Results of Legendre and Related Functions I

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]
14.3.E1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\left(\frac{1+x}{1-x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \FerrersP[\mu]{\nu}@{x} = \left(\frac{1+x}{1-x}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x) = ((1 + x)/(1 - x))^(mu/2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)	LegendreP[\[Nu], \[Mu], x] == (Divide[1 + x,1 - x])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Failed [186 / 300] Result: .299069150e-1-2.924977300*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: 1.647025838-2.840829287*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 2} ... skip entries to safe data	Failed [159 / 300] Result: Complex[0.029906915825256147, -2.924977300264846] 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: Complex[-3.067091398010022, -0.8210135056644174] 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[2, 3]], Pi]]]} ... skip entries to safe data
14.3.E2	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi}{2\sin\left(\mu\pi\right)}\left(\cos\left(\mu\pi\right)\left(\frac{1+x}{1-x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)}\left(\frac{1-x}{1+x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)\right)}}$ \FerrersQ[\mu]{\nu}@{x} = \frac{\pi}{2\sin@{\mu\pi}}\left(\cos@{\mu\pi}\left(\frac{1+x}{1-x}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}-\frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}}\left(\frac{1-x}{1+x}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1+\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}\right)	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(nu, mu, x) = (Pi)/(2*sin(mu*Pi))*(cos(mu*Pi)*((1 + x)/(1 - x))^(mu/2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)-(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1))*((1 - x)/(1 + x))^(mu/2)* hypergeom([nu + 1, - nu], [1 + mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 + mu))	LegendreQ[\[Nu], \[Mu], x] == Divide[Pi,2*Sin[\[Mu]*Pi]]*(Cos[\[Mu]*Pi]*(Divide[1 + x,1 - x])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]-Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]]*(Divide[1 - x,1 + x])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 + \[Mu], Divide[1,2]-Divide[1,2]*x])	Failure	Failure	Failed [52 / 120] Result: -4.859700475+.2639835842*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: -4.893385611-2.430027023*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 2} ... skip entries to safe data	Failed [54 / 135] Result: Complex[-4.859700475422212, 0.2639835832089452] 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: Complex[-3.597069591108201, 8.997773008153189] 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, 3]], Pi]]]} ... skip entries to safe data
14.3.E3	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;x\right)=\frac{1}{\Gamma\left(c\right)}F\left(a,b;c;x\right)}}$ \hyperOlverF@{a}{b}{c}{x} = \frac{1}{\EulerGamma@{c}}\hyperF@{a}{b}{c}{x}	${\displaystyle{\displaystyle\Re c>0,|x|<1}}$	hypergeom([a, b], [c], x)/GAMMA(c) = (1)/(GAMMA(c))*hypergeom([a, b], [c], x)	Hypergeometric2F1Regularized[a, b, c, x] == Divide[1,Gamma[c]]*Hypergeometric2F1[a, b, c, x]	Successful	Successful	-	Successful [Tested: 108]
14.3.E4	${\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(1-x^{2}\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \FerrersP[m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu+m+1}}{2^{m}\EulerGamma@{\nu-m+1}}\left(1-x^{2}\right)^{m/2}\hyperOlverF@{\nu+m+1}{m-\nu}{m+1}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle\Re(\nu+m+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, m, x) = (- 1)^(m)*(GAMMA(nu + m + 1))/((2)^(m)* GAMMA(nu - m + 1))*(1 - (x)^(2))^(m/2)* hypergeom([nu + m + 1, m - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)	LegendreP[\[Nu], m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]+ m + 1],(2)^(m)* Gamma[\[Nu]- m + 1]]*(1 - (x)^(2))^(m/2)* Hypergeometric2F1Regularized[\[Nu]+ m + 1, m - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
14.3.E5	${\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{\Gamma\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}\right)^{m/2}\mathbf{F}\left(\nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \FerrersP[m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu+m+1}}{\EulerGamma@{\nu-m+1}}\left(\frac{1-x}{1+x}\right)^{m/2}\hyperOlverF@{\nu+1}{-\nu}{m+1}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle\Re(\nu+m+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, m, x) = (- 1)^(m)*(GAMMA(nu + m + 1))/(GAMMA(nu - m + 1))*((1 - x)/(1 + x))^(m/2)* hypergeom([nu + 1, - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)	LegendreP[\[Nu], m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]+ m + 1],Gamma[\[Nu]- m + 1]]*(Divide[1 - x,1 + x])^(m/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
