Results of Theta Functions

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DLMF Formula 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)}} JacobiTheta1(z,exp(I*Pi*tau))= JacobiTheta1(z, q) EllipticTheta[1, z, \[Tau]]= EllipticTheta[1, z, q] Failure Failure Error Successful
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)}} JacobiTheta2(z,exp(I*Pi*tau))= JacobiTheta2(z, q) EllipticTheta[2, z, \[Tau]]= EllipticTheta[2, z, q] Failure Failure Error Successful
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)}} JacobiTheta3(z,exp(I*Pi*tau))= JacobiTheta3(z, q) EllipticTheta[3, z, \[Tau]]= EllipticTheta[3, z, q] Failure Failure Error Successful
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)}} JacobiTheta4(z,exp(I*Pi*tau))= JacobiTheta4(z, q) EllipticTheta[4, z, \[Tau]]= EllipticTheta[4, z, q] Failure Failure Error Successful
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)}} 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, \[Tau]]=(- 1)^(m + n)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[1, z, \[Tau]] Failure Failure Error Skip
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)}} 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, \[Tau]]=(- 1)^(m)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[2, z, \[Tau]] Failure Failure Error Skip
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)}} 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, \[Tau]]= (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[3, z, \[Tau]] Failure Failure Error Skip
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)}} 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, \[Tau]]=(- 1)^(n)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[4, z, \[Tau]] Failure Failure Error Skip
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)}} JacobiTheta1(z,exp(I*Pi*tau))= - JacobiTheta2(z +(1)/(2)*Pi,exp(I*Pi*tau)) EllipticTheta[1, z, \[Tau]]= - EllipticTheta[2, z +Divide[1,2]*Pi, \[Tau]] Successful Failure - Successful
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)}} - 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, \[Tau]]= - I*M*EllipticTheta[4, z +Divide[1,2]*Pi*\[Tau], \[Tau]] Failure Failure Error Successful
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)}} - 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], \[Tau]]= - I*M*EllipticTheta[3, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], \[Tau]] Successful Failure - Successful
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)}} JacobiTheta2(z,exp(I*Pi*tau))= JacobiTheta1(z +(1)/(2)*Pi,exp(I*Pi*tau)) EllipticTheta[2, z, \[Tau]]= EllipticTheta[1, z +Divide[1,2]*Pi, \[Tau]] Successful Failure - Successful
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)}} 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, \[Tau]]= M*EllipticTheta[3, z +Divide[1,2]*Pi*\[Tau], \[Tau]] Failure Failure Error Successful
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*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], \[Tau]]= M*EllipticTheta[4, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], \[Tau]] Successful Failure - Successful
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)}} JacobiTheta3(z,exp(I*Pi*tau))= JacobiTheta4(z +(1)/(2)*Pi,exp(I*Pi*tau)) EllipticTheta[3, z, \[Tau]]= EllipticTheta[4, z +Divide[1,2]*Pi, \[Tau]] Successful Failure - Successful
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)}} 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, \[Tau]]= M*EllipticTheta[2, z +Divide[1,2]*Pi*\[Tau], \[Tau]] Failure Failure Error Successful
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*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], \[Tau]]= M*EllipticTheta[1, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], \[Tau]] Successful Failure - Successful
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)}} JacobiTheta4(z,exp(I*Pi*tau))= JacobiTheta3(z +(1)/(2)*Pi,exp(I*Pi*tau)) EllipticTheta[4, z, \[Tau]]= EllipticTheta[3, z +Divide[1,2]*Pi, \[Tau]] Successful Failure - Successful
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)}} 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, \[Tau]]= - I*M*EllipticTheta[1, z +Divide[1,2]*Pi*\[Tau], \[Tau]] Failure Failure Error Successful
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)}} - 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], \[Tau]]= I*M*EllipticTheta[2, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], \[Tau]] Successful Failure - Successful
20.4.E1 θ 1 ⁡ ( 0 , q ) = θ 2 ′ ⁡ ( 0 , q ) Jacobi-theta 1 0 𝑞 diffop Jacobi-theta 2 1 0 𝑞 {\displaystyle{\displaystyle\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q% \right)}} JacobiTheta1(0, q)= subs( temp=0, diff( JacobiTheta2(temp, q), temp$(1) ) ) EllipticTheta[1, 0, q]= (D[EllipticTheta[2, temp, q], {temp, 1}]/.temp-> 0) Successful Successful - -
