Orthogonal Polynomials - 18.39 Physical Applications

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.39.E3	${\displaystyle{\displaystyle V(x)=\tfrac{1}{2}m\omega^{2}x^{2}}}$ V(x) = \tfrac{1}{2}m\omega^{2}x^{2}		V(x) = (1)/(2)*m*(omega)^(2)* (x)^(2)	V[x] == Divide[1,2]*m*\[Omega]^(2)* (x)^(2)	Skipped - no semantic math	Skipped - no semantic math	-	-
18.39.E5	${\displaystyle{\displaystyle\eta_{n}(x)=\pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!\,b)^{-\frac{1}{2}}H_{n}\left(x/b\right)e^{-x^{2}/2b^{2}}}}$ \eta_{n}(x) = \pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!\,b)^{-\frac{1}{2}}\HermitepolyH{n}@{x/b}e^{-x^{2}/2b^{2}}		eta[n](x) = (Pi)^(-(1)/(4))* (2)^(-(1)/(2)*n)*(factorial(n)*b)^(-(1)/(2))* HermiteH(n, x/b)*exp(- (x)^(2)/2*(b)^(2))	Subscript[\[Eta], n][x] == (Pi)^(-Divide[1,4])* (2)^(-Divide[1,2]*n)*((n)!*b)^(-Divide[1,2])* HermiteH[n, x/b]*Exp[- (x)^(2)/2*(b)^(2)]	Failure	Failure	Failed [300 / 300] Result: 1.299038106+.6809960435*I Test Values: {b = -3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, eta[n] = 1/2*3^(1/2)+1/2*I, n = 1} Result: 1.299038106+.7845019783*I Test Values: {b = -3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, eta[n] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.299038105676658, 0.6809960434853285] Test Values: {Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[η, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.299038105676658, 0.7845019782573356] Test Values: {Rule[b, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[η, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data