Elliptic Integrals - 20.2 Definitions and Periodic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
20.2.E1 θ 1 ⁡ ( z | τ ) = θ 1 ⁡ ( z , q ) Jacobi-theta-tau 1 𝑧 𝜏 Jacobi-theta 1 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=\theta_{1}% \left(z,q\right)}}
\Jacobithetatau{1}@{z}{\tau} = \Jacobithetaq{1}@{z}{q}		 

JacobiTheta1(z,exp(I*Pi*tau)) = JacobiTheta1(z, q)

EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[1, z, q]
Failure Failure Error Successful [Tested: 300]
20.2.E2 θ 2 ⁡ ( z | τ ) = θ 2 ⁡ ( z , q ) Jacobi-theta-tau 2 𝑧 𝜏 Jacobi-theta 2 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{2}% \left(z,q\right)}}
\Jacobithetatau{2}@{z}{\tau} = \Jacobithetaq{2}@{z}{q}		 

JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta2(z, q)

EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, z, q]
Failure Failure Error Successful [Tested: 300]
20.2.E3 θ 3 ⁡ ( z | τ ) = θ 3 ⁡ ( z , q ) Jacobi-theta-tau 3 𝑧 𝜏 Jacobi-theta 3 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{3}% \left(z,q\right)}}
\Jacobithetatau{3}@{z}{\tau} = \Jacobithetaq{3}@{z}{q}		 

JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta3(z, q)

EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, z, q]
Failure Failure Error Successful [Tested: 300]
20.2.E4 θ 4 ⁡ ( z | τ ) = θ 4 ⁡ ( z , q ) Jacobi-theta-tau 4 𝑧 𝜏 Jacobi-theta 4 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{4}% \left(z,q\right)}}
\Jacobithetatau{4}@{z}{\tau} = \Jacobithetaq{4}@{z}{q}		 

JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta4(z, q)

EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, z, q]
Failure Failure Error Successful [Tested: 300]
20.2.E5 z m , n = ( m + n ⁢ τ ) ⁢ π subscript 𝑧 𝑚 𝑛 𝑚 𝑛 𝜏 𝜋 {\displaystyle{\displaystyle z_{m,n}=(m+n\tau)\pi}}
z_{m,n} = (m+n\tau)\pi		 

z[m , n] = (m + n*tau)*Pi
 
Subscript[z, m , n] == (m + n*\[Tau])*Pi
 Skipped - no semantic math Skipped - no semantic math - -
20.2.E6 θ 1 ⁡ ( z + ( m + n ⁢ τ ) ⁢ π | τ ) = ( - 1 ) m + n ⁢ q - n 2 ⁢ e - 2 ⁢ i ⁢ n ⁢ z ⁢ θ 1 ⁡ ( z | τ ) Jacobi-theta-tau 1 𝑧 𝑚 𝑛 𝜏 𝜋 𝜏 superscript 1 𝑚 𝑛 superscript 𝑞 superscript 𝑛 2 superscript 𝑒 2 𝑖 𝑛 𝑧 Jacobi-theta-tau 1 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(z+(m+n\tau)\pi\middle|\tau\right)=% (-1)^{m+n}q^{-n^{2}}e^{-2inz}\theta_{1}\left(z\middle|\tau\right)}}
\Jacobithetatau{1}@{z+(m+n\tau)\pi}{\tau} = (-1)^{m+n}q^{-n^{2}}e^{-2inz}\Jacobithetatau{1}@{z}{\tau}		 

JacobiTheta1(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (- 1)^(m + n)* (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta1(z,exp(I*Pi*tau))

EllipticTheta[1, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (- 1)^(m + n)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: -18.62843952+6.320473139*I
Test Values: {q = 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, m = 1, n = 1}


Result: -4187.991134+3174.249087*I
Test Values: {q = 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, m = 1, n = 2}

... skip entries to safe data
Failed [72 / 300]
Result: Complex[-18.628439525286133, 6.320473094431787]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[-4187.991166649552, 3174.249038247393]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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]]]}