20.4.E1 θ 2 ′ ⁡ ( 0 , q ) = θ 3 ′ ⁡ ( 0 , q ) diffop Jacobi-theta 2 1 0 𝑞 diffop Jacobi-theta 3 1 0 𝑞 {\displaystyle{\displaystyle\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)}} subs( temp=0, diff( JacobiTheta2(temp, q), temp$(1) ) )= subs( temp=0, diff( JacobiTheta3(temp, q), temp$(1) ) ) (D[EllipticTheta[2, temp, q], {temp, 1}]/.temp-> 0)= (D[EllipticTheta[3, temp, q], {temp, 1}]/.temp-> 0) Failure Successful Error -
20.4.E1 θ 3 ′ ⁡ ( 0 , q ) = θ 4 ′ ⁡ ( 0 , q ) diffop Jacobi-theta 3 1 0 𝑞 diffop Jacobi-theta 4 1 0 𝑞 {\displaystyle{\displaystyle\theta_{3}'\left(0,q\right)=\theta_{4}'\left(0,q% \right)}} subs( temp=0, diff( JacobiTheta3(temp, q), temp$(1) ) )= subs( temp=0, diff( JacobiTheta4(temp, q), temp$(1) ) ) (D[EllipticTheta[3, temp, q], {temp, 1}]/.temp-> 0)= (D[EllipticTheta[4, temp, q], {temp, 1}]/.temp-> 0) Failure Successful Skip -
20.4.E1 θ 4 ′ ⁡ ( 0 , q ) = 0 diffop Jacobi-theta 4 1 0 𝑞 0 {\displaystyle{\displaystyle\theta_{4}'\left(0,q\right)=0}} subs( temp=0, diff( JacobiTheta4(temp, q), temp$(1) ) )= 0 (D[EllipticTheta[4, temp, q], {temp, 1}]/.temp-> 0)= 0 Successful Successful - -
20.4.E6 θ 1 ′ ⁡ ( 0 , q ) = θ 2 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( 0 , q ) ⁢ θ 4 ⁡ ( 0 , q ) diffop Jacobi-theta 1 1 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 3 0 𝑞 Jacobi-theta 4 0 𝑞 {\displaystyle{\displaystyle\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q% \right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}} subs( temp=0, diff( JacobiTheta1(temp, q), temp$(1) ) )= JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q) (D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> 0)= EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q] Failure Failure Error Successful
20.4.E7 θ 1 ′′ ⁢ ( 0 , q ) = θ 2 ′′′ ⁡ ( 0 , q ) diffop Jacobi-theta 1 2 0 𝑞 diffop Jacobi-theta 2 3 0 𝑞 {\displaystyle{\displaystyle\theta_{1}''(0,q)=\theta_{2}'''\left(0,q\right)}} subs( temp=(0 , q), diff( JacobiTheta1(temp, =), temp$(2) ) )*subs( temp=0, diff( JacobiTheta2(temp, q), temp$(3) ) ) (D[EllipticTheta[1, temp, =], {temp, 2}]/.temp-> (0 , q))*(D[EllipticTheta[2, temp, q], {temp, 3}]/.temp-> 0) Error Failure - Error
20.4.E7 θ 2 ′′′ ⁡ ( 0 , q ) = θ 3 ′′′ ⁡ ( 0 , q ) diffop Jacobi-theta 2 3 0 𝑞 diffop Jacobi-theta 3 3 0 𝑞 {\displaystyle{\displaystyle\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(0% ,q\right)}} subs( temp=0, diff( JacobiTheta2(temp, q), temp$(3) ) )= subs( temp=0, diff( JacobiTheta3(temp, q), temp$(3) ) ) (D[EllipticTheta[2, temp, q], {temp, 3}]/.temp-> 0)= (D[EllipticTheta[3, temp, q], {temp, 3}]/.temp-> 0) Failure Failure Error
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
20.4.E7 θ 3 ′′′ ⁡ ( 0 , q ) = θ 4 ′′′ ⁡ ( 0 , q ) diffop Jacobi-theta 3 3 0 𝑞 diffop Jacobi-theta 4 3 0 𝑞 {\displaystyle{\displaystyle\theta_{3}'''\left(0,q\right)=\theta_{4}'''\left(0% ,q\right)}} subs( temp=0, diff( JacobiTheta3(temp, q), temp$(3) ) )= subs( temp=0, diff( JacobiTheta4(temp, q), temp$(3) ) ) (D[EllipticTheta[3, temp, q], {temp, 3}]/.temp-> 0)= (D[EllipticTheta[4, temp, q], {temp, 3}]/.temp-> 0) Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
20.4.E7 θ 4 ′′′ ⁡ ( 0 , q ) = 0 diffop Jacobi-theta 4 3 0 𝑞 0 {\displaystyle{\displaystyle\theta_{4}'''\left(0,q\right)=0}} subs( temp=0, diff( JacobiTheta4(temp, q), temp$(3) ) )= 0 (D[EllipticTheta[4, temp, q], {temp, 3}]/.temp-> 0)= 0 Successful Failure -
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
20.4.E8 θ 1 ′′′ ⁡ ( 0 , q ) θ 1 ′ ⁡ ( 0 , q ) = - 1 + 24 ⁢ ∑ n = 1 ∞ q 2 ⁢ n ( 1 - q 2 ⁢ n ) 2 diffop Jacobi-theta 1 3 0 𝑞 diffop Jacobi-theta 1 1 0 𝑞 1 24 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 superscript 1 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'% \left(0,q\right)}=-1+24\sum_{n=1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}}} (subs( temp=0, diff( JacobiTheta1(temp, q), temp$(3) ) ))/(subs( temp=0, diff( JacobiTheta1(temp, q), temp$(1) ) ))= - 1 + 24*sum(((q)^(2*n))/((1 - (q)^(2*n))^(2)), n = 1..infinity) Divide[D[EllipticTheta[1, temp, q], {temp, 3}]/.temp-> 0,D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> 0]= - 1 + 24*Sum[Divide[(q)^(2*n),(1 - (q)^(2*n))^(2)], {n, 1, Infinity}] Failure Failure Skip Skip
20.4.E9 θ 2 ′′ ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) = - 1 - 8 ⁢ ∑ n = 1 ∞ q 2 ⁢ n ( 1 + q 2 ⁢ n ) 2 diffop Jacobi-theta 2 2 0 𝑞 Jacobi-theta 2 0 𝑞 1 8 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 superscript 1 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}% \left(0,q\right)}=-1-8\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}}}} (subs( temp=0, diff( JacobiTheta2(temp, q), temp$(2) ) ))/(JacobiTheta2(0, q))= - 1 - 8*sum(((q)^(2*n))/((1 + (q)^(2*n))^(2)), n = 1..infinity) Divide[D[EllipticTheta[2, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[2, 0, q]]= - 1 - 8*Sum[Divide[(q)^(2*n),(1 + (q)^(2*n))^(2)], {n, 1, Infinity}] Failure Failure Skip Skip
