Results of Weierstrass Elliptic and Modular Functions

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23.5#Ex8 ${\displaystyle{\displaystyle K\left(k\right)=K'\left(k\right)}}$ EllipticK(k)= subs( temp=k, diff( EllipticK(temp), temp$(1) ) ) EllipticK[(k)^2]= (D[EllipticK[(temp)^2], {temp, 1}]/.temp-> k) Failure Failure Error Fail Complex[1.3320292471861073, -1.3934110303935494] <- {Rule[k, 2]} Complex[0.7299050661514341, -1.0197357312195425] <- {Rule[k, 3]} 23.5#Ex8 ${\displaystyle{\displaystyle K'\left(k\right)=\ifrac{\left(\Gamma\left(\tfrac{% 1}{4}\right)\right)^{2}}{\left(4\sqrt{\pi}\right)}}}$ subs( temp=k, diff( EllipticK(temp), temp$(1) ) )=((GAMMA((1)/(4)))^(2))/(4*sqrt(Pi)) (D[EllipticK[(temp)^2], {temp, 1}]/.temp-> k)=Divide[(Gamma[Divide[1,4]])^(2),4*Sqrt[Pi]] Failure Failure Error
Fail
Complex[-2.343228747081181, 0.3151532066437278] <- {Rule[k, 2]}
Complex[-2.044850831577895, 0.1768605538132445] <- {Rule[k, 3]}
23.5#Ex11 ${\displaystyle{\displaystyle k^{2}=e^{\mathrm{i}\pi/3}}}$ (k)^(2)= exp(I*Pi/ 3) (k)^(2)= Exp[I*Pi/ 3] Failure Failure
Fail
.5000000000-.8660254040*I <- {k = 1}
3.500000000-.8660254040*I <- {k = 2}
8.500000000-.8660254040*I <- {k = 3}
Fail
Complex[0.4999999999999999, -0.8660254037844386] <- {Rule[k, 1]}
Complex[3.5, -0.8660254037844386] <- {Rule[k, 2]}
Complex[8.5, -0.8660254037844386] <- {Rule[k, 3]}
23.5#Ex12 ${\displaystyle{\displaystyle K\left(k\right)=e^{\mathrm{i}\pi/6}K'\left(k% \right)}}$ EllipticK(k)= exp(I*Pi/ 6)*subs( temp=k, diff( EllipticK(temp), temp$(1) ) ) EllipticK[(k)^2]= Exp[I*Pi/ 6]*(D[EllipticK[(temp)^2], {temp, 1}]/.temp-> k) Failure Failure Error Fail Complex[1.4240716315220228, -1.1066114718975122] <- {Rule[k, 2]} Complex[0.7927761848213015, -0.9006528327976908] <- {Rule[k, 3]} 23.5#Ex12 ${\displaystyle{\displaystyle e^{\mathrm{i}\pi/6}K'\left(k\right)=e^{\mathrm{i}% \pi/12}\frac{3^{1/4}\left(\Gamma\left(\frac{1}{3}\right)\right)^{3}}{2^{7/3}% \pi}}}$ exp(I*Pi/ 6)*subs( temp=k, diff( EllipticK(temp), temp$(1) ) )= exp(I*Pi/ 12)*((3)^(1/ 4)*(GAMMA((1)/(3)))^(3))/((2)^(7/ 3)* Pi) Exp[I*Pi/ 6]*(D[EllipticK[(temp)^2], {temp, 1}]/.temp-> k)= Exp[I*Pi/ 12]*Divide[(3)^(1/ 4)*(Gamma[Divide[1,3]])^(3),(2)^(7/ 3)* Pi] Failure Failure Error
Fail
Complex[-2.1248830880335463, -0.38527593877730804] <- {Rule[k, 2]}
Complex[-1.7973339068642127, -0.3558519315336056] <- {Rule[k, 3]}
23.6.E8 ${\displaystyle{\displaystyle\eta_{1}=-\frac{\pi^{2}}{12\omega_{1}}\frac{\theta% _{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}}}$ eta[1]= -((Pi)^(2))/(12*omega[1])*(subs( temp=0, diff( JacobiTheta1(temp, q), temp$(3) ) ))/(subs( temp=0, diff( JacobiTheta1(temp, q), temp$(1) ) )) Subscript[\[Eta], 1]= -Divide[(Pi)^(2),12*Subscript[\[Omega], 1]]*Divide[D[EllipticTheta[1, temp, q], {temp, 3}]/.temp-> 0,D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> 0] Failure Failure Error Successful
23.6#Ex5 ${\displaystyle{\displaystyle{K^{2}}=(K\left(k\right))^{2}}}$ (EllipticK(k))^(2)=(EllipticK(k))^(2) (EllipticK[(k)^2])^(2)=(EllipticK[(k)^2])^(2) Successful Successful - -
23.8.E5 ${\displaystyle{\displaystyle\eta_{1}=\frac{\pi^{2}}{2\omega_{1}}\left(\frac{1}% {6}+\sum_{n=1}^{\infty}{\csc^{2}}\left(\frac{n\pi\omega_{3}}{\omega_{1}}\right% )\right)}}$ eta[1]=((Pi)^(2))/(2*omega[1])*((1)/(6)+ sum((csc((n*Pi*omega[3])/(omega[1])))^(2), n = 1..infinity)) Subscript[\[Eta], 1]=Divide[(Pi)^(2),2*Subscript[\[Omega], 1]]*(Divide[1,6]+ Sum[(Csc[Divide[n*Pi*Subscript[\[Omega], 3],Subscript[\[Omega], 1]]])^(2), {n, 1, Infinity}]) Failure Failure Skip Skip
