Theta Functions - 20.7 Identities

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DLMF Formula Constraints Maple Mathematica Symbolic
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20.7.E1 θ 3 2 ⁡ ( 0 , q ) ⁢ θ 3 2 ⁡ ( z , q ) = θ 4 2 ⁡ ( 0 , q ) ⁢ θ 4 2 ⁡ ( z , q ) + θ 2 2 ⁡ ( 0 , q ) ⁢ θ 2 2 ⁡ ( z , q ) Jacobi-theta 3 2 0 𝑞 Jacobi-theta 3 2 𝑧 𝑞 Jacobi-theta 4 2 0 𝑞 Jacobi-theta 4 2 𝑧 𝑞 Jacobi-theta 2 2 0 𝑞 Jacobi-theta 2 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{3}^{2}}\left(0,q\right){\theta_{3}^{2}}% \left(z,q\right)={\theta_{4}^{2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q% \right)+{\theta_{2}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}
\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q}+\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}		 

(JacobiTheta3(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta4(z, q))^(2)+ (JacobiTheta2(0, q))^(2)* (JacobiTheta2(z, q))^(2)

(EllipticTheta[3, 0, q])^(2)* (EllipticTheta[3, z, q])^(2) == (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[4, z, q])^(2)+ (EllipticTheta[2, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)
Failure Failure Error Successful [Tested: 70]
20.7.E2 θ 3 2 ⁡ ( 0 , q ) ⁢ θ 4 2 ⁡ ( z , q ) = θ 2 2 ⁡ ( 0 , q ) ⁢ θ 1 2 ⁡ ( z , q ) + θ 4 2 ⁡ ( 0 , q ) ⁢ θ 3 2 ⁡ ( z , q ) Jacobi-theta 3 2 0 𝑞 Jacobi-theta 4 2 𝑧 𝑞 Jacobi-theta 2 2 0 𝑞 Jacobi-theta 1 2 𝑧 𝑞 Jacobi-theta 4 2 0 𝑞 Jacobi-theta 3 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{3}^{2}}\left(0,q\right){\theta_{4}^{2}}% \left(z,q\right)={\theta_{2}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q% \right)+{\theta_{4}^{2}}\left(0,q\right){\theta_{3}^{2}}\left(z,q\right)}}
\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q}		 

(JacobiTheta3(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta2(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta3(z, q))^(2)

(EllipticTheta[3, 0, q])^(2)* (EllipticTheta[4, z, q])^(2) == (EllipticTheta[2, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[3, z, q])^(2)
Failure Failure Error Successful [Tested: 70]
20.7.E3 θ 2 2 ⁡ ( 0 , q ) ⁢ θ 4 2 ⁡ ( z , q ) = θ 3 2 ⁡ ( 0 , q ) ⁢ θ 1 2 ⁡ ( z , q ) + θ 4 2 ⁡ ( 0 , q ) ⁢ θ 2 2 ⁡ ( z , q ) Jacobi-theta 2 2 0 𝑞 Jacobi-theta 4 2 𝑧 𝑞 Jacobi-theta 3 2 0 𝑞 Jacobi-theta 1 2 𝑧 𝑞 Jacobi-theta 4 2 0 𝑞 Jacobi-theta 2 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{2}^{2}}\left(0,q\right){\theta_{4}^{2}}% \left(z,q\right)={\theta_{3}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q% \right)+{\theta_{4}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}
\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}		 

(JacobiTheta2(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta3(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta2(z, q))^(2)

(EllipticTheta[2, 0, q])^(2)* (EllipticTheta[4, z, q])^(2) == (EllipticTheta[3, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)
Failure Failure Error Successful [Tested: 70]
20.7.E4 θ 2 2 ⁡ ( 0 , q ) ⁢ θ 3 2 ⁡ ( z , q ) = θ 4 2 ⁡ ( 0 , q ) ⁢ θ 1 2 ⁡ ( z , q ) + θ 3 2 ⁡ ( 0 , q ) ⁢ θ 2 2 ⁡ ( z , q ) Jacobi-theta 2 2 0 𝑞 Jacobi-theta 3 2 𝑧 𝑞 Jacobi-theta 4 2 0 𝑞 Jacobi-theta 1 2 𝑧 𝑞 Jacobi-theta 3 2 0 𝑞 Jacobi-theta 2 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{2}^{2}}\left(0,q\right){\theta_{3}^{2}}% \left(z,q\right)={\theta_{4}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q% \right)+{\theta_{3}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}
\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}		 

(JacobiTheta2(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta3(0, q))^(2)* (JacobiTheta2(z, q))^(2)

(EllipticTheta[2, 0, q])^(2)* (EllipticTheta[3, z, q])^(2) == (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[3, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)
Failure Failure Error Successful [Tested: 70]
20.7.E5 θ 3 4 ⁡ ( 0 , q ) = θ 2 4 ⁡ ( 0 , q ) + θ 4 4 ⁡ ( 0 , q ) Jacobi-theta 3 4 0 𝑞 Jacobi-theta 2 4 0 𝑞 Jacobi-theta 4 4 0 𝑞 {\displaystyle{\displaystyle{\theta_{3}^{4}}\left(0,q\right)={\theta_{2}^{4}}% \left(0,q\right)+{\theta_{4}^{4}}\left(0,q\right)}}
\Jacobithetaq{3}^{4}@{0}{q} = \Jacobithetaq{2}^{4}@{0}{q}+\Jacobithetaq{4}^{4}@{0}{q}		 