20.4.E10 θ 3 ′′ ⁡ ( 0 , q ) θ 3 ⁡ ( 0 , q ) = - 8 ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 ( 1 + q 2 ⁢ n - 1 ) 2 diffop Jacobi-theta 3 2 0 𝑞 Jacobi-theta 3 0 𝑞 8 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 1 superscript 𝑞 2 𝑛 1 2 {\displaystyle{\displaystyle\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}% \left(0,q\right)}=-8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}}}} (subs( temp=0, diff( JacobiTheta3(temp, q), temp$(2) ) ))/(JacobiTheta3(0, q))= - 8*sum(((q)^(2*n - 1))/((1 + (q)^(2*n - 1))^(2)), n = 1..infinity) Divide[D[EllipticTheta[3, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[3, 0, q]]= - 8*Sum[Divide[(q)^(2*n - 1),(1 + (q)^(2*n - 1))^(2)], {n, 1, Infinity}] Failure Failure Skip Error
20.4.E11 θ 4 ′′ ⁡ ( 0 , q ) θ 4 ⁡ ( 0 , q ) = 8 ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 ( 1 - q 2 ⁢ n - 1 ) 2 diffop Jacobi-theta 4 2 0 𝑞 Jacobi-theta 4 0 𝑞 8 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 1 superscript 𝑞 2 𝑛 1 2 {\displaystyle{\displaystyle\frac{\theta_{4}''\left(0,q\right)}{\theta_{4}% \left(0,q\right)}=8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}}} (subs( temp=0, diff( JacobiTheta4(temp, q), temp$(2) ) ))/(JacobiTheta4(0, q))= 8*sum(((q)^(2*n - 1))/((1 - (q)^(2*n - 1))^(2)), n = 1..infinity) Divide[D[EllipticTheta[4, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[4, 0, q]]= 8*Sum[Divide[(q)^(2*n - 1),(1 - (q)^(2*n - 1))^(2)], {n, 1, Infinity}] Failure Failure Skip Error
20.4.E12 θ 1 ′′′ ⁡ ( 0 , q ) θ 1 ′ ⁡ ( 0 , q ) = θ 2 ′′ ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) + θ 3 ′′ ⁡ ( 0 , q ) θ 3 ⁡ ( 0 , q ) + θ 4 ′′ ⁡ ( 0 , q ) θ 4 ⁡ ( 0 , q ) diffop Jacobi-theta 1 3 0 𝑞 diffop Jacobi-theta 1 1 0 𝑞 diffop Jacobi-theta 2 2 0 𝑞 Jacobi-theta 2 0 𝑞 diffop Jacobi-theta 3 2 0 𝑞 Jacobi-theta 3 0 𝑞 diffop Jacobi-theta 4 2 0 𝑞 Jacobi-theta 4 0 𝑞 {\displaystyle{\displaystyle\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'% \left(0,q\right)}=\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}\left(0,q% \right)}+\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}\left(0,q\right)}+\frac% {\theta_{4}''\left(0,q\right)}{\theta_{4}\left(0,q\right)}}} (subs( temp=0, diff( JacobiTheta1(temp, q), temp$(3) ) ))/(subs( temp=0, diff( JacobiTheta1(temp, q), temp$(1) ) ))=(subs( temp=0, diff( JacobiTheta2(temp, q), temp$(2) ) ))/(JacobiTheta2(0, q))+(subs( temp=0, diff( JacobiTheta3(temp, q), temp$(2) ) ))/(JacobiTheta3(0, q))+(subs( temp=0, diff( JacobiTheta4(temp, q), temp$(2) ) ))/(JacobiTheta4(0, q)) Divide[D[EllipticTheta[1, temp, q], {temp, 3}]/.temp-> 0,D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> 0]=Divide[D[EllipticTheta[2, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[2, 0, q]]+Divide[D[EllipticTheta[3, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[3, 0, q]]+Divide[D[EllipticTheta[4, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[4, 0, q]] Failure Failure Error Successful
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)}}} JacobiTheta1(z,exp(I*Pi*tau))= subs( temp=0, diff( JacobiTheta1(temp,exp(I*Pi*tau)), temp$(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, \[Tau]]= (D[EllipticTheta[1, temp, \[Tau]], {temp, 1}]/.temp-> 0)*Sin[z]*Product[Divide[Sin[n*Pi*\[Tau]+ z]*Sin[n*Pi*\[Tau]- z],(Sin[n*Pi*\[Tau]])^(2)], {n, 1, Infinity}] Failure Failure Skip Error
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)}}} 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, \[Tau]]= EllipticTheta[2, 0, \[Tau]]*Cos[z]*Product[Divide[Cos[n*Pi*\[Tau]+ z]*Cos[n*Pi*\[Tau]- z],(Cos[n*Pi*\[Tau]])^(2)], {n, 1, Infinity}] Failure Failure Skip Skip
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)}}} 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, \[Tau]]= EllipticTheta[3, 0, \[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}] Failure Failure Skip Error
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)}}} 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, \[Tau]]= EllipticTheta[4, 0, \[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}] Failure Failure Skip Error
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((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}]= Product[(1 - (q)^(2*n))*(1 + (q)^(2*n - 1)* (p)^(2))*(1 + (q)^(2*n - 1)* (p)^(- 2)), {n, 1, Infinity}] Error Error - -
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}}}} (subs( temp=z, diff( JacobiTheta1(temp, q), temp$(1) ) ))/(JacobiTheta1(z, q))- cot(z)= 4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity) Divide[D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> z,EllipticTheta[1, z, q]]- Cot[z]= 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}] Failure Failure Skip Skip