... skip entries to safe data
20.2.E7 θ 2 ⁡ ( z + ( m + n ⁢ τ ) ⁢ π | τ ) = ( - 1 ) m ⁢ q - n 2 ⁢ e - 2 ⁢ i ⁢ n ⁢ z ⁢ θ 2 ⁡ ( z | τ ) Jacobi-theta-tau 2 𝑧 𝑚 𝑛 𝜏 𝜋 𝜏 superscript 1 𝑚 superscript 𝑞 superscript 𝑛 2 superscript 𝑒 2 𝑖 𝑛 𝑧 Jacobi-theta-tau 2 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{2}\left(z+(m+n\tau)\pi\middle|\tau\right)=% (-1)^{m}q^{-n^{2}}e^{-2inz}\theta_{2}\left(z\middle|\tau\right)}}
\Jacobithetatau{2}@{z+(m+n\tau)\pi}{\tau} = (-1)^{m}q^{-n^{2}}e^{-2inz}\Jacobithetatau{2}@{z}{\tau}		 

JacobiTheta2(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (- 1)^(m)* (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta2(z,exp(I*Pi*tau))

EllipticTheta[2, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (- 1)^(m)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: 3.950576529-14.16574159*I
Test Values: {q = 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, m = 1, n = 1}


Result: 194.3416227+3923.809342*I
Test Values: {q = 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, m = 1, n = 2}

... skip entries to safe data
Failed [72 / 300]
Result: Complex[3.9505765593957305, -14.165741580817551]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[194.34158297403354, 3923.809350793304]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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]]]}

... skip entries to safe data
20.2.E8 θ 3 ⁡ ( z + ( m + n ⁢ τ ) ⁢ π | τ ) = q - n 2 ⁢ e - 2 ⁢ i ⁢ n ⁢ z ⁢ θ 3 ⁡ ( z | τ ) Jacobi-theta-tau 3 𝑧 𝑚 𝑛 𝜏 𝜋 𝜏 superscript 𝑞 superscript 𝑛 2 superscript 𝑒 2 𝑖 𝑛 𝑧 Jacobi-theta-tau 3 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(z+(m+n\tau)\pi\middle|\tau\right)=% q^{-n^{2}}e^{-2inz}\theta_{3}\left(z\middle|\tau\right)}}
\Jacobithetatau{3}@{z+(m+n\tau)\pi}{\tau} = q^{-n^{2}}e^{-2inz}\Jacobithetatau{3}@{z}{\tau}		 

JacobiTheta3(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta3(z,exp(I*Pi*tau))

EllipticTheta[3, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: -8.181021151+18.44680448*I
Test Values: {q = 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, m = 1, n = 1}


Result: 516.5479372-5365.925849*I
Test Values: {q = 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, m = 1, n = 2}

... skip entries to safe data
Failed [72 / 300]
Result: Complex[-8.181021187984683, 18.446804447343553]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[516.547995329447, -5365.925840115722]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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]]]}

... skip entries to safe data
20.2.E9 θ 4 ⁡ ( z + ( m + n ⁢ τ ) ⁢ π | τ ) = ( - 1 ) n ⁢ q - n 2 ⁢ e - 2 ⁢ i ⁢ n ⁢ z ⁢ θ 4 ⁡ ( z | τ ) Jacobi-theta-tau 4 𝑧 𝑚 𝑛 𝜏 𝜋 𝜏 superscript 1 𝑛 superscript 𝑞 superscript 𝑛 2 superscript 𝑒 2 𝑖 𝑛 𝑧 Jacobi-theta-tau 4 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{4}\left(z+(m+n\tau)\pi\middle|\tau\right)=% (-1)^{n}q^{-n^{2}}e^{-2inz}\theta_{4}\left(z\middle|\tau\right)}}
\Jacobithetatau{4}@{z+(m+n\tau)\pi}{\tau} = (-1)^{n}q^{-n^{2}}e^{-2inz}\Jacobithetatau{4}@{z}{\tau}		 

JacobiTheta4(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (- 1)^(n)* (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta4(z,exp(I*Pi*tau))

EllipticTheta[4, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (- 1)^(n)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: -4.510694228-11.16801166*I
Test Values: {q = 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, m = 1, n = 1}


Result: -2085.869632-2449.864344*I
Test Values: {q = 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, m = 1, n = 2}

... skip entries to safe data
Failed [72 / 300]
Result: Complex[-4.5106942149502025, -11.168011665083736]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[-2085.8696157878926, -2449.864367431773]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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]]]}