(JacobiTheta3(0, q))^(4) = (JacobiTheta2(0, q))^(4)+ (JacobiTheta4(0, q))^(4)

(EllipticTheta[3, 0, q])^(4) == (EllipticTheta[2, 0, q])^(4)+ (EllipticTheta[4, 0, q])^(4)
Successful Failure - Successful [Tested: 10]
20.7.E6 θ 4 2 ⁡ ( 0 , q ) ⁢ θ 1 ⁡ ( w + z , q ) ⁢ θ 1 ⁡ ( w - z , q ) = θ 3 2 ⁡ ( w , q ) ⁢ θ 2 2 ⁡ ( z , q ) - θ 2 2 ⁡ ( w , q ) ⁢ θ 3 2 ⁡ ( z , q ) Jacobi-theta 4 2 0 𝑞 Jacobi-theta 1 𝑤 𝑧 𝑞 Jacobi-theta 1 𝑤 𝑧 𝑞 Jacobi-theta 3 2 𝑤 𝑞 Jacobi-theta 2 2 𝑧 𝑞 Jacobi-theta 2 2 𝑤 𝑞 Jacobi-theta 3 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{1}\left(w+% z,q\right)\theta_{1}\left(w-z,q\right)={\theta_{3}^{2}}\left(w,q\right){\theta% _{2}^{2}}\left(z,q\right)-{\theta_{2}^{2}}\left(w,q\right){\theta_{3}^{2}}% \left(z,q\right)}}
\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}@{w+z}{q}\Jacobithetaq{1}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}		 

(JacobiTheta4(0, q))^(2)* JacobiTheta1(w + z, q)*JacobiTheta1(w - z, q) = (JacobiTheta3(w, q))^(2)* (JacobiTheta2(z, q))^(2)- (JacobiTheta2(w, q))^(2)* (JacobiTheta3(z, q))^(2)

(EllipticTheta[4, 0, q])^(2)* EllipticTheta[1, w + z, q]*EllipticTheta[1, w - z, q] == (EllipticTheta[3, w, q])^(2)* (EllipticTheta[2, z, q])^(2)- (EllipticTheta[2, w, q])^(2)* (EllipticTheta[3, z, q])^(2)
Failure Failure Error Successful [Tested: 300]
20.7.E7 θ 4 2 ⁡ ( 0 , q ) ⁢ θ 2 ⁡ ( w + z , q ) ⁢ θ 2 ⁡ ( w - z , q ) = θ 4 2 ⁡ ( w , q ) ⁢ θ 2 2 ⁡ ( z , q ) - θ 1 2 ⁡ ( w , q ) ⁢ θ 3 2 ⁡ ( z , q ) Jacobi-theta 4 2 0 𝑞 Jacobi-theta 2 𝑤 𝑧 𝑞 Jacobi-theta 2 𝑤 𝑧 𝑞 Jacobi-theta 4 2 𝑤 𝑞 Jacobi-theta 2 2 𝑧 𝑞 Jacobi-theta 1 2 𝑤 𝑞 Jacobi-theta 3 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{2}\left(w+% z,q\right)\theta_{2}\left(w-z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta% _{2}^{2}}\left(z,q\right)-{\theta_{1}^{2}}\left(w,q\right){\theta_{3}^{2}}% \left(z,q\right)}}
\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}@{w+z}{q}\Jacobithetaq{2}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}		 

(JacobiTheta4(0, q))^(2)* JacobiTheta2(w + z, q)*JacobiTheta2(w - z, q) = (JacobiTheta4(w, q))^(2)* (JacobiTheta2(z, q))^(2)- (JacobiTheta1(w, q))^(2)* (JacobiTheta3(z, q))^(2)

(EllipticTheta[4, 0, q])^(2)* EllipticTheta[2, w + z, q]*EllipticTheta[2, w - z, q] == (EllipticTheta[4, w, q])^(2)* (EllipticTheta[2, z, q])^(2)- (EllipticTheta[1, w, q])^(2)* (EllipticTheta[3, z, q])^(2)
Failure Failure Error Successful [Tested: 300]
20.7.E8 θ 4 2 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( w + z , q ) ⁢ θ 3 ⁡ ( w - z , q ) = θ 4 2 ⁡ ( w , q ) ⁢ θ 3 2 ⁡ ( z , q ) - θ 1 2 ⁡ ( w , q ) ⁢ θ 2 2 ⁡ ( z , q ) Jacobi-theta 4 2 0 𝑞 Jacobi-theta 3 𝑤 𝑧 𝑞 Jacobi-theta 3 𝑤 𝑧 𝑞 Jacobi-theta 4 2 𝑤 𝑞 Jacobi-theta 3 2 𝑧 𝑞 Jacobi-theta 1 2 𝑤 𝑞 Jacobi-theta 2 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{3}\left(w+% z,q\right)\theta_{3}\left(w-z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta% _{3}^{2}}\left(z,q\right)-{\theta_{1}^{2}}\left(w,q\right){\theta_{2}^{2}}% \left(z,q\right)}}
\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}@{w+z}{q}\Jacobithetaq{3}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}		 