20.5.E10 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - 2 ⁢ q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n = 4 ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 2 superscript 𝑞 2 𝑛 2 𝑧 superscript 𝑞 4 𝑛 4 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle 4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n}}{1-2q^{2n}\cos\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q% ^{2n}}\sin\left(2nz\right)}} 4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity)= 4*sum(((q)^(2*n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity) 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}]= 4*Sum[Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}] Failure Failure Skip Skip
20.5.E11 θ 2 ′ ⁡ ( z , q ) θ 2 ⁡ ( z , q ) + tan ⁡ z = - 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 + 2 ⁢ q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n diffop Jacobi-theta 2 1 𝑧 𝑞 Jacobi-theta 2 𝑧 𝑞 𝑧 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 2 superscript 𝑞 2 𝑛 2 𝑧 superscript 𝑞 4 𝑛 {\displaystyle{\displaystyle\frac{\theta_{2}'\left(z,q\right)}{\theta_{2}\left% (z,q\right)}+\tan z=-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q% ^{2n}\cos\left(2z\right)+q^{4n}}}} (subs( temp=z, diff( JacobiTheta2(temp, q), temp$(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, temp, q], {temp, 1}]/.temp-> z,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}] Failure Failure Skip Skip
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(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}]= 4*Sum[(- 1)^(n)*Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}] Failure Failure Skip Skip
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}}}} (subs( temp=z, diff( JacobiTheta3(temp, q), temp$(1) ) ))/(JacobiTheta3(z, q))= - 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity) Divide[D[EllipticTheta[3, temp, q], {temp, 1}]/.temp-> z,EllipticTheta[3, z, q]]= - 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}] Failure Failure Skip Error
20.5.E12 - 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 1 + 2 ⁢ q 2 ⁢ n - 1 ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n - 2 = 4 ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 4 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n-1}}{1+2q^{2n-1}\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}(-1)^{n}% \frac{q^{n}}{1-q^{2n}}\sin\left(2nz\right)}} - 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity)= 4*sum((- 1)^(n)*((q)^(n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity) - 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}]= 4*Sum[(- 1)^(n)*Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}] Failure Failure Skip Skip
20.5.E13 θ 4 ′ ⁡ ( z , q ) θ 4 ⁡ ( z , q ) = 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 1 - 2 ⁢ q 2 ⁢ n - 1 ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n - 2 diffop Jacobi-theta 4 1 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 {\displaystyle{\displaystyle\frac{\theta_{4}'\left(z,q\right)}{\theta_{4}\left% (z,q\right)}=4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1% }\cos\left(2z\right)+q^{4n-2}}}} (subs( temp=z, diff( JacobiTheta4(temp, q), temp$(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, temp, q], {temp, 1}]/.temp-> z,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}] Failure Failure Skip Error
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(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}]= 4*Sum[Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}] Failure Failure Skip Error
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)}} 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, \[Tau]]= EllipticTheta[2, 0, \[Tau]]* Limit[Product[Limit[Product[1 +Divide[z,(m -Divide[1,2]+ n*\[Tau])* Pi], {m, 1 - M, M}], M -> Infinity], {n, - N, N}], N -> Infinity] Failure Failure Skip Error
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)}} 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, \[Tau]]= EllipticTheta[3, 0, \[Tau]]* Limit[Product[Limit[Product[1 +Divide[z,(m -Divide[1,2]+(n -Divide[1,2])*\[Tau])* Pi], {m, 1 - M, M}], M -> Infinity], {n, 1 - N, N}], N -> Infinity] Failure Failure Skip Error
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)}} 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, \[Tau]]= EllipticTheta[4, 0, \[Tau]]* Limit[Product[Limit[Product[1 +Divide[z,(m +(n -Divide[1,2])*\[Tau])* Pi], {m, - M, M}], M -> Infinity], {n, 1 - N, N}], N -> Infinity] Failure Failure Skip Error
20.6.E2 θ 1 ⁡ ( π ⁢ z | τ ) = π ⁢ z ⁢ θ 1 ′ ⁡ ( 0 | τ ) ⁢ exp ⁡ ( - ∑ j = 1 ∞ 1 2 ⁢ j ⁢ δ 2 ⁢ j ⁢ ( τ ) ⁢ z 2 ⁢ j ) Jacobi-theta-tau 1 𝜋 𝑧 𝜏 𝜋 𝑧 diffop Jacobi-theta-tau 1 1 0 𝜏 superscript subscript 𝑗 1 1 2 𝑗 subscript 𝛿 2 𝑗 𝜏 superscript 𝑧 2 𝑗 {\displaystyle{\displaystyle\theta_{1}\left(\pi z\middle|\tau\right)=\pi z% \theta_{1}'\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j% }\delta_{2j}(\tau)z^{2j}\right)}} JacobiTheta1(Pi*z,exp(I*Pi*tau))= Pi*z*subs( temp=0, diff( JacobiTheta1(temp,exp(I*Pi*tau)), temp$(1) ) )*exp(- sum((1)/(2*j)*delta[2*j]*(tau)* (z)^(2*j), j = 1..infinity)) EllipticTheta[1, Pi*z, \[Tau]]= Pi*z*(D[EllipticTheta[1, temp, \[Tau]], {temp, 1}]/.temp-> 0)*Exp[- Sum[Divide[1,2*j]*Subscript[\[Delta], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}]] Failure Failure Skip Skip