23.10.E15 ${\displaystyle{\displaystyle A_{n}=\left(\frac{\pi^{2}G^{2}}{\omega_{1}}\right% )^{n^{2}-1}\frac{q^{n(n-1)/2}}{i^{n-1}}\exp\left(-\frac{(n-1)\eta_{1}}{3\omega% _{1}}\left((2n-1)(\omega_{1}^{2}+\omega_{3}^{2})+3(n-1)\omega_{1}\omega_{3}% \right)\right)}}$ A[n]=exp(-((n - 1)* eta[1])/(3*omega[1])*((2*n - 1)*(omega(omega[1])^(2)+ omega(omega[3])^(2))+ 3*(n - 1)*omega[1]*omega[3])) Subscript[A, n]=Exp[-Divide[(n - 1)* Subscript[\[Eta], 1],3*Subscript[\[Omega], 1]]*((2*n - 1)*(\[Omega](Subscript[\[Omega], 1])^(2)+ \[Omega](Subscript[\[Omega], 3])^(2))+ 3*(n - 1)*Subscript[\[Omega], 1]*Subscript[\[Omega], 3])] Failure Failure Error Successful
23.11#Ex1 ${\displaystyle{\displaystyle f_{1}(s,\tau)=\frac{{\cosh^{2}}\left(\tfrac{1}{2}% \tau s\right)}{1-2e^{-s}\cosh\left(\tau s\right)+e^{-2s}}}}$ f[1]*(s , tau)=((cosh((1)/(2)*tau*s))^(2))/(1 - 2*exp(- s)*cosh(tau*s)+ exp(- 2*s)) Subscript[f, 1]*(s , \[Tau])=Divide[(Cosh[Divide[1,2]*\[Tau]*s])^(2),1 - 2*Exp[- s]*Cosh[\[Tau]*s]+ Exp[- 2*s]] Failure Failure Error Error
23.11#Ex2 ${\displaystyle{\displaystyle f_{2}(s,\tau)=\frac{{\cos^{2}}\left(\tfrac{1}{2}s% \right)}{1-2e^{i\tau s}\cos s+e^{2i\tau s}}}}$ f[2]*(s , tau)=((cos((1)/(2)*s))^(2))/(1 - 2*exp(I*tau*s)*cos(s)+ exp(2*I*tau*s)) Subscript[f, 2]*(s , \[Tau])=Divide[(Cos[Divide[1,2]*s])^(2),1 - 2*Exp[I*\[Tau]*s]*Cos[s]+ Exp[2*I*\[Tau]*s]] Failure Failure Error Error
23.15.E1 ${\displaystyle{\displaystyle q=\exp\left(-\pi\frac{{K^{\prime}}\left(k\right)}% {K\left(k\right)}\right)}}$ q = exp(- Pi*(EllipticCK(k))/(EllipticK(k))) q = Exp[- Pi*Divide[EllipticK[1-(k)^2],EllipticK[(k)^2]]] Failure Failure Error
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[k, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4948305828127435, 1.61650806185012] <- {Rule[k, 2], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.5612675359881547, 1.6042143845245938] <- {Rule[k, 3], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[k, 1], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
23.15#Ex1 ${\displaystyle{\displaystyle k=\frac{{\theta_{2}^{2}}\left(0,q\right)}{{\theta% _{3}^{2}}\left(0,q\right)}}}$ k =((JacobiTheta2(0, q))^(2))/((JacobiTheta3(0, q))^(2)) k =Divide[(EllipticTheta[2, 0, q])^(2),(EllipticTheta[3, 0, q])^(2)] Failure Failure Error Successful
23.15.E9 ${\displaystyle{\displaystyle\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)% \right)^{1/3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle|3% \tau\right)}}$ ((1)/(2)*subs( temp=0, diff( JacobiTheta1(temp, q), temp\$(1) ) ))^(1/ 3)= exp(I*Pi*tau/ 12)*JacobiTheta3((1)/(2)*Pi*(1 + tau),exp(I*Pi*3*tau)) ((Divide[1,2]*(D[EllipticTheta[1, temp, q], {temp, 1}]/.temp-> 0)))^(1/ 3)= Exp[I*Pi*\[Tau]/ 12]*EllipticTheta[3, Divide[1,2]*Pi*(1 + \[Tau]), 3*\[Tau]] Failure Failure Error Successful
23.18.E6 ${\displaystyle{\displaystyle\varepsilon(\mathcal{A})=\exp\left(\pi i\left(% \frac{a+d}{12c}+s(-d,c)\right)\right)}}$ varepsilon*(A)= exp(Pi*I*((a + d)/(12*c)+ s*(- d , c))) \[CurlyEpsilon]*(A)= Exp[Pi*I*(Divide[a + d,12*c]+ s*(- d , c))] Failure Failure Skip Error
23.22.E2 ${\displaystyle{\displaystyle 2\omega_{1}=-2i\omega_{3}}}$ 2*omega[1]= - 2*I*omega[3] 2*Subscript[\[Omega], 1]= - 2*I*Subscript[\[Omega], 3] Failure Failure
Fail
0.+5.656854248*I <- {omega[1] = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)+I*2^(1/2)}
5.656854248+5.656854248*I <- {omega[1] = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)-I*2^(1/2)}
5.656854248+0.*I <- {omega[1] = 2^(1/2)+I*2^(1/2), omega[3] = -2^(1/2)-I*2^(1/2)}
5.656854248+0.*I <- {omega[1] = 2^(1/2)-I*2^(1/2), omega[3] = 2^(1/2)-I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.0, 5.656854249492381] <- {Rule[Subscript[ω, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[5.656854249492381, 5.656854249492381] <- {Rule[Subscript[ω, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