(JacobiTheta4(0, q))^(2)* JacobiTheta3(w + z, q)*JacobiTheta3(w - z, q) = (JacobiTheta4(w, q))^(2)* (JacobiTheta3(z, q))^(2)- (JacobiTheta1(w, q))^(2)* (JacobiTheta2(z, q))^(2)

(EllipticTheta[4, 0, q])^(2)* EllipticTheta[3, w + z, q]*EllipticTheta[3, w - z, q] == (EllipticTheta[4, w, q])^(2)* (EllipticTheta[3, z, q])^(2)- (EllipticTheta[1, w, q])^(2)* (EllipticTheta[2, z, q])^(2)
Failure Failure Error Successful [Tested: 300]
20.7.E9 θ 4 2 ⁡ ( 0 , q ) ⁢ θ 4 ⁡ ( w + z , q ) ⁢ θ 4 ⁡ ( w - z , q ) = θ 3 2 ⁡ ( w , q ) ⁢ θ 3 2 ⁡ ( z , q ) - θ 2 2 ⁡ ( w , q ) ⁢ θ 2 2 ⁡ ( z , q ) Jacobi-theta 4 2 0 𝑞 Jacobi-theta 4 𝑤 𝑧 𝑞 Jacobi-theta 4 𝑤 𝑧 𝑞 Jacobi-theta 3 2 𝑤 𝑞 Jacobi-theta 3 2 𝑧 𝑞 Jacobi-theta 2 2 𝑤 𝑞 Jacobi-theta 2 2 𝑧 𝑞 {\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{4}\left(w+% z,q\right)\theta_{4}\left(w-z,q\right)={\theta_{3}^{2}}\left(w,q\right){\theta% _{3}^{2}}\left(z,q\right)-{\theta_{2}^{2}}\left(w,q\right){\theta_{2}^{2}}% \left(z,q\right)}}
\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}@{w+z}{q}\Jacobithetaq{4}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}		 

(JacobiTheta4(0, q))^(2)* JacobiTheta4(w + z, q)*JacobiTheta4(w - z, q) = (JacobiTheta3(w, q))^(2)* (JacobiTheta3(z, q))^(2)- (JacobiTheta2(w, q))^(2)* (JacobiTheta2(z, q))^(2)

(EllipticTheta[4, 0, q])^(2)* EllipticTheta[4, w + z, q]*EllipticTheta[4, w - z, q] == (EllipticTheta[3, w, q])^(2)* (EllipticTheta[3, z, q])^(2)- (EllipticTheta[2, w, q])^(2)* (EllipticTheta[2, z, q])^(2)
Failure Failure Error Successful [Tested: 300]
20.7.E10 θ 1 ⁡ ( 2 ⁢ z , q ) = 2 ⁢ θ 1 ⁡ ( z , q ) ⁢ θ 2 ⁡ ( z , q ) ⁢ θ 3 ⁡ ( z , q ) ⁢ θ 4 ⁡ ( z , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( 0 , q ) ⁢ θ 4 ⁡ ( 0 , q ) Jacobi-theta 1 2 𝑧 𝑞 2 Jacobi-theta 1 𝑧 𝑞 Jacobi-theta 2 𝑧 𝑞 Jacobi-theta 3 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 3 0 𝑞 Jacobi-theta 4 0 𝑞 {\displaystyle{\displaystyle\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left% (z,q\right)\theta_{2}\left(z,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left% (z,q\right)}{\theta_{2}\left(0,q\right)\theta_{3}\left(0,q\right)\theta_{4}% \left(0,q\right)}}}
\Jacobithetaq{1}@{2z}{q} = 2\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{0}{q}\Jacobithetaq{4}@{0}{q}}		 

JacobiTheta1(2*z, q) = 2*(JacobiTheta1(z, q)*JacobiTheta2(z, q)*JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q))

EllipticTheta[1, 2*z, q] == 2*Divide[EllipticTheta[1, z, q]*EllipticTheta[2, z, q]*EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q]]
Failure Failure Error Successful [Tested: 70]
20.7.E11 θ 1 ⁡ ( z , q ) ⁢ θ 2 ⁡ ( z , q ) θ 1 ⁡ ( 2 ⁢ z , q 2 ) = θ 3 ⁡ ( z , q ) ⁢ θ 4 ⁡ ( z , q ) θ 4 ⁡ ( 2 ⁢ z , q 2 ) Jacobi-theta 1 𝑧 𝑞 Jacobi-theta 2 𝑧 𝑞 Jacobi-theta 1 2 𝑧 superscript 𝑞 2 Jacobi-theta 3 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 Jacobi-theta 4 2 𝑧 superscript 𝑞 2 {\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z,% q\right)}{\theta_{1}\left(2z,q^{2}\right)}=\frac{\theta_{3}\left(z,q\right)% \theta_{4}\left(z,q\right)}{\theta_{4}\left(2z,q^{2}\right)}}}
\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}}{\Jacobithetaq{1}@{2z}{q^{2}}} = \frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}}		 

(JacobiTheta1(z, q)*JacobiTheta2(z, q))/(JacobiTheta1(2*z, (q)^(2))) = (JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2)))