20.6.E3 θ 2 ⁡ ( π ⁢ z | τ ) = θ 2 ⁡ ( 0 | τ ) ⁢ exp ⁡ ( - ∑ j = 1 ∞ 1 2 ⁢ j ⁢ α 2 ⁢ j ⁢ ( τ ) ⁢ z 2 ⁢ j ) Jacobi-theta-tau 2 𝜋 𝑧 𝜏 Jacobi-theta-tau 2 0 𝜏 superscript subscript 𝑗 1 1 2 𝑗 subscript 𝛼 2 𝑗 𝜏 superscript 𝑧 2 𝑗 {\displaystyle{\displaystyle\theta_{2}\left(\pi z\middle|\tau\right)=\theta_{2% }\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\alpha_{2% j}(\tau)z^{2j}\right)}} JacobiTheta2(Pi*z,exp(I*Pi*tau))= JacobiTheta2(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*alpha[2*j]*(tau)* (z)^(2*j), j = 1..infinity)) EllipticTheta[2, Pi*z, \[Tau]]= EllipticTheta[2, 0, \[Tau]]*Exp[- Sum[Divide[1,2*j]*Subscript[\[Alpha], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}]] Failure Failure Skip Skip
20.6.E4 θ 3 ⁡ ( π ⁢ z | τ ) = θ 3 ⁡ ( 0 | τ ) ⁢ exp ⁡ ( - ∑ j = 1 ∞ 1 2 ⁢ j ⁢ β 2 ⁢ j ⁢ ( τ ) ⁢ z 2 ⁢ j ) Jacobi-theta-tau 3 𝜋 𝑧 𝜏 Jacobi-theta-tau 3 0 𝜏 superscript subscript 𝑗 1 1 2 𝑗 subscript 𝛽 2 𝑗 𝜏 superscript 𝑧 2 𝑗 {\displaystyle{\displaystyle\theta_{3}\left(\pi z\middle|\tau\right)=\theta_{3% }\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\beta_{2j% }(\tau)z^{2j}\right)}} JacobiTheta3(Pi*z,exp(I*Pi*tau))= JacobiTheta3(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*beta[2*j]*(tau)* (z)^(2*j), j = 1..infinity)) EllipticTheta[3, Pi*z, \[Tau]]= EllipticTheta[3, 0, \[Tau]]*Exp[- Sum[Divide[1,2*j]*Subscript[\[Beta], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}]] Failure Failure Skip Skip
20.6.E5 θ 4 ⁡ ( π ⁢ z | τ ) = θ 4 ⁡ ( 0 | τ ) ⁢ exp ⁡ ( - ∑ j = 1 ∞ 1 2 ⁢ j ⁢ γ 2 ⁢ j ⁢ ( τ ) ⁢ z 2 ⁢ j ) Jacobi-theta-tau 4 𝜋 𝑧 𝜏 Jacobi-theta-tau 4 0 𝜏 superscript subscript 𝑗 1 1 2 𝑗 subscript 𝛾 2 𝑗 𝜏 superscript 𝑧 2 𝑗 {\displaystyle{\displaystyle\theta_{4}\left(\pi z\middle|\tau\right)=\theta_{4% }\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\gamma_{2% j}(\tau)z^{2j}\right)}} JacobiTheta4(Pi*z,exp(I*Pi*tau))= JacobiTheta4(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*gamma[2*j]*(tau)* (z)^(2*j), j = 1..infinity)) EllipticTheta[4, Pi*z, \[Tau]]= EllipticTheta[4, 0, \[Tau]]*Exp[- Sum[Divide[1,2*j]*Subscript[\[Gamma], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}]] Failure Failure Skip Skip
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)}} (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
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)}} (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
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)}} (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
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)}} (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
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)}} (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
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)}} (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
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)}} (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
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)}} (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
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)}} (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
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)}}} 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
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)}}} (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 Successful
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)}} (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
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)}}} (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
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)}} (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 Successful
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)}} 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
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)}} 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
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)}} 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, 2*\[Tau]]= A*EllipticTheta[1, z, \[Tau]]*EllipticTheta[2, z, \[Tau]] Failure Failure Error Successful
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)}} 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, 2*\[Tau]]= A*EllipticTheta[1, Divide[1,4]*Pi - z, \[Tau]]*EllipticTheta[1, Divide[1,4]*Pi + z, \[Tau]] Error Failure - Successful
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)}} 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, 2*\[Tau]]= A*EllipticTheta[3, Divide[1,4]*Pi - z, \[Tau]]*EllipticTheta[3, Divide[1,4]*Pi + z, \[Tau]] Error Failure - Successful
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)}} 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, 2*\[Tau]]= A*EllipticTheta[3, z, \[Tau]]*EllipticTheta[4, z, \[Tau]] Failure Failure Error Successful
