23.15

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
23.15.E1	${\displaystyle{\displaystyle q=\exp\left(-\pi\frac{{K^{\prime}}\left(k\right)}{K\left(k\right)}\right)}}$ q = \exp@{-\pi\frac{\ccompellintKk@{k}}{\compellintKk@{k}}}		q = exp(- Pi*(EllipticCK(k))/(EllipticK(k)))	q == Exp[- Pi*Divide[EllipticK[1-(k)^2],EllipticK[(k)^2]]]	Failure	Failure	Error	Failed [30 / 30] Result: Complex[-0.1339745962155613, 0.49999999999999994] Test Values: {Rule[k, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.9466424242240871, 0.7022944994770247] Test Values: {Rule[k, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... 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 = \frac{\Jacobithetaq{2}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}		k = ((JacobiTheta2(0, q))^(2))/((JacobiTheta3(0, q))^(2))	k == Divide[(EllipticTheta[2, 0, q])^(2),(EllipticTheta[3, 0, q])^(2)]	Failure	Failure	Error	Failed [5 / 30] Result: Complex[1.0, -308.9309168668012] Test Values: {Rule[k, 1], Rule[q, -0.5]} Result: Complex[2.0, -308.9309168668012] Test Values: {Rule[k, 2], Rule[q, -0.5]} ... skip entries to safe data
23.15.E3	${\displaystyle{\displaystyle\mathcal{A}\tau=\frac{a\tau+b}{c\tau+d}}}$ \mathcal{A}\tau = \frac{a\tau+b}{c\tau+d}		A*tau = (a*tau + b)/(c*tau + d)	A*\[Tau] == Divide[a*\[Tau]+ b,c*\[Tau]+ d]	Skipped - no semantic math	Skipped - no semantic math	-	-
23.15.E4	${\displaystyle{\displaystyle ad-bc=1}}$ ad-bc = 1		a*d - b*c = 1	a*d - b*c == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
23.15.E6	${\displaystyle{\displaystyle\lambda\left(\tau\right)=\frac{{\theta_{2}^{4}}\left(0,q\right)}{{\theta_{3}^{4}}\left(0,q\right)}}}$ \modularlambdatau@{\tau} = \frac{\Jacobithetaq{2}^{4}@{0}{q}}{\Jacobithetaq{3}^{4}@{0}{q}}		Error	ModularLambda[\[Tau]] == Divide[(EllipticTheta[2, 0, q])^(4),(EllipticTheta[3, 0, q])^(4)]	Missing Macro Error	Failure	-	Failed [4 / 100] Result: Complex[95438.81139616246, 21.966995277463894] Test Values: {Rule[q, -0.5], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[95438.81139616246, -0.8660254037844387] Test Values: {Rule[q, -0.5], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
23.15.E7	${\displaystyle{\displaystyle J\left(\tau\right)=\frac{\left({\theta_{2}^{8}}\left(0,q\right)+{\theta_{3}^{8}}\left(0,q\right)+{\theta_{4}^{8}}\left(0,q\right)\right)^{3}}{54\left(\theta_{1}'\left(0,q\right)\right)^{8}}}}$ \KleincompinvarJtau@{\tau} = \frac{\left(\Jacobithetaq{2}^{8}@{0}{q}+\Jacobithetaq{3}^{8}@{0}{q}+\Jacobithetaq{4}^{8}@{0}{q}\right)^{3}}{54\left(\Jacobithetaq{1}'@{0}{q}\right)^{8}}		Error	KleinInvariantJ[\[Tau]] == Divide[((EllipticTheta[2, 0, q])^(8)+ (EllipticTheta[3, 0, q])^(8)+ (EllipticTheta[4, 0, q])^(8))^(3),54*(D[EllipticTheta[1, 0, q], {0, 1}])^(8)]	Missing Macro Error	Failure	-	Failed [100 / 100] Result: Plus[Complex[-71.08223570333668, -2.1851275073468844*^-14], Times[-0.018518518518518517, Power[D[0.0 Test Values: {0.0, 1.0}], -8], Power[Plus[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8]], 3]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Times[-0.018518518518518517, Power[D[0.0 Test Values: {0.0, 1.0}], -8], Power[Plus[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8]], 3]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
23.15.E9	${\displaystyle{\displaystyle\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)\right)^{1/3}}}$ \Dedekindeta@{\tau} = \left(\tfrac{1}{2}\Jacobithetaq{1}'@{0}{q}\right)^{1/3}		Error	DedekindEta[\[Tau]] == (Divide[1,2]*D[EllipticTheta[1, 0, q], {0, 1}])^(1/3)	Missing Macro Error	Failure	-	Failed [10 / 10] Result: Plus[0.7682254223260567, Times[-0.7937005259840998, Power[D[0.0 Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]} Result: Plus[0.7682254223260567, Times[-0.7937005259840998, Power[D[0.0 Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]} ... skip entries to safe data
23.15.E9	${\displaystyle{\displaystyle\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)\right)^{1/3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle|3\tau\right)}}$ \left(\tfrac{1}{2}\Jacobithetaq{1}'@{0}{q}\right)^{1/3} = e^{i\pi\tau/12}\Jacobithetatau{3}@{\tfrac{1}{2}\pi(1+\tau)}{3\tau}		((1)/(2)*diff( JacobiTheta1(0, q), 0$(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, 0, q], {0, 1}])^(1/3) == Exp[I*Pi*\[Tau]/12]*EllipticTheta[3, Divide[1,2]*Pi*(1 + \[Tau]), Exp[I*Pi*(3*\[Tau])]]	Error	Failure	-	Failed [10 / 10] Result: Plus[Complex[-0.7682254223260567, 1.7569052324234997*^-19], Times[0.7937005259840998, Power[D[0.0 Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]} Result: Plus[Complex[-0.7682254223260567, 1.7569052324234997*^-19], Times[0.7937005259840998, Power[D[0.0 Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]} ... skip entries to safe data