Divide[EllipticTheta[1, z, q]*EllipticTheta[2, z, q],EllipticTheta[1, 2*z, (q)^(2)]] == Divide[EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[4, 2*z, (q)^(2)]]
Failure Failure Error
Failed [7 / 70]
Result: Complex[-0.5078048710711283, 0.5078048710711279]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.5078048710711284, 0.5078048710711281]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.7.E11 θ 3 ⁡ ( z , q ) ⁢ θ 4 ⁡ ( z , q ) θ 4 ⁡ ( 2 ⁢ z , q 2 ) = θ 4 ⁡ ( 0 , q 2 ) Jacobi-theta 3 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 Jacobi-theta 4 2 𝑧 superscript 𝑞 2 Jacobi-theta 4 0 superscript 𝑞 2 {\displaystyle{\displaystyle\frac{\theta_{3}\left(z,q\right)\theta_{4}\left(z,% q\right)}{\theta_{4}\left(2z,q^{2}\right)}=\theta_{4}\left(0,q^{2}\right)}}
\frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}} = \Jacobithetaq{4}@{0}{q^{2}}		 

(JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2))) = JacobiTheta4(0, (q)^(2))

Divide[EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[4, 2*z, (q)^(2)]] == EllipticTheta[4, 0, (q)^(2)]
Failure Failure Error Successful [Tested: 70]
20.7.E12 θ 1 ⁡ ( z , q 2 ) ⁢ θ 4 ⁡ ( z , q 2 ) θ 1 ⁡ ( z , q ) = θ 2 ⁡ ( z , q 2 ) ⁢ θ 3 ⁡ ( z , q 2 ) θ 2 ⁡ ( z , q ) Jacobi-theta 1 𝑧 superscript 𝑞 2 Jacobi-theta 4 𝑧 superscript 𝑞 2 Jacobi-theta 1 𝑧 𝑞 Jacobi-theta 2 𝑧 superscript 𝑞 2 Jacobi-theta 3 𝑧 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑞 {\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q^{2}\right)\theta_{4}% \left(z,q^{2}\right)}{\theta_{1}\left(z,q\right)}=\frac{\theta_{2}\left(z,q^{2% }\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{2}\left(z,q\right)}}}
\frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{q}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}}		 

(JacobiTheta1(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta1(z, q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q))

Divide[EllipticTheta[1, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[1, z, q]] == Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[2, z, q]]
Failure Failure Error Successful [Tested: 70]
20.7.E12 θ 2 ⁡ ( z , q 2 ) ⁢ θ 3 ⁡ ( z , q 2 ) θ 2 ⁡ ( z , q ) = 1 2 ⁢ θ 2 ⁡ ( 0 , q ) Jacobi-theta 2 𝑧 superscript 𝑞 2 Jacobi-theta 3 𝑧 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑞 1 2 Jacobi-theta 2 0 𝑞 {\displaystyle{\displaystyle\frac{\theta_{2}\left(z,q^{2}\right)\theta_{3}% \left(z,q^{2}\right)}{\theta_{2}\left(z,q\right)}=\tfrac{1}{2}\theta_{2}\left(% 0,q\right)}}
\frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}} = \tfrac{1}{2}\Jacobithetaq{2}@{0}{q}		 

(JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q)) = (1)/(2)*JacobiTheta2(0, q)

Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[2, z, q]] == Divide[1,2]*EllipticTheta[2, 0, q]
Failure Failure Error
Failed [7 / 70]
Result: Complex[1.1102230246251565*^-16, -1.5053817239177183]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.3306690738754696*^-16, -1.5053817239177185]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
20.7.E13 θ 1 ⁡ ( z , q ) ⁢ θ 1 ⁡ ( w , q ) = θ 3 ⁡ ( z + w , q 2 ) ⁢ θ 2 ⁡ ( z - w , q 2 ) - θ 2 ⁡ ( z + w , q 2 ) ⁢ θ 3 ⁡ ( z - w , q 2 ) Jacobi-theta 1 𝑧 𝑞 Jacobi-theta 1 𝑤 𝑞 Jacobi-theta 3 𝑧 𝑤 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑤 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑤 superscript 𝑞 2 Jacobi-theta 3 𝑧 𝑤 superscript 𝑞 2 {\displaystyle{\displaystyle\theta_{1}\left(z,q\right)\theta_{1}\left(w,q% \right)=\theta_{3}\left(z+w,q^{2}\right)\theta_{2}\left(z-w,q^{2}\right)-% \theta_{2}\left(z+w,q^{2}\right)\theta_{3}\left(z-w,q^{2}\right)}}
\Jacobithetaq{1}@{z}{q}\Jacobithetaq{1}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}-\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}		 

JacobiTheta1(z, q)*JacobiTheta1(w, q) = JacobiTheta3(z + w, (q)^(2))*JacobiTheta2(z - w, (q)^(2))- JacobiTheta2(z + w, (q)^(2))*JacobiTheta3(z - w, (q)^(2))