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)}} 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, 4*\[Tau]]= B*EllipticTheta[1, z, \[Tau]]*EllipticTheta[1, Divide[1,4]*Pi - z, \[Tau]]* EllipticTheta[1, Divide[1,4]*Pi + z, \[Tau]]*EllipticTheta[2, z, \[Tau]] Error Failure - Successful
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)}} 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, 4*\[Tau]]= B*EllipticTheta[2, Divide[1,8]*Pi - z, \[Tau]]*EllipticTheta[2, Divide[1,8]*Pi + z, \[Tau]]* EllipticTheta[2, Divide[3,8]*Pi - z, \[Tau]]*EllipticTheta[2, Divide[3,8]*Pi + z, \[Tau]] Error Failure - Successful
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)}} 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, 4*\[Tau]]= B*EllipticTheta[3, Divide[1,8]*Pi - z, \[Tau]]*EllipticTheta[3, Divide[1,8]*Pi + z, \[Tau]]* EllipticTheta[3, Divide[3,8]*Pi - z, \[Tau]]*EllipticTheta[3, Divide[3,8]*Pi + z, \[Tau]] Error Failure - Successful
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)}} 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, 4*\[Tau]]= B*EllipticTheta[4, z, \[Tau]]*EllipticTheta[4, Divide[1,4]*Pi - z, \[Tau]]* EllipticTheta[4, Divide[1,4]*Pi + z, \[Tau]]*EllipticTheta[3, z, \[Tau]] Error Failure - Successful
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)}}} 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, \[Tau]],EllipticTheta[4, z, \[Tau]]], z]= -Divide[(EllipticTheta[3, 0, \[Tau]])^(2)* EllipticTheta[1, z, \[Tau]]*EllipticTheta[3, z, \[Tau]],(EllipticTheta[4, z, \[Tau]])^(2)] Failure Failure Error Successful
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)}} JacobiTheta1(z,exp(I*Pi*tau + 1))= exp(I*Pi/ 4)*JacobiTheta1(z,exp(I*Pi*tau)) EllipticTheta[1, z, \[Tau]+ 1]= Exp[I*Pi/ 4]*EllipticTheta[1, z, \[Tau]] Failure Failure Error Successful
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)}} JacobiTheta2(z,exp(I*Pi*tau + 1))= exp(I*Pi/ 4)*JacobiTheta2(z,exp(I*Pi*tau)) EllipticTheta[2, z, \[Tau]+ 1]= Exp[I*Pi/ 4]*EllipticTheta[2, z, \[Tau]] Failure Failure Error Successful
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)}} JacobiTheta3(z,exp(I*Pi*tau + 1))= JacobiTheta4(z,exp(I*Pi*tau)) EllipticTheta[3, z, \[Tau]+ 1]= EllipticTheta[4, z, \[Tau]] Failure Failure Error Successful
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)}} JacobiTheta4(z,exp(I*Pi*tau + 1))= JacobiTheta3(z,exp(I*Pi*tau)) EllipticTheta[4, z, \[Tau]+ 1]= EllipticTheta[3, z, \[Tau]] Failure Failure Error Successful
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)}}} (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
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}}}} (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 Successful
20.8.E1 θ 2 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( z , q ) ⁢ θ 4 ⁡ ( z , q ) θ 2 ⁡ ( z , q ) = 2 ⁢ ∑ n = - ∞ ∞ ( - 1 ) n ⁢ q n 2 ⁢ e i ⁢ 2 ⁢ n ⁢ z q - n ⁢ e - i ⁢ z + q n ⁢ e i ⁢ z Jacobi-theta 2 0 𝑞 Jacobi-theta 3 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 Jacobi-theta 2 𝑧 𝑞 2 superscript subscript 𝑛 superscript 1 𝑛 superscript 𝑞 superscript 𝑛 2 superscript 𝑒 𝑖 2 𝑛 𝑧 superscript 𝑞 𝑛 superscript 𝑒 𝑖 𝑧 superscript 𝑞 𝑛 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,% q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-% \infty}^{\infty}\frac{(-1)^{n}q^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}}} (JacobiTheta2(0, q)*JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta2(z, q))= 2*sum(((- 1)^(n)* (q)^((n)^(2))* exp(I*2*n*z))/((q)^(- n)* exp(- I*z)+ (q)^(n)* exp(I*z)), n = - infinity..infinity) Divide[EllipticTheta[2, 0, q]*EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[2, z, q]]= 2*Sum[Divide[(- 1)^(n)* (q)^((n)^(2))* Exp[I*2*n*z],(q)^(- n)* Exp[- I*z]+ (q)^(n)* Exp[I*z]], {n, - Infinity, Infinity}] Failure Failure Skip Error
20.9#Ex1 K ⁡ ( k ) = 1 2 ⁢ π ⁢ θ 3 2 ⁡ ( 0 | τ ) complete-elliptic-integral-first-kind-K 𝑘 1 2 𝜋 Jacobi-theta-tau 3 2 0 𝜏 {\displaystyle{\displaystyle K\left(k\right)=\tfrac{1}{2}\pi{\theta_{3}^{2}}% \left(0\middle|\tau\right)}} EllipticK(k)=(1)/(2)*Pi*(JacobiTheta3(0,exp(I*Pi*tau)))^(2) EllipticK[(k)^2]=Divide[1,2]*Pi*(EllipticTheta[3, 0, \[Tau]])^(2) Failure Failure Error
Fail
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
20.9#Ex2 K ′ ⁡ ( k ) = - i ⁢ τ ⁢ K ⁡ ( k ) diffop complete-elliptic-integral-first-kind-K 1 𝑘 𝑖 𝜏 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K'\left(k\right)=-i\tau K\left(k\right)}} subs( temp=k, diff( EllipticK(temp), temp$(1) ) )= - I*tau*EllipticK(k) (D[EllipticK[(temp)^2], {temp, 1}]/.temp-> k)= - I*\[Tau]*EllipticK[(k)^2] Failure Failure Error
Fail
Complex[-0.1562727389735284, 3.03204555200124] <- {Rule[k, 2], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.23878593395814385, 2.131309480129808] <- {Rule[k, 3], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.227738275577703, -0.017728124162552983] <- {Rule[k, 2], Rule[τ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.7636727720400398, -0.25270153442142274] <- {Rule[k, 3], Rule[τ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
