14.2

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.2.E1	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}+\nu(\nu+1)w=0}}$ \left(1-x^{2}\right)\deriv[2]{w}{x}-2x\deriv{w}{x}+\nu(\nu+1)w = 0		(1 - (x)^(2))*diff(w, [x$(2)])- 2*x*diff(w, x)+ nu*(nu + 1)*w = 0	(1 - (x)^(2))*D[w, {x, 2}]- 2*x*D[w, x]+ \[Nu]*(\[Nu]+ 1)*w == 0	Failure	Failure	Failed [300 / 300] Result: .5000000007+1.866025405*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: .5000000007+1.866025405*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.5000000000000004, 1.8660254037844386] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.8660254037844388, -0.5] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.2.E2	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-x^{2}}\right)w=0}}$ \left(1-x^{2}\right)\deriv[2]{w}{x}-2x\deriv{w}{x}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-x^{2}}\right)w = 0		(1 - (x)^(2))*diff(w, [x$(2)])- 2*x*diff(w, x)+(nu*(nu + 1)-((mu)^(2))/(1 - (x)^(2)))*w = 0	(1 - (x)^(2))*D[w, {x, 2}]- 2*x*D[w, x]+(\[Nu]*(\[Nu]+ 1)-Divide[\[Mu]^(2),1 - (x)^(2)])*w == 0	Failure	Failure	Failed [300 / 300] Result: .5000000005+2.666025404*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, x = 3/2} Result: .4999999998+.5326920710*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, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.5000000000000007, 2.666025403784439] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.8660254037844387, 0.30000000000000043] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], 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
14.2.E3	${\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{-\mu}_{\nu}\left(-x\right)\right\}=\frac{2}{\Gamma\left(\mu-\nu\right)\Gamma\left(\nu+\mu+1\right)\left(1-x^{2}\right)}}}$ \Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersP[-\mu]{\nu}@{-x}} = \frac{2}{\EulerGamma@{\mu-\nu}\EulerGamma@{\nu+\mu+1}\left(1-x^{2}\right)}	${\displaystyle{\displaystyle\Re(\mu-\nu)>0,\Re(\nu+\mu+1)>0}}$	(LegendreP(nu, - mu, x))*diff(LegendreP(nu, - mu, - x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, - mu, - x)) = (2)/(GAMMA(mu - nu)*GAMMA(nu + mu + 1)*(1 - (x)^(2)))	Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreP[\[Nu], - \[Mu], - x]}, x] == Divide[2,Gamma[\[Mu]- \[Nu]]*Gamma[\[Nu]+ \[Mu]+ 1]*(1 - (x)^(2))]	Failure	Failure	Successful [Tested: 87]	Successful [Tested: 96]
14.2.E4	${\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{\mu}_{\nu}\left(x\right),\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right\}=\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\left(1-x^{2}\right)}}}$ \Wronskian@{\FerrersP[\mu]{\nu}@{x},\FerrersQ[\mu]{\nu}@{x}} = \frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\left(1-x^{2}\right)}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}}$	(LegendreP(nu, mu, x))*diff(LegendreQ(nu, mu, x), x)-diff(LegendreP(nu, mu, x), x)*(LegendreQ(nu, mu, x)) = (GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*(1 - (x)^(2)))	Wronskian[{LegendreP[\[Nu], \[Mu], x], LegendreQ[\[Nu], \[Mu], x]}, x] == Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*(1 - (x)^(2))]	Failure	Failure	Successful [Tested: 120]	Successful [Tested: 135]
14.2.E5	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)=\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)}}}$ \FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x}-\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x} = \frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+2}}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+2)>0}}$	LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x)- LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x) = (GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 2))	LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu], \[Mu], x]- LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x] == Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 2]]	Failure	Failure	Successful [Tested: 162]	Successful [Tested: 174]
14.2.E6	${\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right\}=\frac{\cos\left(\mu\pi\right)}{1-x^{2}}}}$ \Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersQ[\mu]{\nu}@{x}} = \frac{\cos@{\mu\pi}}{1-x^{2}}		(LegendreP(nu, - mu, x))*diff(LegendreQ(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreQ(nu, mu, x)) = (cos(mu*Pi))/(1 - (x)^(2))	Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreQ[\[Nu], \[Mu], x]}, x] == Divide[Cos[\[Mu]*Pi],1 - (x)^(2)]	Failure	Failure	Error	Failed [21 / 300] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -0.5]} ... skip entries to safe data
