# Results of Theta Functions

DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
20.2.E1 ${\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=\theta_{1}% \left(z,q\right)}}$ JacobiTheta1(z,exp(I*Pi*tau))= JacobiTheta1(z, q) EllipticTheta[1, z, \[Tau]]= EllipticTheta[1, z, q] Failure Failure Error Successful
20.2.E2 ${\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{2}% \left(z,q\right)}}$ JacobiTheta2(z,exp(I*Pi*tau))= JacobiTheta2(z, q) EllipticTheta[2, z, \[Tau]]= EllipticTheta[2, z, q] Failure Failure Error Successful
20.2.E3 ${\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{3}% \left(z,q\right)}}$ JacobiTheta3(z,exp(I*Pi*tau))= JacobiTheta3(z, q) EllipticTheta[3, z, \[Tau]]= EllipticTheta[3, z, q] Failure Failure Error Successful
20.2.E4 ${\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{4}% \left(z,q\right)}}$ JacobiTheta4(z,exp(I*Pi*tau))= JacobiTheta4(z, q) EllipticTheta[4, z, \[Tau]]= EllipticTheta[4, z, q] Failure Failure Error Successful
20.2.E6 ${\displaystyle{\displaystyle\theta_{1}\left(z+(m+n\tau)\pi\middle|\tau\right)=% (-1)^{m+n}q^{-n^{2}}e^{-2inz}\theta_{1}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}\left(z+(m+n\tau)\pi\middle|\tau\right)=% (-1)^{m}q^{-n^{2}}e^{-2inz}\theta_{2}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(z+(m+n\tau)\pi\middle|\tau\right)=% q^{-n^{2}}e^{-2inz}\theta_{3}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(z+(m+n\tau)\pi\middle|\tau\right)=% (-1)^{n}q^{-n^{2}}e^{-2inz}\theta_{4}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=-\theta_{2}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle-\theta_{2}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=-iM\theta_{4}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}$ - 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 ${\displaystyle{\displaystyle-iM\theta_{4}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=-iM\theta_{3}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|% \tau\right)}}$ - 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 ${\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{1}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{1}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=M\theta_{3}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle M\theta_{3}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=M\theta_{4}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau% \right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{4}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=M\theta_{2}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle M\theta_{2}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=M\theta_{1}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau% \right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{3}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(z+\tfrac{1}{2}\pi\middle|\tau% \right)=-iM\theta_{1}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle-iM\theta_{1}\left(z+\tfrac{1}{2}\pi\tau\middle|% \tau\right)=iM\theta_{2}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|% \tau\right)}}$ - 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 ${\displaystyle{\displaystyle\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q% \right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}'\left(0,q\right)=\theta_{4}'\left(0,q% \right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}'\left(0,q\right)=0}}$ 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 ${\displaystyle{\displaystyle\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q% \right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{1}''(0,q)=\theta_{2}'''\left(0,q\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(0% ,q\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}'''\left(0,q\right)=\theta_{4}'''\left(0% ,q\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}'''\left(0,q\right)=0}}$ 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 ${\displaystyle{\displaystyle\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'% \left(0,q\right)}=-1+24\sum_{n=1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}}}$ (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 ${\displaystyle{\displaystyle\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}% \left(0,q\right)}=-1-8\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}}}}$ (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 ${\displaystyle{\displaystyle\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}% \left(0,q\right)}=-8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}}}}$ (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 ${\displaystyle{\displaystyle\frac{\theta_{4}''\left(0,q\right)}{\theta_{4}% \left(0,q\right)}=8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}}}$ (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 ${\displaystyle{\displaystyle\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'% \left(0,q\right)}=\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}\left(0,q% \right)}+\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}\left(0,q\right)}+\frac% {\theta_{4}''\left(0,q\right)}{\theta_{4}\left(0,q\right)}}}$ (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 ${\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=\theta_{1}'% \left(0\middle|\tau\right)\sin z\prod_{n=1}^{\infty}\frac{\sin\left(n\pi\tau+z% \right)\sin\left(n\pi\tau-z\right)}{{\sin^{2}}\left(n\pi\tau\right)}}}$ 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 ${\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{2}% \left(0\middle|\tau\right)\cos z\prod_{n=1}^{\infty}\frac{\cos\left(n\pi\tau+z% \right)\cos\left(n\pi\tau-z\right)}{{\cos^{2}}\left(n\pi\tau\right)}}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{3}% \left(0\middle|\tau\right)\prod_{n=1}^{\infty}\frac{\cos\left((n-\tfrac{1}{2})% \pi\tau+z\right)\cos\left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\cos^{2}}\left((n% -\tfrac{1}{2})\pi\tau\right)}}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{4}% \left(0\middle|\tau\right)\prod_{n=1}^{\infty}\frac{\sin\left((n-\tfrac{1}{2})% \pi\tau+z\right)\sin\left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\sin^{2}}\left((n% -\tfrac{1}{2})\pi\tau\right)}}}$ 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 ${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}p^{2n}q^{n^{2}}\\ =\prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+% q^{2n-1}p^{-2}\right)}}$ 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 ${\displaystyle{\displaystyle\frac{\theta_{1}'\left(z,q\right)}{\theta_{1}\left% (z,q\right)}-\cot z=4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^% {2n}\cos\left(2z\right)+q^{4n}}}}$ (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 ${\displaystyle{\displaystyle 4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n}}{1-2q^{2n}\cos\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q% ^{2n}}\sin\left(2nz\right)}}$ 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 ${\displaystyle{\displaystyle\frac{\theta_{2}'\left(z,q\right)}{\theta_{2}\left% (z,q\right)}+\tan z=-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q% ^{2n}\cos\left(2z\right)+q^{4n}}}}$ (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 ${\displaystyle{\displaystyle-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n}}{1+2q^{2n}\cos\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{% 2n}}{1-q^{2n}}\sin\left(2nz\right)}}$ - 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 ${\displaystyle{\displaystyle\frac{\theta_{3}'\left(z,q\right)}{\theta_{3}\left% (z,q\right)}=-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-% 1}\cos\left(2z\right)+q^{4n-2}}}}$ (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 ${\displaystyle{\displaystyle-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n-1}}{1+2q^{2n-1}\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}(-1)^{n}% \frac{q^{n}}{1-q^{2n}}\sin\left(2nz\right)}}$ - 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 ${\displaystyle{\displaystyle\frac{\theta_{4}'\left(z,q\right)}{\theta_{4}\left% (z,q\right)}=4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1% }\cos\left(2z\right)+q^{4n-2}}}}$ (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 ${\displaystyle{\displaystyle 4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2% n-1}}{1-2q^{2n-1}\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}\frac{q^{n}% }{1-q^{2n}}\sin\left(2nz\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau\right)=\theta_{2}% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}% \prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+n\tau)\pi}\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{3}% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty% }\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)\pi}% \right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{4}% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty% }\prod_{m=-M}^{M}\left(1+\frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{1}\left(\pi z\middle|\tau\right)=\pi z% \theta_{1}'\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j% }\delta_{2j}(\tau)z^{2j}\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}\left(\pi z\middle|\tau\right)=\theta_{2% }\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\alpha_{2% j}(\tau)z^{2j}\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(\pi z\middle|\tau\right)=\theta_{3% }\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\beta_{2j% }(\tau)z^{2j}\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(\pi z\middle|\tau\right)=\theta_{4% }\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\gamma_{2% j}(\tau)z^{2j}\right)}}$ 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 ${\displaystyle{\displaystyle{\theta_{3}^{2}}\left(0,q\right){\theta_{3}^{2}}% \left(z,q\right)={\theta_{4}^{2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q% \right)+{\theta_{2}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{3}^{2}}\left(0,q\right){\theta_{4}^{2}}% \left(z,q\right)={\theta_{2}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q% \right)+{\theta_{4}^{2}}\left(0,q\right){\theta_{3}^{2}}\left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{2}^{2}}\left(0,q\right){\theta_{4}^{2}}% \left(z,q\right)={\theta_{3}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q% \right)+{\theta_{4}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{2}^{2}}\left(0,q\right){\theta_{3}^{2}}% \left(z,q\right)={\theta_{4}^{2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q% \right)+{\theta_{3}^{2}}\left(0,q\right){\theta_{2}^{2}}\left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{3}^{4}}\left(0,q\right)={\theta_{2}^{4}}% \left(0,q\right)+{\theta_{4}^{4}}\left(0,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{1}\left(w+% z,q\right)\theta_{1}\left(w-z,q\right)={\theta_{3}^{2}}\left(w,q\right){\theta% _{2}^{2}}\left(z,q\right)-{\theta_{2}^{2}}\left(w,q\right){\theta_{3}^{2}}% \left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{2}\left(w+% z,q\right)\theta_{2}\left(w-z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta% _{2}^{2}}\left(z,q\right)-{\theta_{1}^{2}}\left(w,q\right){\theta_{3}^{2}}% \left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{3}\left(w+% z,q\right)\theta_{3}\left(w-z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta% _{3}^{2}}\left(z,q\right)-{\theta_{1}^{2}}\left(w,q\right){\theta_{2}^{2}}% \left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle{\theta_{4}^{2}}\left(0,q\right)\theta_{4}\left(w+% z,q\right)\theta_{4}\left(w-z,q\right)={\theta_{3}^{2}}\left(w,q\right){\theta% _{3}^{2}}\left(z,q\right)-{\theta_{2}^{2}}\left(w,q\right){\theta_{2}^{2}}% \left(z,q\right)}}$ (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 ${\displaystyle{\displaystyle\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left% (z,q\right)\theta_{2}\left(z,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left% (z,q\right)}{\theta_{2}\left(0,q\right)\theta_{3}\left(0,q\right)\theta_{4}% \left(0,q\right)}}}$ 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 ${\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z,% q\right)}{\theta_{1}\left(2z,q^{2}\right)}=\frac{\theta_{3}\left(z,q\right)% \theta_{4}\left(z,q\right)}{\theta_{4}\left(2z,q^{2}\right)}}}$ (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 ${\displaystyle{\displaystyle\frac{\theta_{3}\left(z,q\right)\theta_{4}\left(z,% q\right)}{\theta_{4}\left(2z,q^{2}\right)}=\theta_{4}\left(0,q^{2}\right)}}$ (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 ${\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q^{2}\right)\theta_{4}% \left(z,q^{2}\right)}{\theta_{1}\left(z,q\right)}=\frac{\theta_{2}\left(z,q^{2% }\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{2}\left(z,q\right)}}}$ (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 ${\displaystyle{\displaystyle\frac{\theta_{2}\left(z,q^{2}\right)\theta_{3}% \left(z,q^{2}\right)}{\theta_{2}\left(z,q\right)}=\tfrac{1}{2}\theta_{2}\left(% 0,q\right)}}$ (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 ${\displaystyle{\displaystyle\theta_{1}\left(z,q\right)\theta_{1}\left(w,q% \right)=\theta_{3}\left(z+w,q^{2}\right)\theta_{2}\left(z-w,q^{2}\right)-% \theta_{2}\left(z+w,q^{2}\right)\theta_{3}\left(z-w,q^{2}\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(z,q\right)\theta_{3}\left(w,q% \right)=\theta_{3}\left(z+w,q^{2}\right)\theta_{3}\left(z-w,q^{2}\right)+% \theta_{2}\left(z+w,q^{2}\right)\theta_{2}\left(z-w,q^{2}\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{1}\left(2z\middle|2\tau\right)=A\theta_{1}% \left(z\middle|\tau\right)\theta_{2}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}\left(2z\middle|2\tau\right)=A\theta_{1}% \left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{1}\left(\tfrac{1}{4}\pi+z% \middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(2z\middle|2\tau\right)=A\theta_{3}% \left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{3}\left(\tfrac{1}{4}\pi+z% \middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(2z\middle|2\tau\right)=A\theta_{3}% \left(z\middle|\tau\right)\theta_{4}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{1}\left(4z\middle|4\tau\right)=B\theta_{1}% \left(z\middle|\tau\right)\theta_{1}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)% \*\theta_{1}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\theta_{2}\left(z\middle% |\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}\left(4z\middle|4\tau\right)=B\theta_{2}% \left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{2}\left(\tfrac{1}{8}\pi+z% \middle|\tau\right)\*\theta_{2}\left(\tfrac{3}{8}\pi-z\middle|\tau\right)% \theta_{2}\left(\tfrac{3}{8}\pi+z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(4z\middle|4\tau\right)=B\theta_{3}% \left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{3}\left(\tfrac{1}{8}\pi+z% \middle|\tau\right)\*\theta_{3}\left(\tfrac{3}{8}\pi-z\middle|\tau\right)% \theta_{3}\left(\tfrac{3}{8}\pi+z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(4z\middle|4\tau\right)=B\theta_{4}% \left(z\middle|\tau\right)\theta_{4}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)% \*\theta_{4}\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\theta_{3}\left(z\middle% |\tau\right)}}$ 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 ${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\theta_{% 2}\left(z\middle|\tau\right)}{\theta_{4}\left(z\middle|\tau\right)}\right)=-% \frac{{\theta_{3}^{2}}\left(0\middle|\tau\right)\theta_{1}\left(z\middle|\tau% \right)\theta_{3}\left(z\middle|\tau\right)}{{\theta_{4}^{2}}\left(z\middle|% \tau\right)}}}$ 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 ${\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau+1\right)=e^{i\pi/4}% \theta_{1}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{2}\left(z\middle|\tau+1\right)=e^{i\pi/4}% \theta_{2}\left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau+1\right)=\theta_{4}% \left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau+1\right)=\theta_{3}% \left(z\middle|\tau\right)}}$ 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 ${\displaystyle{\displaystyle\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}% \left(z,q^{2}\right)}{\theta_{1}\left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{% 2}\right)\theta_{4}\left(z,q^{2}\right)}{\theta_{2}\left(z,iq\right)}}}$ (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