20.9.E3 R F ⁡ ( θ 2 2 ⁡ ( z , q ) θ 2 2 ⁡ ( 0 , q ) , θ 3 2 ⁡ ( z , q ) θ 3 2 ⁡ ( 0 , q ) , θ 4 2 ⁡ ( z , q ) θ 4 2 ⁡ ( 0 , q ) ) = θ 1 ′ ⁡ ( 0 , q ) θ 1 ⁡ ( z , q ) ⁢ z Carlson-integral-RF Jacobi-theta 2 2 𝑧 𝑞 Jacobi-theta 2 2 0 𝑞 Jacobi-theta 3 2 𝑧 𝑞 Jacobi-theta 3 2 0 𝑞 Jacobi-theta 4 2 𝑧 𝑞 Jacobi-theta 4 2 0 𝑞 diffop Jacobi-theta 1 1 0 𝑞 Jacobi-theta 1 𝑧 𝑞 𝑧 {\displaystyle{\displaystyle R_{F}\left(\frac{{\theta_{2}^{2}}\left(z,q\right)% }{{\theta_{2}^{2}}\left(0,q\right)},\frac{{\theta_{3}^{2}}\left(z,q\right)}{{% \theta_{3}^{2}}\left(0,q\right)},\frac{{\theta_{4}^{2}}\left(z,q\right)}{{% \theta_{4}^{2}}\left(0,q\right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{% \theta_{1}\left(z,q\right)}z}} 0.5*int(1/(sqrt(t+((JacobiTheta2(z, q))^(2))/((JacobiTheta2(0, q))^(2)))*sqrt(t+((JacobiTheta3(z, q))^(2))/((JacobiTheta3(0, q))^(2)))*sqrt(t+((JacobiTheta4(z, q))^(2))/((JacobiTheta4(0, q))^(2)))), t = 0..infinity)=(subs( temp=0, diff( JacobiTheta1(temp, q), temp$(1) ) ))/(JacobiTheta1(z, q))*z Error Error Error - -
20.9.E4 R F ⁡ ( 0 , θ 3 4 ⁡ ( 0 , q ) , θ 4 4 ⁡ ( 0 , q ) ) = 1 2 ⁢ π Carlson-integral-RF 0 Jacobi-theta 3 4 0 𝑞 Jacobi-theta 4 4 0 𝑞 1 2 𝜋 {\displaystyle{\displaystyle R_{F}\left(0,{\theta_{3}^{4}}\left(0,q\right),{% \theta_{4}^{4}}\left(0,q\right)\right)=\tfrac{1}{2}\pi}} 0.5*int(1/(sqrt(t+0)*sqrt(t+(JacobiTheta3(0, q))^(4))*sqrt(t+(JacobiTheta4(0, q))^(4))), t = 0..infinity)=(1)/(2)*Pi Error Error Error - -
20.10.E1 ∫ 0 ∞ x s - 1 ⁢ θ 2 ⁡ ( 0 | i ⁢ x 2 ) ⁢ d x = 2 s ⁢ ( 1 - 2 - s ) ⁢ π - s / 2 ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ ζ ⁡ ( s ) superscript subscript 0 superscript 𝑥 𝑠 1 Jacobi-theta-tau 2 0 𝑖 superscript 𝑥 2 𝑥 superscript 2 𝑠 1 superscript 2 𝑠 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|% ix^{2}\right)\mathrm{d}x=2^{s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s% \right)\zeta\left(s\right)}} int((x)^(s - 1)* JacobiTheta2(0,exp(I*Pi*I*(x)^(2))), x = 0..infinity)= (2)^(s)*(1 - (2)^(- s))* (Pi)^(- s/ 2)* GAMMA((1)/(2)*s)*Zeta(s) Integrate[(x)^(s - 1)* EllipticTheta[2, 0, I*(x)^(2)], {x, 0, Infinity}]= (2)^(s)*(1 - (2)^(- s))* (Pi)^(- s/ 2)* Gamma[Divide[1,2]*s]*Zeta[s] Failure Failure Skip Error
20.10.E2 ∫ 0 ∞ x s - 1 ⁢ ( θ 3 ⁡ ( 0 | i ⁢ x 2 ) - 1 ) ⁢ d x = π - s / 2 ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ ζ ⁡ ( s ) superscript subscript 0 superscript 𝑥 𝑠 1 Jacobi-theta-tau 3 0 𝑖 superscript 𝑥 2 1 𝑥 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|% ix^{2}\right)-1)\mathrm{d}x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta% \left(s\right)}} int((x)^(s - 1)*(JacobiTheta3(0,exp(I*Pi*I*(x)^(2)))- 1), x = 0..infinity)= (Pi)^(- s/ 2)* GAMMA((1)/(2)*s)*Zeta(s) Integrate[(x)^(s - 1)*(EllipticTheta[3, 0, I*(x)^(2)]- 1), {x, 0, Infinity}]= (Pi)^(- s/ 2)* Gamma[Divide[1,2]*s]*Zeta[s] Failure Failure Skip Error
20.10.E3 ∫ 0 ∞ x s - 1 ⁢ ( 1 - θ 4 ⁡ ( 0 | i ⁢ x 2 ) ) ⁢ d x = ( 1 - 2 1 - s ) ⁢ π - s / 2 ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ ζ ⁡ ( s ) superscript subscript 0 superscript 𝑥 𝑠 1 1 Jacobi-theta-tau 4 0 𝑖 superscript 𝑥 2 𝑥 1 superscript 2 1 𝑠 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0% \middle|ix^{2}\right))\mathrm{d}x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2% }s\right)\zeta\left(s\right)}} int((x)^(s - 1)*(1 - JacobiTheta4(0,exp(I*Pi*I*(x)^(2)))), x = 0..infinity)=(1 - (2)^(1 - s))* (Pi)^(- s/ 2)* GAMMA((1)/(2)*s)*Zeta(s) Integrate[(x)^(s - 1)*(1 - EllipticTheta[4, 0, I*(x)^(2)]), {x, 0, Infinity}]=(1 - (2)^(1 - s))* (Pi)^(- s/ 2)* Gamma[Divide[1,2]*s]*Zeta[s] Failure Failure Skip Error
20.10.E4 ∫ 0 ∞ e - s ⁢ t ⁢ θ 1 ⁡ ( β ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ∫ 0 ∞ e - s ⁢ t ⁢ θ 2 ⁡ ( ( 1 + β ) ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 1 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 2 1 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{% \beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{% \infty}e^{-st}\theta_{2}\left(\frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{% \ell^{2}}\right)\mathrm{d}t}} - stint(exp(1)*JacobiTheta1((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)= - stint(exp(1)*JacobiTheta2(((1 + beta)* Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) - stIntegrate[E*EllipticTheta[1, Divide[\[Beta]*Pi,2*\[ScriptL]], Divide[I*Pi*t,(\[ScriptL])^(2)]], {t, 0, Infinity}]= - stIntegrate[E*EllipticTheta[2, Divide[(1 + \[Beta])* Pi,2*\[ScriptL]], Divide[I*Pi*t,(\[ScriptL])^(2)]], {t, 0, Infinity}] Failure Failure Skip
Fail
Complex[0.0, -2.8284271247461903] <- {Rule[stIntegrate[Times[E, EllipticTheta[1, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[2, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[stIntegrate[Times[E, EllipticTheta[1, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[2, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
