Results of Theta Functions

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20.2.E1 JacobiTheta1(z,exp(I*Pi*tau))= JacobiTheta1(z, q) EllipticTheta[1, z, \[Tau]]= EllipticTheta[1, z, q] Failure Failure Error Successful
20.2.E2 JacobiTheta2(z,exp(I*Pi*tau))= JacobiTheta2(z, q) EllipticTheta[2, z, \[Tau]]= EllipticTheta[2, z, q] Failure Failure Error Successful
20.2.E3 JacobiTheta3(z,exp(I*Pi*tau))= JacobiTheta3(z, q) EllipticTheta[3, z, \[Tau]]= EllipticTheta[3, z, q] Failure Failure Error Successful
20.2.E4 JacobiTheta4(z,exp(I*Pi*tau))= JacobiTheta4(z, q) EllipticTheta[4, z, \[Tau]]= EllipticTheta[4, z, q] Failure Failure Error Successful
20.2.E6 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 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 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 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 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 - 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*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 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 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*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 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 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*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 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 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*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 JacobiTheta1(0, q)= subs( temp=0, diff( JacobiTheta2(temp, q), temp$(1) ) ) EllipticTheta[1, 0, q]= (D[EllipticTheta[2, temp, q], {temp, 1}]/.temp-> 0) Successful Successful - -
20.4.E1 subs( temp=0, diff( JacobiTheta2(temp, q), temp$(1) ) )= subs( temp=0, diff( JacobiTheta3(temp, q), temp$(1) ) ) (D[EllipticTheta[2, temp, q], {temp, 1}]/.temp-> 0)= (D[EllipticTheta[3, temp, q], {temp, 1}]/.temp-> 0) Failure Successful Error -
20.4.E1 subs( temp=0, diff( JacobiTheta3(temp, q), temp$(1) ) )= subs( temp=0, diff( JacobiTheta4(temp, q), temp$(1) ) ) (D[EllipticTheta[3, temp, q], {temp, 1}]/.temp-> 0)= (D[EllipticTheta[4, temp, q], {temp, 1}]/.temp-> 0) Failure Successful Skip -
20.4.E1 subs( temp=0, diff( JacobiTheta4(temp, q), temp$(1) ) )= 0 (D[EllipticTheta[4, temp, q], {temp, 1}]/.temp-> 0)= 0 Successful Successful - -
20.4.E6 subs( temp=0, diff( JacobiTheta1(temp, q), temp$(1) ) )= JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q) (D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> 0)= EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q] Failure Failure Error Successful
20.4.E7 subs( temp=(0 , q), diff( JacobiTheta1(temp, =), temp$(2) ) )*subs( temp=0, diff( JacobiTheta2(temp, q), temp$(3) ) ) (D[EllipticTheta[1, temp, =], {temp, 2}]/.temp-> (0 , q))*(D[EllipticTheta[2, temp, q], {temp, 3}]/.temp-> 0) Error Failure - Error
20.4.E7 subs( temp=0, diff( JacobiTheta2(temp, q), temp$(3) ) )= subs( temp=0, diff( JacobiTheta3(temp, q), temp$(3) ) ) (D[EllipticTheta[2, temp, q], {temp, 3}]/.temp-> 0)= (D[EllipticTheta[3, temp, q], {temp, 3}]/.temp-> 0) Failure Failure Error
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][2, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
20.4.E7 subs( temp=0, diff( JacobiTheta3(temp, q), temp$(3) ) )= subs( temp=0, diff( JacobiTheta4(temp, q), temp$(3) ) ) (D[EllipticTheta[3, temp, q], {temp, 3}]/.temp-> 0)= (D[EllipticTheta[4, temp, q], {temp, 3}]/.temp-> 0) Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[Derivative[0, 2, 0][EllipticThetaPrime][3, 0, q], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Derivative[0, 2, 0][EllipticThetaPrime][4, 0, q], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
20.4.E7 subs( temp=0, diff( JacobiTheta4(temp, q), temp$(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 (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 (subs( temp=0, diff( JacobiTheta2(temp, q), temp$(2) ) ))/(JacobiTheta2(0, q))= - 1 - 8*sum(((q)^(2*n))/((1 + (q)^(2*n))^(2)), n = 1..infinity) Divide[D[EllipticTheta[2, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[2, 0, q]]= - 1 - 8*Sum[Divide[(q)^(2*n),(1 + (q)^(2*n))^(2)], {n, 1, Infinity}] Failure Failure Skip Skip
20.4.E10 (subs( temp=0, diff( JacobiTheta3(temp, q), temp$(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 (subs( temp=0, diff( JacobiTheta4(temp, q), temp$(2) ) ))/(JacobiTheta4(0, q))= 8*sum(((q)^(2*n - 1))/((1 - (q)^(2*n - 1))^(2)), n = 1..infinity) Divide[D[EllipticTheta[4, temp, q], {temp, 2}]/.temp-> 0,EllipticTheta[4, 0, q]]= 8*Sum[Divide[(q)^(2*n - 1),(1 - (q)^(2*n - 1))^(2)], {n, 1, Infinity}] Failure Failure Skip Error
20.4.E12 (subs( temp=0, diff( JacobiTheta1(temp, q), temp$(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 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 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 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 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 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 (subs( temp=z, diff( JacobiTheta1(temp, q), temp$(1) ) ))/(JacobiTheta1(z, q))- cot(z)= 4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity) Divide[D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> z,EllipticTheta[1, z, q]]- Cot[z]= 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}] Failure Failure Skip Skip
20.5.E10 4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity)= 4*sum(((q)^(2*n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity) 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}]= 4*Sum[Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}] Failure Failure Skip Skip
20.5.E11 (subs( temp=z, diff( JacobiTheta2(temp, q), temp$(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)*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 (subs( temp=z, diff( JacobiTheta3(temp, q), temp$(1) ) ))/(JacobiTheta3(z, q))= - 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity) Divide[D[EllipticTheta[3, temp, q], {temp, 1}]/.temp-> z,EllipticTheta[3, z, q]]= - 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}] Failure Failure Skip Error
20.5.E12 - 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity)= 4*sum((- 1)^(n)*((q)^(n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity) - 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}]= 4*Sum[(- 1)^(n)*Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}] Failure Failure Skip Skip
20.5.E13 (subs( temp=z, diff( JacobiTheta4(temp, q), temp$(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)*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 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 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 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 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 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 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 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 (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 (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 (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 (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 (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 (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 (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 (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 (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 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 (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 (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 (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 (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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 (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