... skip entries to safe data
20.2.E11 θ 1 ⁡ ( z | τ ) = - θ 2 ⁡ ( z + 1 2 ⁢ π | τ ) Jacobi-theta-tau 1 𝑧 𝜏 Jacobi-theta-tau 2 𝑧 1 2 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=-\theta_{2}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}
\Jacobithetatau{1}@{z}{\tau} = -\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi}{\tau}		 

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

EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == - EllipticTheta[2, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 70]
20.2.E11 - θ 2 ⁡ ( z + 1 2 ⁢ π | τ ) = - i ⁢ M ⁢ θ 4 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) Jacobi-theta-tau 2 𝑧 1 2 𝜋 𝜏 𝑖 𝑀 Jacobi-theta-tau 4 𝑧 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle-\theta_{2}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=-iM\theta_{4}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}
-\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi}{\tau} = -iM\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi\tau}{\tau}		 

- JacobiTheta2(z +(1)/(2)*Pi,exp(I*Pi*tau)) = - I*M*JacobiTheta4(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))

- EllipticTheta[2, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == - I*M*EllipticTheta[4, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: -2.656130280+.8441101403*I
Test Values: {M = 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: 2.726401771+.6812031274*I
Test Values: {M = 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[-2.65613027348202, 0.8441101301235214]
Test Values: {Rule[M, 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[-2.1127027753782777, -0.09362434622808774]
Test Values: {Rule[M, 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
20.2.E11 - i ⁢ M ⁢ θ 4 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) = - i ⁢ M ⁢ θ 3 ⁡ ( z + 1 2 ⁢ π + 1 2 ⁢ π ⁢ τ | τ ) 𝑖 𝑀 Jacobi-theta-tau 4 𝑧 1 2 𝜋 𝜏 𝜏 𝑖 𝑀 Jacobi-theta-tau 3 𝑧 1 2 𝜋 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle-iM\theta_{4}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=-iM\theta_{3}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|% \tau\right)}}
-iM\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi\tau}{\tau} = -iM\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}		 

- I*M*JacobiTheta4(z +(1)/(2)*Pi*tau,exp(I*Pi*tau)) = - I*M*JacobiTheta3(z +(1)/(2)*Pi +(1)/(2)*Pi*tau,exp(I*Pi*tau))

- I*M*EllipticTheta[4, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == - I*M*EllipticTheta[3, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 300]
20.2.E12 θ 2 ⁡ ( z | τ ) = θ 1 ⁡ ( z + 1 2 ⁢ π | τ ) Jacobi-theta-tau 2 𝑧 𝜏 Jacobi-theta-tau 1 𝑧 1 2 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{1}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}
\Jacobithetatau{2}@{z}{\tau} = \Jacobithetatau{1}@{z+\tfrac{1}{2}\pi}{\tau}		 

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

EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[1, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 70]
20.2.E12 θ 1 ⁡ ( z + 1 2 ⁢ π | τ ) = M ⁢ θ 3 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) Jacobi-theta-tau 1 𝑧 1 2 𝜋 𝜏 𝑀 Jacobi-theta-tau 3 𝑧 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=M\theta_{3}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}
\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi}{\tau} = M\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi\tau}{\tau}		 

JacobiTheta1(z +(1)/(2)*Pi,exp(I*Pi*tau)) = M*JacobiTheta3(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))

EllipticTheta[1, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[3, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: -.5985410657+1.995750316*I
Test Values: {M = 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: -.54909285e-2-5.605651596*I
Test Values: {M = 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[-0.5985410729973577, 1.9957503125524838]
Test Values: {Rule[M, 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[-1.3220776692839347, 1.382964384147599]
Test Values: {Rule[M, 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
20.2.E12 M ⁢ θ 3 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) = M ⁢ θ 4 ⁡ ( z + 1 2 ⁢ π + 1 2 ⁢ π ⁢ τ | τ ) 𝑀 Jacobi-theta-tau 3 𝑧 1 2 𝜋 𝜏 𝜏 𝑀 Jacobi-theta-tau 4 𝑧 1 2 𝜋 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle M\theta_{3}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=M\theta_{4}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau% \right)}}
M\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi\tau}{\tau} = M\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}		 

M*JacobiTheta3(z +(1)/(2)*Pi*tau,exp(I*Pi*tau)) = M*JacobiTheta4(z +(1)/(2)*Pi +(1)/(2)*Pi*tau,exp(I*Pi*tau))