-2.8284271247461903 <- {Rule[stIntegrate[Times[E, EllipticTheta[1, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[2, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 2.8284271247461903] <- {Rule[stIntegrate[Times[E, EllipticTheta[1, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[2, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
20.10.E4 ∫ 0 ∞ e - s ⁢ t ⁢ θ 2 ⁡ ( ( 1 + β ) ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = - ℓ s ⁢ sinh ⁡ ( β ⁢ s ) ⁢ sech ⁡ ( ℓ ⁢ s ) superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 2 1 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 ℓ 𝑠 𝛽 𝑠 ℓ 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-st}\theta_{2}\left(\frac{(1+% \beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=-\frac{\ell% }{\sqrt{s}}\sinh\left(\beta\sqrt{s}\right)\operatorname{sech}\left(\ell\sqrt{s% }\right)}} - stint(exp(1)*JacobiTheta2(((1 + beta)* Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)= -(ell)/(sqrt(s))*sinh(beta*sqrt(s))*sech(ell*sqrt(s)) - stIntegrate[E*EllipticTheta[2, Divide[(1 + \[Beta])* Pi,2*\[ScriptL]], Divide[I*Pi*t,(\[ScriptL])^(2)]], {t, 0, Infinity}]= -Divide[\[ScriptL],Sqrt[s]]*Sinh[\[Beta]*Sqrt[s]]*Sech[\[ScriptL]*Sqrt[s]] Failure Failure Skip Successful
20.10.E5 ∫ 0 ∞ e - s ⁢ t ⁢ θ 3 ⁡ ( ( 1 + β ) ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ∫ 0 ∞ e - s ⁢ t ⁢ θ 4 ⁡ ( β ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 3 1 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 4 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-st}\theta_{3}\left(\frac{(1+% \beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{% \infty}e^{-st}\theta_{4}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell% ^{2}}\right)\mathrm{d}t}} - stint(exp(1)*JacobiTheta3(((1 + beta)* Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)= - stint(exp(1)*JacobiTheta4((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) - stIntegrate[E*EllipticTheta[3, Divide[(1 + \[Beta])* Pi,2*\[ScriptL]], Divide[I*Pi*t,(\[ScriptL])^(2)]], {t, 0, Infinity}]= - stIntegrate[E*EllipticTheta[4, Divide[\[Beta]*Pi,2*\[ScriptL]], Divide[I*Pi*t,(\[ScriptL])^(2)]], {t, 0, Infinity}] Failure Failure Skip
Fail
Complex[0.0, -2.8284271247461903] <- {Rule[stIntegrate[Times[E, EllipticTheta[3, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[4, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[stIntegrate[Times[E, EllipticTheta[3, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[4, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
-2.8284271247461903 <- {Rule[stIntegrate[Times[E, EllipticTheta[3, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[4, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 2.8284271247461903] <- {Rule[stIntegrate[Times[E, EllipticTheta[3, Times[Rational[1, 2], Pi, Power[ℓ, -1], Plus[1, β]], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[stIntegrate[Times[E, EllipticTheta[4, Times[Rational[1, 2], Pi, Power[ℓ, -1], β], Times[Complex[0, 1], Pi, t, Power[ℓ, -2]]]], {t, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
20.10.E5 ∫ 0 ∞ e - s ⁢ t ⁢ θ 4 ⁡ ( β ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ℓ s ⁢ cosh ⁡ ( β ⁢ s ) ⁢ csch ⁡ ( ℓ ⁢ s ) superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 4 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 ℓ 𝑠 𝛽 𝑠 ℓ 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-st}\theta_{4}\left(\frac{% \beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\frac{\ell}{% \sqrt{s}}\cosh\left(\beta\sqrt{s}\right)\operatorname{csch}\left(\ell\sqrt{s}% \right)}} - stint(exp(1)*JacobiTheta4((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)=(ell)/(sqrt(s))*cosh(beta*sqrt(s))*csch(ell*sqrt(s)) - stIntegrate[E*EllipticTheta[4, Divide[\[Beta]*Pi,2*\[ScriptL]], Divide[I*Pi*t,(\[ScriptL])^(2)]], {t, 0, Infinity}]=Divide[\[ScriptL],Sqrt[s]]*Cosh[\[Beta]*Sqrt[s]]*Csch[\[ScriptL]*Sqrt[s]] Failure Failure Skip Successful
20.11.E5 F 1 2 ⁡ ( 1 2 , 1 2 ; 1 ; k 2 ) = θ 3 2 ⁡ ( 0 | τ ) Gauss-hypergeometric-F-as-2F1 1 2 1 2 1 superscript 𝑘 2 Jacobi-theta-tau 3 2 0 𝜏 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{% 2}\right)={\theta_{3}^{2}}\left(0\middle|\tau\right)}} hypergeom([(1)/(2),(1)/(2)], [1], (k)^(2))= (JacobiTheta3(0,exp(I*Pi*tau)))^(2) HypergeometricPFQ[{Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]= (EllipticTheta[3, 0, \[Tau]])^(2) Failure Failure Error
Fail
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[τ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
20.13.E2 ∂ ⁡ θ / ∂ ⁡ t = α ⁢ ∂ 2 ⁡ θ / ∂ ⁡ z 2 partial-derivative 𝜃 𝑡 𝛼 partial-derivative 𝜃 𝑧 2 {\displaystyle{\displaystyle\ifrac{\partial\theta}{\partial t}=\alpha\ifrac{{% \partial}^{2}\theta}{{\partial z}^{2}}}} diff(theta, t)= alpha*diff(theta, [z$(2)]) D[\[Theta], t]= \[Alpha]*D[\[Theta], {z, 2}] Successful Successful - -
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