4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n}}{1-2q^{2n}\cos\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q% ^{2n}}\sin\left(2nz\right)}}
4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\cos@{2z}+q^{4n}} = 4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}}\sin@{2nz}		 

4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity) = 4*sum(((q)^(2*n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)

4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
20.5.E11 θ 2 ′ ⁡ ( z , q ) θ 2 ⁡ ( z , q ) + tan ⁡ z = - 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 + 2 ⁢ q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n diffop Jacobi-theta 2 1 𝑧 𝑞 Jacobi-theta 2 𝑧 𝑞 𝑧 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 2 superscript 𝑞 2 𝑛 2 𝑧 superscript 𝑞 4 𝑛 {\displaystyle{\displaystyle\frac{\theta_{2}'\left(z,q\right)}{\theta_{2}\left% (z,q\right)}+\tan z=-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q% ^{2n}\cos\left(2z\right)+q^{4n}}}}
\frac{\Jacobithetaq{2}'@{z}{q}}{\Jacobithetaq{2}@{z}{q}}+\tan@@{z} = -4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\cos@{2z}+q^{4n}}		 

(diff( JacobiTheta2(z, q), z$(1) ))/(JacobiTheta2(z, q))+ tan(z) = - 4*sin(2*z)*sum(((q)^(2*n))/(1 + 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity)

Divide[D[EllipticTheta[2, z, q], {z, 1}],EllipticTheta[2, z, q]]+ Tan[z] == - 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 + 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
20.5.E11 - 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 + 2 ⁢ q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n = 4 ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q 2 ⁢ n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 2 superscript 𝑞 2 𝑛 2 𝑧 superscript 𝑞 4 𝑛 4 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 2 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n}}{1+2q^{2n}\cos\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{% 2n}}{1-q^{2n}}\sin\left(2nz\right)}}
-4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\cos@{2z}+q^{4n}} = 4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{2n}}{1-q^{2n}}\sin@{2nz}		 

- 4*sin(2*z)*sum(((q)^(2*n))/(1 + 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity) = 4*sum((- 1)^(n)*((q)^(2*n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)

- 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 + 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[(- 1)^(n)*Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
20.5.E12 θ 3 ′ ⁡ ( z , q ) θ 3 ⁡ ( z , q ) = - 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 1 + 2 ⁢ q 2 ⁢ n - 1 ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n - 2 diffop Jacobi-theta 3 1 𝑧 𝑞 Jacobi-theta 3 𝑧 𝑞 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 {\displaystyle{\displaystyle\frac{\theta_{3}'\left(z,q\right)}{\theta_{3}\left% (z,q\right)}=-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-% 1}\cos\left(2z\right)+q^{4n-2}}}}
\frac{\Jacobithetaq{3}'@{z}{q}}{\Jacobithetaq{3}@{z}{q}} = -4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\cos@{2z}+q^{4n-2}}		 

(diff( JacobiTheta3(z, q), z$(1) ))/(JacobiTheta3(z, q)) = - 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity)

Divide[D[EllipticTheta[3, z, q], {z, 1}],EllipticTheta[3, z, q]] == - 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
20.5.E12 - 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 1 + 2 ⁢ q 2 ⁢ n - 1 ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n - 2 = 4 ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 4 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n-1}}{1+2q^{2n-1}\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}(-1)^{n}% \frac{q^{n}}{1-q^{2n}}\sin\left(2nz\right)}}
-4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\cos@{2z}+q^{4n-2}} = 4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{2n}}\sin@{2nz}		 

- 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity) = 4*sum((- 1)^(n)*((q)^(n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)

- 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[(- 1)^(n)*Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
20.5.E13 θ 4 ′ ⁡ ( z , q ) θ 4 ⁡ ( z , q ) = 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 1 - 2 ⁢ q 2 ⁢ n - 1 ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n - 2 diffop Jacobi-theta 4 1 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 {\displaystyle{\displaystyle\frac{\theta_{4}'\left(z,q\right)}{\theta_{4}\left% (z,q\right)}=4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1% }\cos\left(2z\right)+q^{4n-2}}}}
\frac{\Jacobithetaq{4}'@{z}{q}}{\Jacobithetaq{4}@{z}{q}} = 4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\cos@{2z}+q^{4n-2}}		 

(diff( JacobiTheta4(z, q), z$(1) ))/(JacobiTheta4(z, q)) = 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 - 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity)