M*EllipticTheta[3, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[4, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 300]
20.2.E13 θ 3 ⁡ ( z | τ ) = θ 4 ⁡ ( z + 1 2 ⁢ π | τ ) Jacobi-theta-tau 3 𝑧 𝜏 Jacobi-theta-tau 4 𝑧 1 2 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{4}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}
\Jacobithetatau{3}@{z}{\tau} = \Jacobithetatau{4}@{z+\tfrac{1}{2}\pi}{\tau}		 

JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta4(z +(1)/(2)*Pi,exp(I*Pi*tau))

EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 70]
20.2.E13 θ 4 ⁡ ( z + 1 2 ⁢ π | τ ) = M ⁢ θ 2 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) Jacobi-theta-tau 4 𝑧 1 2 𝜋 𝜏 𝑀 Jacobi-theta-tau 2 𝑧 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle\theta_{4}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=M\theta_{2}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}
\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi}{\tau} = M\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi\tau}{\tau}		 

JacobiTheta4(z +(1)/(2)*Pi,exp(I*Pi*tau)) = M*JacobiTheta2(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))

EllipticTheta[4, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[2, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: -1.209558888+2.590545189*I
Test Values: {M = 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: -.8537358258+1.281173247*I
Test Values: {M = 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[-1.2095588901959111, 2.5905451776573183]
Test Values: {Rule[M, 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[-1.901447370098885, -0.21207958455265288]
Test Values: {Rule[M, 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
20.2.E13 M ⁢ θ 2 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) = M ⁢ θ 1 ⁡ ( z + 1 2 ⁢ π + 1 2 ⁢ π ⁢ τ | τ ) 𝑀 Jacobi-theta-tau 2 𝑧 1 2 𝜋 𝜏 𝜏 𝑀 Jacobi-theta-tau 1 𝑧 1 2 𝜋 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle M\theta_{2}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=M\theta_{1}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau% \right)}}
M\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi\tau}{\tau} = M\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}		 

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

M*EllipticTheta[2, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[1, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 300]
20.2.E14 θ 4 ⁡ ( z | τ ) = θ 3 ⁡ ( z + 1 2 ⁢ π | τ ) Jacobi-theta-tau 4 𝑧 𝜏 Jacobi-theta-tau 3 𝑧 1 2 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{3}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}
\Jacobithetatau{4}@{z}{\tau} = \Jacobithetatau{3}@{z+\tfrac{1}{2}\pi}{\tau}		 

JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta3(z +(1)/(2)*Pi,exp(I*Pi*tau))

EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 70]
20.2.E14 θ 3 ⁡ ( z + 1 2 ⁢ π | τ ) = - i ⁢ M ⁢ θ 1 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) Jacobi-theta-tau 3 𝑧 1 2 𝜋 𝜏 𝑖 𝑀 Jacobi-theta-tau 1 𝑧 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=-iM\theta_{1}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}
\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi}{\tau} = -iM\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi\tau}{\tau}		 

JacobiTheta3(z +(1)/(2)*Pi,exp(I*Pi*tau)) = - I*M*JacobiTheta1(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))

EllipticTheta[3, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == - I*M*EllipticTheta[1, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Failure Failure
Failed [300 / 300]
Result: .6082553523+1.594370406*I
Test Values: {M = 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: -3.995156823-3.872683361*I
Test Values: {M = 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[0.6082553477594059, 1.594370409676146]
Test Values: {Rule[M, 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[-2.5995966648178563, -0.1152010311326023]
Test Values: {Rule[M, 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
20.2.E14 - i ⁢ M ⁢ θ 1 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) = i ⁢ M ⁢ θ 2 ⁡ ( z + 1 2 ⁢ π + 1 2 ⁢ π ⁢ τ | τ ) 𝑖 𝑀 Jacobi-theta-tau 1 𝑧 1 2 𝜋 𝜏 𝜏 𝑖 𝑀 Jacobi-theta-tau 2 𝑧 1 2 𝜋 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle-iM\theta_{1}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=iM\theta_{2}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|% \tau\right)}}
-iM\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi\tau}{\tau} = iM\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}		 

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

- I*M*EllipticTheta[1, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == I*M*EllipticTheta[2, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 300]
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