EllipticTheta[1, z, q]*EllipticTheta[1, w, q] == EllipticTheta[3, z + w, (q)^(2)]*EllipticTheta[2, z - w, (q)^(2)]- EllipticTheta[2, z + w, (q)^(2)]*EllipticTheta[3, z - w, (q)^(2)]
Failure Failure Error Successful [Tested: 300]
20.7.E14 θ 3 ⁡ ( z , q ) ⁢ θ 3 ⁡ ( w , q ) = θ 3 ⁡ ( z + w , q 2 ) ⁢ θ 3 ⁡ ( z - w , q 2 ) + θ 2 ⁡ ( z + w , q 2 ) ⁢ θ 2 ⁡ ( z - w , q 2 ) Jacobi-theta 3 𝑧 𝑞 Jacobi-theta 3 𝑤 𝑞 Jacobi-theta 3 𝑧 𝑤 superscript 𝑞 2 Jacobi-theta 3 𝑧 𝑤 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑤 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑤 superscript 𝑞 2 {\displaystyle{\displaystyle\theta_{3}\left(z,q\right)\theta_{3}\left(w,q% \right)=\theta_{3}\left(z+w,q^{2}\right)\theta_{3}\left(z-w,q^{2}\right)+% \theta_{2}\left(z+w,q^{2}\right)\theta_{2}\left(z-w,q^{2}\right)}}
\Jacobithetaq{3}@{z}{q}\Jacobithetaq{3}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}+\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}		 

JacobiTheta3(z, q)*JacobiTheta3(w, q) = JacobiTheta3(z + w, (q)^(2))*JacobiTheta3(z - w, (q)^(2))+ JacobiTheta2(z + w, (q)^(2))*JacobiTheta2(z - w, (q)^(2))

EllipticTheta[3, z, q]*EllipticTheta[3, w, q] == EllipticTheta[3, z + w, (q)^(2)]*EllipticTheta[3, z - w, (q)^(2)]+ EllipticTheta[2, z + w, (q)^(2)]*EllipticTheta[2, z - w, (q)^(2)]
Failure Failure Error Successful [Tested: 300]
20.7.E16 θ 1 ⁡ ( 2 ⁢ z | 2 ⁢ τ ) = A ⁢ θ 1 ⁡ ( z | τ ) ⁢ θ 2 ⁡ ( z | τ ) Jacobi-theta-tau 1 2 𝑧 2 𝜏 𝐴 Jacobi-theta-tau 1 𝑧 𝜏 Jacobi-theta-tau 2 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(2z\middle|2\tau\right)=A\theta_{1}% \left(z\middle|\tau\right)\theta_{2}\left(z\middle|\tau\right)}}
\Jacobithetatau{1}@{2z}{2\tau} = A\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{2}@{z}{\tau}		 

JacobiTheta1(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta2(z,exp(I*Pi*tau))

EllipticTheta[1, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: 1.631641333-1.744983248*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -1.353330373+4.008308689*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [60 / 300]
Result: Complex[1.6316413333035786, -1.7449832486391479]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.25205232655780907, -0.3227610482702816]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E17 θ 2 ⁡ ( 2 ⁢ z | 2 ⁢ τ ) = A ⁢ θ 1 ⁡ ( 1 4 ⁢ π - z | τ ) ⁢ θ 1 ⁡ ( 1 4 ⁢ π + z | τ ) Jacobi-theta-tau 2 2 𝑧 2 𝜏 𝐴 Jacobi-theta-tau 1 1 4 𝜋 𝑧 𝜏 Jacobi-theta-tau 1 1 4 𝜋 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{2}\left(2z\middle|2\tau\right)=A\theta_{1}% \left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{1}\left(\tfrac{1}{4}\pi+z% \middle|\tau\right)}}
\Jacobithetatau{2}@{2z}{2\tau} = A\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}		 

JacobiTheta2(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta1((1)/(4)*Pi - z,exp(I*Pi*tau))*JacobiTheta1((1)/(4)*Pi + z,exp(I*Pi*tau))

EllipticTheta[2, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[1, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[1, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]
Error Failure -
Failed [60 / 300]
Result: Complex[-1.4403734484961686, -1.1891981543571708]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.23150096143650367, 0.21570115304796234]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E18 θ 3 ⁡ ( 2 ⁢ z | 2 ⁢ τ ) = A ⁢ θ 3 ⁡ ( 1 4 ⁢ π - z | τ ) ⁢ θ 3 ⁡ ( 1 4 ⁢ π + z | τ ) Jacobi-theta-tau 3 2 𝑧 2 𝜏 𝐴 Jacobi-theta-tau 3 1 4 𝜋 𝑧 𝜏 Jacobi-theta-tau 3 1 4 𝜋 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(2z\middle|2\tau\right)=A\theta_{3}% \left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{3}\left(\tfrac{1}{4}\pi+z% \middle|\tau\right)}}
\Jacobithetatau{3}@{2z}{2\tau} = A\Jacobithetatau{3}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{4}\pi+z}{\tau}		 

JacobiTheta3(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta3((1)/(4)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((1)/(4)*Pi + z,exp(I*Pi*tau))

EllipticTheta[3, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[3, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]
Error Failure -
Failed [60 / 300]
Result: Complex[0.3438479503598899, -0.39372543999621956]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.12535543238516544, -0.5211900545642698]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E19 θ 4 ⁡ ( 2 ⁢ z | 2 ⁢ τ ) = A ⁢ θ 3 ⁡ ( z | τ ) ⁢ θ 4 ⁡ ( z | τ ) Jacobi-theta-tau 4 2 𝑧 2 𝜏 𝐴 Jacobi-theta-tau 3 𝑧 𝜏 Jacobi-theta-tau 4 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{4}\left(2z\middle|2\tau\right)=A\theta_{3}% \left(z\middle|\tau\right)\theta_{4}\left(z\middle|\tau\right)}}
\Jacobithetatau{4}@{2z}{2\tau} = A\Jacobithetatau{3}@{z}{\tau}\Jacobithetatau{4}@{z}{\tau}		 