Divide[D[EllipticTheta[4, z, q], {z, 1}],EllipticTheta[4, z, q]] == 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 - 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
20.5.E13 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 1 - 2 ⁢ q 2 ⁢ n - 1 ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n - 2 = 4 ⁢ ∑ n = 1 ∞ q n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 4 superscript subscript 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle 4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n-1}}{1-2q^{2n-1}\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}\frac{q^{n}% }{1-q^{2n}}\sin\left(2nz\right)}}
4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\cos@{2z}+q^{4n-2}} = 4\sum_{n=1}^{\infty}\frac{q^{n}}{1-q^{2n}}\sin@{2nz}		 

4*sin(2*z)*sum(((q)^(2*n - 1))/(1 - 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity) = 4*sum(((q)^(n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)

4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 - 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
20.5.E15 θ 2 ⁡ ( z | τ ) = θ 2 ⁡ ( 0 | τ ) ⁢ lim N → ∞ ⁡ ∏ n = - N N lim M → ∞ ⁡ ∏ m = 1 - M M ( 1 + z ( m - 1 2 + n ⁢ τ ) ⁢ π ) Jacobi-theta-tau 2 𝑧 𝜏 Jacobi-theta-tau 2 0 𝜏 subscript → 𝑁 superscript subscript product 𝑛 𝑁 𝑁 subscript → 𝑀 superscript subscript product 𝑚 1 𝑀 𝑀 1 𝑧 𝑚 1 2 𝑛 𝜏 𝜋 {\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{2}% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}% \prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+n\tau)\pi}\right)}}
\Jacobithetatau{2}@{z}{\tau} = \Jacobithetatau{2}@{0}{\tau}\*\lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+n\tau)\pi}\right)		 

JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta2(0,exp(I*Pi*tau))* limit(product(limit(product(1 +(z)/((m -(1)/(2)+ n*tau)*Pi), m = 1 - M..M), M = infinity), n = - N..N), N = infinity)

EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]]* Limit[Product[Limit[Product[1 +Divide[z,(m -Divide[1,2]+ n*\[Tau])*Pi], {m, 1 - 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
20.5.E16 θ 3 ⁡ ( z | τ ) = θ 3 ⁡ ( 0 | τ ) ⁢ lim N → ∞ ⁡ ∏ n = 1 - N N lim M → ∞ ⁡ ∏ m = 1 - M M ( 1 + z ( m - 1 2 + ( n - 1 2 ) ⁢ τ ) ⁢ π ) Jacobi-theta-tau 3 𝑧 𝜏 Jacobi-theta-tau 3 0 𝜏 subscript → 𝑁 superscript subscript product 𝑛 1 𝑁 𝑁 subscript → 𝑀 superscript subscript product 𝑚 1 𝑀 𝑀 1 𝑧 𝑚 1 2 𝑛 1 2 𝜏 𝜋 {\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{3}% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty% }\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)\pi}% \right)}}
\Jacobithetatau{3}@{z}{\tau} = \Jacobithetatau{3}@{0}{\tau}\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)\pi}\right)		 

JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta3(0,exp(I*Pi*tau))* limit(product(limit(product(1 +(z)/((m -(1)/(2)+(n -(1)/(2))*tau)*Pi), m = 1 - M..M), M = infinity), n = 1 - N..N), N = infinity)

EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]]* Limit[Product[Limit[Product[1 +Divide[z,(m -Divide[1,2]+(n -Divide[1,2])*\[Tau])*Pi], {m, 1 - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None], {n, 1 - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
20.5.E17 θ 4 ⁡ ( z | τ ) = θ 4 ⁡ ( 0 | τ ) ⁢ lim N → ∞ ⁡ ∏ n = 1 - N N lim M → ∞ ⁡ ∏ m = - M M ( 1 + z ( m + ( n - 1 2 ) ⁢ τ ) ⁢ π ) Jacobi-theta-tau 4 𝑧 𝜏 Jacobi-theta-tau 4 0 𝜏 subscript → 𝑁 superscript subscript product 𝑛 1 𝑁 𝑁 subscript → 𝑀 superscript subscript product 𝑚 𝑀 𝑀 1 𝑧 𝑚 𝑛 1 2 𝜏 𝜋 {\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{4}% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty% }\prod_{m=-M}^{M}\left(1+\frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right)}}
\Jacobithetatau{4}@{z}{\tau} = \Jacobithetatau{4}@{0}{\tau}\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+\frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right)		 

JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta4(0,exp(I*Pi*tau))* limit(product(limit(product(1 +(z)/((m +(n -(1)/(2))*tau)*Pi), m = - M..M), M = infinity), n = 1 - N..N), N = infinity)

EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, 0, Exp[I*Pi*(\[Tau])]]* Limit[Product[Limit[Product[1 +Divide[z,(m +(n -Divide[1,2])*\[Tau])*Pi], {m, - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None], {n, 1 - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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