14.2.E7	${\displaystyle{\displaystyle\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),P^{\mu}_{\nu}\left(x\right)\right\}=\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{\mu}_{\nu}\left(x\right)\right\}}}$ \Wronskian@{\assLegendreP[-\mu]{\nu}@{x},\assLegendreP[\mu]{\nu}@{x}} = \Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersP[\mu]{\nu}@{x}}		(LegendreP(nu, - mu, x))*diff(LegendreP(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, mu, x)) = (LegendreP(nu, - mu, x))*diff(LegendreP(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, mu, x))	Wronskian[{LegendreP[\[Nu], - \[Mu], 3, x], LegendreP[\[Nu], \[Mu], 3, x]}, x] == Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreP[\[Nu], \[Mu], x]}, x]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 300]
14.2.E7	${\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{\mu}_{\nu}\left(x\right)\right\}=\frac{2\sin\left(\mu\pi\right)}{\pi\left(1-x^{2}\right)}}}$ \Wronskian@{\FerrersP[-\mu]{\nu}@{x},\FerrersP[\mu]{\nu}@{x}} = \frac{2\sin@{\mu\pi}}{\pi\left(1-x^{2}\right)}		(LegendreP(nu, - mu, x))*diff(LegendreP(nu, mu, x), x)-diff(LegendreP(nu, - mu, x), x)*(LegendreP(nu, mu, x)) = (2*sin(mu*Pi))/(Pi*(1 - (x)^(2)))	Wronskian[{LegendreP[\[Nu], - \[Mu], x], LegendreP[\[Nu], \[Mu], x]}, x] == Divide[2*Sin[\[Mu]*Pi],Pi*(1 - (x)^(2))]	Failure	Failure	Successful [Tested: 300]	Successful [Tested: 300]
14.2.E8	${\displaystyle{\displaystyle\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)\right\}=-\frac{1}{\Gamma\left(\nu+\mu+1\right)\left(x^{2}-1\right)}}}$ \Wronskian@{\assLegendreP[-\mu]{\nu}@{x},\assLegendreOlverQ[\mu]{\nu}@{x}} = -\frac{1}{\EulerGamma@{\nu+\mu+1}\left(x^{2}-1\right)}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0}}$	(LegendreP(nu, - mu, x))*diff(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1), x)-diff(LegendreP(nu, - mu, x), x)*(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1)) = -(1)/(GAMMA(nu + mu + 1)*((x)^(2)- 1))	Wronskian[{LegendreP[\[Nu], - \[Mu], 3, x], Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1]}, x] == -Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]*((x)^(2)- 1)]	Failure	Failure	Successful [Tested: 195]	Successful [Tested: 207]
14.2.E9	${\displaystyle{\displaystyle\mathscr{W}\left\{\boldsymbol{Q}^{\mu}_{\nu}\left(x\right),\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)\right\}=\frac{\cos\left(\nu\pi\right)}{x^{2}-1}}}$ \Wronskian@{\assLegendreOlverQ[\mu]{\nu}@{x},\assLegendreOlverQ[\mu]{-\nu-1}@{x}} = \frac{\cos@{\nu\pi}}{x^{2}-1}		(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1))*diff(exp(-(mu)*Pi*I)*LegendreQ(- nu - 1,mu,x)/GAMMA(- nu - 1+mu+1), x)-diff(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1), x)*(exp(-(mu)*Pi*I)*LegendreQ(- nu - 1,mu,x)/GAMMA(- nu - 1+mu+1)) = (cos(nu*Pi))/((x)^(2)- 1)	Wronskian[{Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1], Exp[-(\[Mu]) Pi I] LegendreQ[- \[Nu]- 1, \[Mu], 3, x]/Gamma[- \[Nu]- 1 + \[Mu] + 1]}, x] == Divide[Cos[\[Nu]*Pi],(x)^(2)- 1]	Failure	Aborted	Failed [39 / 300] Result: 1.832150333+.7522048283*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -3.053583887-1.253674714*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [57 / 300] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.2.E10	${\displaystyle{\displaystyle\mathscr{W}\left\{P^{\mu}_{\nu}\left(x\right),Q^{\mu}_{\nu}\left(x\right)\right\}=-e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\left(x^{2}-1\right)}}}$ \Wronskian@{\assLegendreP[\mu]{\nu}@{x},\assLegendreQ[\mu]{\nu}@{x}} = -e^{\mu\pi i}\frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\left(x^{2}-1\right)}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}}$	(LegendreP(nu, mu, x))*diff(LegendreQ(nu, mu, x), x)-diff(LegendreP(nu, mu, x), x)*(LegendreQ(nu, mu, x)) = - exp(mu*Pi*I)*(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*((x)^(2)- 1))	Wronskian[{LegendreP[\[Nu], \[Mu], 3, x], LegendreQ[\[Nu], \[Mu], 3, x]}, x] == - Exp[\[Mu]*Pi*I]*Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*((x)^(2)- 1)]	Failure	Failure	Successful [Tested: 120]	Successful [Tested: 135]
14.2.E11	${\displaystyle{\displaystyle P^{\mu}_{\nu+1}\left(x\right)Q^{\mu}_{\nu}\left(x\right)-P^{\mu}_{\nu}\left(x\right)Q^{\mu}_{\nu+1}\left(x\right)=e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)}}}$ \assLegendreP[\mu]{\nu+1}@{x}\assLegendreQ[\mu]{\nu}@{x}-\assLegendreP[\mu]{\nu}@{x}\assLegendreQ[\mu]{\nu+1}@{x} = e^{\mu\pi i}\frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+2}}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+2)>0}}$	LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x)- LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x) = exp(mu*Pi*I)*(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 2))	LegendreP[\[Nu]+ 1, \[Mu], 3, x]*LegendreQ[\[Nu], \[Mu], 3, x]- LegendreP[\[Nu], \[Mu], 3, x]*LegendreQ[\[Nu]+ 1, \[Mu], 3, x] == Exp[\[Mu]*Pi*I]*Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 2]]	Failure	Failure	Successful [Tested: 162]	Successful [Tested: 174]