JacobiTheta4(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta3(z,exp(I*Pi*tau))*JacobiTheta4(z,exp(I*Pi*tau))

EllipticTheta[4, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: .88393938e-1-.6601554491*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.5678871113-.5102031247*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [60 / 300]
Result: Complex[0.08839393747885427, -0.6601554493410663]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.12758234205780994, -0.4874768056112989]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E21 θ 1 ⁡ ( 4 ⁢ z | 4 ⁢ τ ) = B ⁢ θ 1 ⁡ ( z | τ ) ⁢ θ 1 ⁡ ( 1 4 ⁢ π - z | τ ) ⁢ θ 1 ⁡ ( 1 4 ⁢ π + z | τ ) ⁢ θ 2 ⁡ ( z | τ ) Jacobi-theta-tau 1 4 𝑧 4 𝜏 𝐵 Jacobi-theta-tau 1 𝑧 𝜏 Jacobi-theta-tau 1 1 4 𝜋 𝑧 𝜏 Jacobi-theta-tau 1 1 4 𝜋 𝑧 𝜏 Jacobi-theta-tau 2 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(4z\middle|4\tau\right)=B\theta_{1}% \left(z\middle|\tau\right)\theta_{1}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)% \*\theta_{1}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\theta_{2}\left(z\middle% |\tau\right)}}
\Jacobithetatau{1}@{4z}{4\tau} = B\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{2}@{z}{\tau}		 

JacobiTheta1(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta1((1)/(4)*Pi - z,exp(I*Pi*tau))* JacobiTheta1((1)/(4)*Pi + z,exp(I*Pi*tau))*JacobiTheta2(z,exp(I*Pi*tau))

EllipticTheta[1, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[1, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[1, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]
Error Failure -
Failed [60 / 300]
Result: Complex[-1.1596846442931608, -2.448595776474227]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.3218907084595235, -0.36082838804303224]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E22 θ 2 ⁡ ( 4 ⁢ z | 4 ⁢ τ ) = B ⁢ θ 2 ⁡ ( 1 8 ⁢ π - z | τ ) ⁢ θ 2 ⁡ ( 1 8 ⁢ π + z | τ ) ⁢ θ 2 ⁡ ( 3 8 ⁢ π - z | τ ) ⁢ θ 2 ⁡ ( 3 8 ⁢ π + z | τ ) Jacobi-theta-tau 2 4 𝑧 4 𝜏 𝐵 Jacobi-theta-tau 2 1 8 𝜋 𝑧 𝜏 Jacobi-theta-tau 2 1 8 𝜋 𝑧 𝜏 Jacobi-theta-tau 2 3 8 𝜋 𝑧 𝜏 Jacobi-theta-tau 2 3 8 𝜋 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{2}\left(4z\middle|4\tau\right)=B\theta_{2}% \left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{2}\left(\tfrac{1}{8}\pi+z% \middle|\tau\right)\*\theta_{2}\left(\tfrac{3}{8}\pi-z\middle|\tau\right)% \theta_{2}\left(\tfrac{3}{8}\pi+z\middle|\tau\right)}}
\Jacobithetatau{2}@{4z}{4\tau} = B\Jacobithetatau{2}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{2}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{3}{8}\pi+z}{\tau}		 

JacobiTheta2(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta2((1)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta2((1)/(8)*Pi + z,exp(I*Pi*tau))* JacobiTheta2((3)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta2((3)/(8)*Pi + z,exp(I*Pi*tau))

EllipticTheta[2, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[2, Divide[1,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, Divide[1,8]*Pi + z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[2, Divide[3,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, Divide[3,8]*Pi + z, Exp[I*Pi*(\[Tau])]]
Error Failure -
Failed [60 / 300]
Result: Complex[-2.54672123948714, 1.1372871673366372]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.36415557562453404, -0.3395547407401721]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E23 θ 3 ⁡ ( 4 ⁢ z | 4 ⁢ τ ) = B ⁢ θ 3 ⁡ ( 1 8 ⁢ π - z | τ ) ⁢ θ 3 ⁡ ( 1 8 ⁢ π + z | τ ) ⁢ θ 3 ⁡ ( 3 8 ⁢ π - z | τ ) ⁢ θ 3 ⁡ ( 3 8 ⁢ π + z | τ ) Jacobi-theta-tau 3 4 𝑧 4 𝜏 𝐵 Jacobi-theta-tau 3 1 8 𝜋 𝑧 𝜏 Jacobi-theta-tau 3 1 8 𝜋 𝑧 𝜏 Jacobi-theta-tau 3 3 8 𝜋 𝑧 𝜏 Jacobi-theta-tau 3 3 8 𝜋 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(4z\middle|4\tau\right)=B\theta_{3}% \left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{3}\left(\tfrac{1}{8}\pi+z% \middle|\tau\right)\*\theta_{3}\left(\tfrac{3}{8}\pi-z\middle|\tau\right)% \theta_{3}\left(\tfrac{3}{8}\pi+z\middle|\tau\right)}}
\Jacobithetatau{3}@{4z}{4\tau} = B\Jacobithetatau{3}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{3}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{3}{8}\pi+z}{\tau}		 

