Weierstrass Elliptic and Modular Functions - 23.18 Modular Transformations

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
23.18.E3	${\displaystyle{\displaystyle\lambda\left(\mathcal{A}\tau\right)=\lambda\left(\tau\right)}}$ \modularlambdatau@{\mathcal{A}\tau} = \modularlambdatau@{\tau}		Error	ModularLambda[A*\[Tau]] == ModularLambda[\[Tau]]	Missing Macro Error	Failure	-	Failed [10 / 100] Result: Complex[-5.551115123125783*^-17, -21.100969873679457] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-4.440892098500626*^-16, -21.100969873679432] Test Values: {Rule[A, 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
23.18.E4	${\displaystyle{\displaystyle J\left(\mathcal{A}\tau\right)=J\left(\tau\right)}}$ \KleincompinvarJtau@{\mathcal{A}\tau} = \KleincompinvarJtau@{\tau}		Error	KleinInvariantJ[A*\[Tau]] == KleinInvariantJ[\[Tau]]	Missing Macro Error	Failure	-	Failed [8 / 100] Result: Complex[71.08223570333668, 2.1851275073468844*^-14] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-71.08223570333656, -1.2998925520285436*^-13] Test Values: {Rule[A, 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
23.18.E5	${\displaystyle{\displaystyle\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)\right)^{1/2}\eta\left(\tau\right)}}$ \Dedekindeta@{\mathcal{A}\tau} = \varepsilon(\mathcal{A})\left(-i(c\tau+d)\right)^{1/2}\Dedekindeta@{\tau}		Error	DedekindEta[A*\[Tau]] == \[CurlyEpsilon][A]*(- I*(c*\[Tau]+ d))^(1/2)* DedekindEta[\[Tau]]	Missing Macro Error	Failure	-	Failed [180 / 300] Result: Complex[0.11245781368984653, 0.4581664384510718] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 1], Rule[τ, Complex[0, 1]]} Result: Complex[-0.5688147076679476, 1.020829457922046] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 2], Rule[τ, Complex[0, 1]]} ... skip entries to safe data
23.18.E6	${\displaystyle{\displaystyle\varepsilon(\mathcal{A})=\exp\left(\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)\right)}}$ \varepsilon(\mathcal{A}) = \exp@{\pi i\left(\frac{a+d}{12c}+s(-d,c)\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	Error	Error
23.18.E7	${\displaystyle{\displaystyle s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}\right\rfloor-\frac{1}{2}\right),}}$ s(d,c) = \sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\floor{\frac{dr}{c}}-\frac{1}{2}\right),	${\displaystyle{\displaystyle c>0}}$	s(d , c) = sum((r)/(c)*((d*r)/(c)- floor((d*r)/(c))-(1)/(2)), r = 1..c - 1)	s[d , c] == Sum[Divide[r,c]*(Divide[d*r,c]- Floor[Divide[d*r,c]]-Divide[1,2]), {r, 1, c - 1}, GenerateConditions->None]	Skipped - Unable to analyze test case: Null	Skipped - Unable to analyze test case: Null	-	-