5.656854249492381 <- {Rule[Subscript[ω, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
5.656854249492381 <- {Rule[Subscript[ω, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
23.22.E2 ${\displaystyle{\displaystyle-2i\omega_{3}=\frac{\left(\Gamma\left(\frac{1}{4}% \right)\right)^{2}}{2\sqrt{\pi}c^{1/4}}}}$ - 2*I*omega[3]=((GAMMA((1)/(4)))^(2))/(2*sqrt(Pi)*(c)^(1/ 4)) - 2*I*Subscript[\[Omega], 3]=Divide[(Gamma[Divide[1,4]])^(2),2*Sqrt[Pi]*(c)^(1/ 4)] Failure Failure
Fail
-.229827625-2.220102432*I <- {c = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)+I*2^(1/2)}
-5.886681873-2.220102432*I <- {c = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)-I*2^(1/2)}
-5.886681873+3.436751816*I <- {c = 2^(1/2)+I*2^(1/2), omega[3] = -2^(1/2)-I*2^(1/2)}
-.229827625+3.436751816*I <- {c = 2^(1/2)+I*2^(1/2), omega[3] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.22982762217633157, -2.2201024329857546] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.886681871668712, -2.2201024329857546] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.886681871668712, 3.436751816506626] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.22982762217633157, 3.436751816506626] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
23.22.E3 ${\displaystyle{\displaystyle 2\omega_{1}=2e^{-\pi i/3}\omega_{3}}}$ 2*omega[1]= 2*exp(- Pi*I/ 3)*omega[3] 2*Subscript[\[Omega], 1]= 2*Exp[- Pi*I/ 3]*Subscript[\[Omega], 3] Failure Failure
Fail
-1.035276181+3.863703305*I <- {omega[1] = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)+I*2^(1/2)}
3.863703305+6.692130429*I <- {omega[1] = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)-I*2^(1/2)}
6.692130429+1.793150943*I <- {omega[1] = 2^(1/2)+I*2^(1/2), omega[3] = -2^(1/2)-I*2^(1/2)}
1.793150943-1.035276181*I <- {omega[1] = 2^(1/2)+I*2^(1/2), omega[3] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-1.0352761804100834, 3.8637033051562732] <- {Rule[Subscript[ω, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.8637033051562732, 6.692130429902464] <- {Rule[Subscript[ω, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[6.692130429902464, 1.7931509443361073] <- {Rule[Subscript[ω, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.7931509443361073, -1.0352761804100834] <- {Rule[Subscript[ω, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
23.22.E3 ${\displaystyle{\displaystyle 2e^{-\pi i/3}\omega_{3}=\frac{\left(\Gamma\left(% \frac{1}{3}\right)\right)^{3}}{2\pi d^{1/6}}}}$ 2*exp(- Pi*I/ 3)*omega[3]=((GAMMA((1)/(3)))^(3))/(2*Pi*(d)^(1/ 6)) 2*Exp[- Pi*I/ 3]*Subscript[\[Omega], 3]=Divide[(Gamma[Divide[1,3]])^(3),2*Pi*(d)^(1/ 6)] Failure Failure
Fail
1.160957012-.6794528810*I <- {d = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)+I*2^(1/2)}
-3.738022474-3.507880005*I <- {d = 2^(1/2)+I*2^(1/2), omega[3] = 2^(1/2)-I*2^(1/2)}
-6.566449598+1.391099481*I <- {d = 2^(1/2)+I*2^(1/2), omega[3] = -2^(1/2)-I*2^(1/2)}
-1.667470112+4.219526605*I <- {d = 2^(1/2)+I*2^(1/2), omega[3] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.160957015492158, -0.6794528810307349] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.7380224700741986, -3.5078800057769257] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-6.566449594820389, 1.391099479789431] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6674701092540327, 4.219526604535622] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ω, 3], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data