JacobiTheta3(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta3((1)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((1)/(8)*Pi + z,exp(I*Pi*tau))* JacobiTheta3((3)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((3)/(8)*Pi + z,exp(I*Pi*tau))

EllipticTheta[3, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[3, Divide[1,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[1,8]*Pi + z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[3, Divide[3,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[3,8]*Pi + z, Exp[I*Pi*(\[Tau])]]
Error Failure -
Failed [60 / 300]
Result: Complex[0.2353615104715142, -0.5335293147703523]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.11871524589758675, -0.5091754766273449]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E24 θ 4 ⁡ ( 4 ⁢ z | 4 ⁢ τ ) = B ⁢ θ 4 ⁡ ( z | τ ) ⁢ θ 4 ⁡ ( 1 4 ⁢ π - z | τ ) ⁢ θ 4 ⁡ ( 1 4 ⁢ π + z | τ ) ⁢ θ 3 ⁡ ( z | τ ) Jacobi-theta-tau 4 4 𝑧 4 𝜏 𝐵 Jacobi-theta-tau 4 𝑧 𝜏 Jacobi-theta-tau 4 1 4 𝜋 𝑧 𝜏 Jacobi-theta-tau 4 1 4 𝜋 𝑧 𝜏 Jacobi-theta-tau 3 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{4}\left(4z\middle|4\tau\right)=B\theta_{4}% \left(z\middle|\tau\right)\theta_{4}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)% \*\theta_{4}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\theta_{3}\left(z\middle% |\tau\right)}}
\Jacobithetatau{4}@{4z}{4\tau} = B\Jacobithetatau{4}@{z}{\tau}\Jacobithetatau{4}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{4}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{3}@{z}{\tau}		 

JacobiTheta4(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta4(z,exp(I*Pi*tau))*JacobiTheta4((1)/(4)*Pi - z,exp(I*Pi*tau))* JacobiTheta4((1)/(4)*Pi + z,exp(I*Pi*tau))*JacobiTheta3(z,exp(I*Pi*tau))

EllipticTheta[4, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[4, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[4, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]
Error Failure -
Failed [60 / 300]
Result: Complex[0.3584730563399423, -0.5666107505620169]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.11914720780154586, -0.5081951100786072]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], 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
20.7.E25 d d z ⁡ ( θ 2 ⁡ ( z | τ ) θ 4 ⁡ ( z | τ ) ) = - θ 3 2 ⁡ ( 0 | τ ) ⁢ θ 1 ⁡ ( z | τ ) ⁢ θ 3 ⁡ ( z | τ ) θ 4 2 ⁡ ( z | τ ) derivative 𝑧 Jacobi-theta-tau 2 𝑧 𝜏 Jacobi-theta-tau 4 𝑧 𝜏 Jacobi-theta-tau 3 2 0 𝜏 Jacobi-theta-tau 1 𝑧 𝜏 Jacobi-theta-tau 3 𝑧 𝜏 Jacobi-theta-tau 4 2 𝑧 𝜏 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\theta_{% 2}\left(z\middle|\tau\right)}{\theta_{4}\left(z\middle|\tau\right)}\right)=-% \frac{{\theta_{3}^{2}}\left(0\middle|\tau\right)\theta_{1}\left(z\middle|\tau% \right)\theta_{3}\left(z\middle|\tau\right)}{{\theta_{4}^{2}}\left(z\middle|% \tau\right)}}}
\deriv{}{z}\left(\frac{\Jacobithetatau{2}@{z}{\tau}}{\Jacobithetatau{4}@{z}{\tau}}\right) = -\frac{\Jacobithetatau{3}^{2}@{0}{\tau}\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{3}@{z}{\tau}}{\Jacobithetatau{4}^{2}@{z}{\tau}}		 

diff((JacobiTheta2(z,exp(I*Pi*tau)))/(JacobiTheta4(z,exp(I*Pi*tau))), z) = -((JacobiTheta3(0,exp(I*Pi*tau)))^(2)* JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta3(z,exp(I*Pi*tau)))/((JacobiTheta4(z,exp(I*Pi*tau)))^(2))

D[Divide[EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]],EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]], z] == -Divide[(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)* EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]],(EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]])^(2)]
Failure Failure Error Successful [Tested: 70]
20.7.E26 θ 1 ⁡ ( z | τ + 1 ) = e i ⁢ π / 4 ⁢ θ 1 ⁡ ( z | τ ) Jacobi-theta-tau 1 𝑧 𝜏 1 superscript 𝑒 𝑖 𝜋 4 Jacobi-theta-tau 1 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau+1\right)=e^{i\pi/4}% \theta_{1}\left(z\middle|\tau\right)}}
\Jacobithetatau{1}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{1}@{z}{\tau}		 

JacobiTheta1(z,exp(I*Pi*tau + 1)) = exp(I*Pi/4)*JacobiTheta1(z,exp(I*Pi*tau))
 
