Weierstrass Elliptic and Modular Functions - 23.5 Special Lattices

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
23.5.E2	${\displaystyle{\displaystyle\eta_{1}=i\eta_{3}}}$ \eta_{1} = i\eta_{3}		eta[1] = I*eta[3]	Subscript[\[Eta], 1] == I*Subscript[\[Eta], 3]	Skipped - no semantic math	Skipped - no semantic math	-	-
23.5#Ex7	${\displaystyle{\displaystyle k^{2}=\tfrac{1}{2}}}$ k^{2} = \tfrac{1}{2}		(k)^(2) = (1)/(2)	(k)^(2) == Divide[1,2]	Skipped - no semantic math	Skipped - no semantic math	-	-
23.5#Ex8	${\displaystyle{\displaystyle K\left(k\right)=K'\left(k\right)}}$ \compellintKk@{k} = \compellintKk'@{k}		EllipticK(k) = diff( EllipticK(k), k$(1) )	EllipticK[(k)^2] == D[EllipticK[(k)^2], {k, 1}]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[1.3320292471861073, -1.3934110303935494] Test Values: {Rule[k, 2]} ... skip entries to safe data
23.5#Ex8	${\displaystyle{\displaystyle K'\left(k\right)=\ifrac{\left(\Gamma\left(\tfrac{1}{4}\right)\right)^{2}}{\left(4\sqrt{\pi}\right)}}}$ \compellintKk'@{k} = \ifrac{\left(\EulerGamma@{\tfrac{1}{4}}\right)^{2}}{\left(4\sqrt{\pi}\right)}		diff( EllipticK(k), k$(1) ) = ((GAMMA((1)/(4)))^(2))/(4*sqrt(Pi))	D[EllipticK[(k)^2], {k, 1}] == Divide[(Gamma[Divide[1,4]])^(2),4*Sqrt[Pi]]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.343228747081181, 0.3151532066437278] Test Values: {Rule[k, 2]} ... skip entries to safe data
23.5.E6	${\displaystyle{\displaystyle\eta_{1}=e^{\pi i/3}\eta_{3}}}$ \eta_{1} = e^{\pi i/3}\eta_{3}		eta[1] = exp(Pi*I/3)*eta[3]	Subscript[\[Eta], 1] == Exp[Pi*I/3]*Subscript[\[Eta], 3]	Skipped - no semantic math	Skipped - no semantic math	-	-
23.5#Ex11	${\displaystyle{\displaystyle k^{2}=e^{\mathrm{i}\pi/3}}}$ k^{2} = e^{\iunit\pi/3}		(k)^(2) = exp(I*Pi/3)	(k)^(2) == Exp[I*Pi/3]	Failure	Failure	Failed [3 / 3] Result: .5000000000-.8660254040*I Test Values: {k = 1} Result: 3.500000000-.8660254040*I Test Values: {k = 2} ... skip entries to safe data	Failed [3 / 3] Result: Complex[0.4999999999999999, -0.8660254037844386] Test Values: {Rule[k, 1]} Result: Complex[3.5, -0.8660254037844386] Test Values: {Rule[k, 2]} ... skip entries to safe data
23.5#Ex12	${\displaystyle{\displaystyle K\left(k\right)=e^{\mathrm{i}\pi/6}K'\left(k\right)}}$ \compellintKk@{k} = e^{\iunit\pi/6}\compellintKk'@{k}		EllipticK(k) = exp(I*Pi/6)*diff( EllipticK(k), k$(1) )	EllipticK[(k)^2] == Exp[I*Pi/6]*D[EllipticK[(k)^2], {k, 1}]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[1.4240716315220228, -1.1066114718975122] Test Values: {Rule[k, 2]} ... skip entries to safe data
23.5#Ex12	${\displaystyle{\displaystyle e^{\mathrm{i}\pi/6}K'\left(k\right)=e^{\mathrm{i}\pi/12}\frac{3^{1/4}\left(\Gamma\left(\frac{1}{3}\right)\right)^{3}}{2^{7/3}\pi}}}$ e^{\iunit\pi/6}\compellintKk'@{k} = e^{\iunit\pi/12}\frac{3^{1/4}\left(\EulerGamma@{\frac{1}{3}}\right)^{3}}{2^{7/3}\pi}		exp(I*Pi/6)*diff( EllipticK(k), k$(1) ) = exp(I*Pi/12)*((3)^(1/4)*(GAMMA((1)/(3)))^(3))/((2)^(7/3)* Pi)	Exp[I*Pi/6]*D[EllipticK[(k)^2], {k, 1}] == Exp[I*Pi/12]*Divide[(3)^(1/4)*(Gamma[Divide[1,3]])^(3),(2)^(7/3)* Pi]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.1248830880335463, -0.38527593877730804] Test Values: {Rule[k, 2]} ... skip entries to safe data