Legendre and Related Functions - 14.21 Definitions and Basic Properties

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.21.E1	${\displaystyle{\displaystyle\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0}}$ \left(1-z^{2}\right)\deriv[2]{w}{z}-2z\deriv{w}{z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w = 0		(1 - (z)^(2))*diff(w, [z$(2)])- 2*z*diff(w, z)+(nu*(nu + 1)-((mu)^(2))/(1 - (z)^(2)))*w = 0	(1 - (z)^(2))*D[w, {z, 2}]- 2*z*D[w, z]+(\[Nu]*(\[Nu]+ 1)-Divide[\[Mu]^(2),1 - (z)^(2)])*w == 0	Failure	Failure	Failed [300 / 300] Result: 1.366025404+1.366025404*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .2113248651+1.366025405*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.3660254037844388, 1.3660254037844386] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.7755575615628914*^-16, -0.9999999999999997] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data