EllipticTheta[1, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/4]*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]
 Failure Failure
Failed [70 / 70]
Result: .7294764132+1.608567858*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: 1.107791050+1.561378050*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 70]
Result: Complex[1.6985877827537141, -0.7949460182709149]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.345921896794935, 1.4881712816971224]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
20.7.E27 θ 2 ⁡ ( z | τ + 1 ) = e i ⁢ π / 4 ⁢ θ 2 ⁡ ( z | τ ) Jacobi-theta-tau 2 𝑧 𝜏 1 superscript 𝑒 𝑖 𝜋 4 Jacobi-theta-tau 2 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau+1\right)=e^{i\pi/4}% \theta_{2}\left(z\middle|\tau\right)}}
\Jacobithetatau{2}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{2}@{z}{\tau}		 

JacobiTheta2(z,exp(I*Pi*tau + 1)) = exp(I*Pi/4)*JacobiTheta2(z,exp(I*Pi*tau))
 
EllipticTheta[2, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/4]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]
 Failure Failure
Failed [70 / 70]
Result: -.369621756e-1-.9012887423*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: 4.590414642+4.526034042*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 70]
Result: Complex[0.22524015718924872, -1.3838317643459628]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[2.711359141795916, -1.3916787489924032]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
20.7.E28 θ 3 ⁡ ( z | τ + 1 ) = θ 4 ⁡ ( z | τ ) Jacobi-theta-tau 3 𝑧 𝜏 1 Jacobi-theta-tau 4 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau+1\right)=\theta_{4}% \left(z\middle|\tau\right)}}
\Jacobithetatau{3}@{z}{\tau+1} = \Jacobithetatau{4}@{z}{\tau}		 

JacobiTheta3(z,exp(I*Pi*tau + 1)) = JacobiTheta4(z,exp(I*Pi*tau))
 
EllipticTheta[3, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]
 Failure Failure
Failed [70 / 70]
Result: 1.500564535+2.208881092*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: -1.492914692-.5532090072*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 70]
20.7.E29 θ 4 ⁡ ( z | τ + 1 ) = θ 3 ⁡ ( z | τ ) Jacobi-theta-tau 4 𝑧 𝜏 1 Jacobi-theta-tau 3 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau+1\right)=\theta_{3}% \left(z\middle|\tau\right)}}
\Jacobithetatau{4}@{z}{\tau+1} = \Jacobithetatau{3}@{z}{\tau}		 

JacobiTheta4(z,exp(I*Pi*tau + 1)) = JacobiTheta3(z,exp(I*Pi*tau))
 
EllipticTheta[4, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]
 Failure Failure
Failed [70 / 70]
Result: -.8770870366-.8516489897*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: 7.362801863+2.459098613*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 70]
20.7.E34 θ 1 ⁡ ( z , q 2 ) ⁢ θ 3 ⁡ ( z , q 2 ) θ 1 ⁡ ( z , i ⁢ q ) = θ 2 ⁡ ( z , q 2 ) ⁢ θ 4 ⁡ ( z , q 2 ) θ 2 ⁡ ( z , i ⁢ q ) Jacobi-theta 1 𝑧 superscript 𝑞 2 Jacobi-theta 3 𝑧 superscript 𝑞 2 Jacobi-theta 1 𝑧 𝑖 𝑞 Jacobi-theta 2 𝑧 superscript 𝑞 2 Jacobi-theta 4 𝑧 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑖 𝑞 {\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}% \left(z,q^{2}\right)}{\theta_{1}\left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{% 2}\right)\theta_{4}\left(z,q^{2}\right)}{\theta_{2}\left(z,iq\right)}}}
\frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{iq}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}}		 

(JacobiTheta1(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta1(z, I*q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta2(z, I*q))
 
Divide[EllipticTheta[1, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[1, z, I*q]] == Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[2, z, I*q]]
 Failure Failure Error Successful [Tested: 70]
20.7.E34 θ 2 ⁡ ( z , q 2 ) ⁢ θ 4 ⁡ ( z , q 2 ) θ 2 ⁡ ( z , i ⁢ q ) = i - 1 / 4 ⁢ θ 2 ⁡ ( 0 , q 2 ) ⁢ θ 4 ⁡ ( 0 , q 2 ) 2 Jacobi-theta 2 𝑧 superscript 𝑞 2 Jacobi-theta 4 𝑧 superscript 𝑞 2 Jacobi-theta 2 𝑧 𝑖 𝑞 superscript 𝑖 1 4 Jacobi-theta 2 0 superscript 𝑞 2 Jacobi-theta 4 0 superscript 𝑞 2 2 {\displaystyle{\displaystyle\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}% \left(z,q^{2}\right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_% {2}\left(0,q^{2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}}}
\frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}} = i^{-1/4}\sqrt{\frac{\Jacobithetaq{2}@{0}{q^{2}}\Jacobithetaq{4}@{0}{q^{2}}}{2}}		 

(JacobiTheta2(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta2(z, I*q)) = (I)^(- 1/4)*sqrt((JacobiTheta2(0, (q)^(2))*JacobiTheta4(0, (q)^(2)))/(2))
 
Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[2, z, I*q]] == (I)^(- 1/4)*Sqrt[Divide[EllipticTheta[2, 0, (q)^(2)]*EllipticTheta[4, 0, (q)^(2)],2]]
 Failure Failure Error
Failed [7 / 70]
Result: Complex[-1.1102230246251565*^-16, 0.47279727016045703]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[4.440892098500626*^-16, 0.4727972701604571]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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