Weierstrass Elliptic and Modular Functions - 23.6 Relations to Other Functions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
23.6#Ex1	${\displaystyle{\displaystyle q=e^{i\pi\tau}}}$ q = e^{i\pi\tau}		q = exp(I*Pi*tau)	q == Exp[I*Pi*\[Tau]]	Skipped - no semantic math	Skipped - no semantic math	-	-
23.6#Ex2	${\displaystyle{\displaystyle\tau=\omega_{3}/\omega_{1}}}$ \tau = \omega_{3}/\omega_{1}		tau = omega[3]/omega[1]	\[Tau] == Subscript[\[Omega], 3]/Subscript[\[Omega], 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
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} = -\frac{\pi^{2}}{12\omega_{1}}\frac{\Jacobithetaq{1}'''@{0}{q}}{\Jacobithetaq{1}'@{0}{q}}		eta[1] = -((Pi)^(2))/(12*omega[1])*(diff( JacobiTheta1(0, q), 0$(3) ))/(diff( JacobiTheta1(0, q), 0$(1) ))	Subscript[\[Eta], 1] == -Divide[(Pi)^(2),12*Subscript[\[Omega], 1]]*Divide[D[EllipticTheta[1, 0, q], {0, 3}],D[EllipticTheta[1, 0, q], {0, 1}]]	Error	Failure	-	Failed [300 / 300] Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[0.712277344720507, -0.4112335167120565], Power[D[0.0 Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[η, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[-0.4112335167120564, -0.712277344720507], Power[D[0.0 Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[η, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
23.6#Ex5	${\displaystyle{\displaystyle{K^{2}}=(K\left(k\right))^{2}}}$ \compellintKk^{2}@@{k} = (\compellintKk@{k})^{2}		(EllipticK(k))^(2) = (EllipticK(k))^(2)	(EllipticK[(k)^2])^(2) == (EllipticK[(k)^2])^(2)	Successful	Successful	-	Successful [Tested: 3]