Weierstrass Elliptic and Modular Functions - 23.17 Elementary Properties

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
23.17#Ex1	${\displaystyle{\displaystyle\lambda\left(i\right)=\tfrac{1}{2}}}$ \modularlambdatau@{i} = \tfrac{1}{2}		Error	ModularLambda[I] == Divide[1,2]	Missing Macro Error	Successful	-	Successful [Tested: 1]
23.17#Ex3	${\displaystyle{\displaystyle J\left(i\right)=1}}$ \KleincompinvarJtau@{i} = 1		Error	KleinInvariantJ[I] == 1	Missing Macro Error	Successful	-	Successful [Tested: 1]
23.17#Ex5	${\displaystyle{\displaystyle\eta\left(i\right)=\frac{\Gamma\left(\tfrac{1}{4}\right)}{2\pi^{3/4}}}}$ \Dedekindeta@{i} = \frac{\EulerGamma@{\tfrac{1}{4}}}{2\pi^{3/4}}		Error	DedekindEta[I] == Divide[Gamma[Divide[1,4]],2*(Pi)^(3/4)]	Missing Macro Error	Successful	-	Successful [Tested: 1]
23.17.E6	${\displaystyle{\displaystyle\eta\left(\tau\right)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(6n+1)^{2}/12}}}$ \Dedekindeta@{\tau} = \sum_{n=-\infty}^{\infty}(-1)^{n}q^{(6n+1)^{2}/12}		Error	DedekindEta[\[Tau]] == Sum[(- 1)^(n)* (q)^((6*n + 1)^(2)/12), {n, - Infinity, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [10 / 10] Result: Plus[0.7682254223260567, Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[Rational[1, 12], Power[Plus[1, Times[6, n]], 2]]]] Test Values: {n, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]} Result: Plus[0.7682254223260567, Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Times[Rational[1, 12], Power[Plus[1, Times[6, n]], 2]]]] Test Values: {n, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]} ... skip entries to safe data
23.17.E7	${\displaystyle{\displaystyle\lambda\left(\tau\right)=16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)^{8}}}$ \modularlambdatau@{\tau} = 16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)^{8}		Error	ModularLambda[\[Tau]] == 16*q*Product[(Divide[1 + (q)^(2*n),1 + (q)^(2*n - 1)])^(8), {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [24 / 100] Result: Indeterminate Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
23.17.E8	${\displaystyle{\displaystyle\eta\left(\tau\right)=q^{1/12}\prod_{n=1}^{\infty}(1-q^{2n})}}$ \Dedekindeta@{\tau} = q^{1/12}\prod_{n=1}^{\infty}(1-q^{2n})		Error	DedekindEta[\[Tau]] == (q)^(1/12)* Product[1 - (q)^(2*n), {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [4 / 10] Result: DirectedInfinity[] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]} Result: DirectedInfinity[] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[τ, Complex[0, 1]]} ... skip entries to safe data