14.3.E6	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \assLegendreP[\mu]{\nu}@{x} = \left(\frac{x+1}{x-1}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x) = ((x + 1)/(x - 1))^(mu/2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)	LegendreP[\[Nu], \[Mu], 3, x] == (Divide[x + 1,x - 1])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Failed [106 / 300] Result: -4.719014115+.3779003255*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2} Result: -1.667629478-3.026452547*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -1/2+1/2*I*3^(1/2), x = 1/2} ... skip entries to safe data	Failed [79 / 300] Result: Complex[-4.719014112853729, 0.37790032166140924] Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.667629477217065, -3.026452547389477] Test Values: {Rule[x, 0.5], 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
14.3.E7	${\displaystyle{\displaystyle Q^{\mu}_{\nu}\left(x\right)=e^{\mu\pi i}\frac{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{\nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right)}}$ \assLegendreQ[\mu]{\nu}@{x} = e^{\mu\pi i}\frac{\pi^{1/2}\EulerGamma@{\nu+\mu+1}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{\nu+\mu+1}}\hyperOlverF@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1}{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}}{\nu+\tfrac{3}{2}}{\frac{1}{x^{2}}}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0}}$	LegendreQ(nu, mu, x) = exp(mu*Pi*I)*((Pi)^(1/2)* GAMMA(nu + mu + 1)*((x)^(2)- 1)^(mu/2))/((2)^(nu + 1)* (x)^(nu + mu + 1))*hypergeom([(1)/(2)*nu +(1)/(2)*mu + 1, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [nu +(3)/(2)], (1)/((x)^(2)))/GAMMA(nu +(3)/(2))	LegendreQ[\[Nu], \[Mu], 3, x] == Exp[\[Mu]*Pi*I]*Divide[(Pi)^(1/2)* Gamma[\[Nu]+ \[Mu]+ 1]*((x)^(2)- 1)^(\[Mu]/2),(2)^(\[Nu]+ 1)* (x)^(\[Nu]+ \[Mu]+ 1)]*Hypergeometric2F1Regularized[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Divide[1,(x)^(2)]]	Failure	Failure	Failed [28 / 200] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -3/2, x = 3/2, nu+mu = 1} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -3/2, x = 2, nu+mu = 1} ... skip entries to safe data	Successful [Tested: 138]
14.3.E8	${\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \assLegendreP[m]{\nu}@{x} = \frac{\EulerGamma@{\nu+m+1}}{2^{m}\EulerGamma@{\nu-m+1}}\left(x^{2}-1\right)^{m/2}\hyperOlverF@{\nu+m+1}{m-\nu}{m+1}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle\Re(\nu+m+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, m, x) = (GAMMA(nu + m + 1))/((2)^(m)* GAMMA(nu - m + 1))*((x)^(2)- 1)^(m/2)* hypergeom([nu + m + 1, m - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)	LegendreP[\[Nu], m, 3, x] == Divide[Gamma[\[Nu]+ m + 1],(2)^(m)* Gamma[\[Nu]- m + 1]]*((x)^(2)- 1)^(m/2)* Hypergeometric2F1Regularized[\[Nu]+ m + 1, m - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
14.3.E9	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \assLegendreP[-\mu]{\nu}@{x} = \left(\frac{x-1}{x+1}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{\mu+1}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, - mu, x) = ((x - 1)/(x + 1))^(mu/2)* hypergeom([nu + 1, - nu], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)	LegendreP[\[Nu], - \[Mu], 3, x] == (Divide[x - 1,x + 1])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]	Failure	Successful	Failed [27 / 300] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Successful [Tested: 300]
14.3.E10	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i}\frac{Q^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}}}$ \assLegendreOlverQ[\mu]{\nu}@{x} = e^{-\mu\pi i}\frac{\assLegendreQ[\mu]{\nu}@{x}}{\EulerGamma@{\nu+\mu+1}}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = exp(- mu*Pi*I)*(LegendreQ(nu, mu, x))/(GAMMA(nu + mu + 1))	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Exp[- \[Mu]*Pi*I]*Divide[LegendreQ[\[Nu], \[Mu], 3, x],Gamma[\[Nu]+ \[Mu]+ 1]]	Successful	Successful	-	Successful [Tested: 207]
14.3.E11	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x)}}$ \FerrersP[\mu]{\nu}@{x} = \cos@{\tfrac{1}{2}(\nu+\mu)\pi}w_{1}(\nu,\mu,x)+\sin@{\tfrac{1}{2}(\nu+\mu)\pi}w_{2}(\nu,\mu,x)	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x) = cos((1)/(2)*(nu + mu)*Pi)*w[1](nu , mu , x)+ sin((1)/(2)*(nu + mu)*Pi)*w[2](nu , mu , x)	LegendreP[\[Nu], \[Mu], x] == Cos[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 1][\[Nu], \[Mu], x]+ Sin[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 2][\[Nu], \[Mu], x]	Failure	Failure	Failed [300 / 300] Result: .1996315555-2.444256460*I+(-.424833882+3.265828322*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000) Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = 1/2*3^(1/2)+1/2*I} Result: .1996315555-2.444256460*I+(.206784146+.21312792e-1*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000) Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Error
14.3.E12	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}\pi\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\tfrac{1}{2}\pi\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x)}}$ \FerrersQ[\mu]{\nu}@{x} = -\tfrac{1}{2}\pi\sin@{\tfrac{1}{2}(\nu+\mu)\pi}w_{1}(\nu,\mu,x)+\tfrac{1}{2}\pi\cos@{\tfrac{1}{2}(\nu+\mu)\pi}w_{2}(\nu,\mu,x)	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(nu, mu, x) = -(1)/(2)*Pi*sin((1)/(2)*(nu + mu)*Pi)*w[1](nu , mu , x)+(1)/(2)*Pi*cos((1)/(2)*(nu + mu)*Pi)*w[2](nu , mu , x)	LegendreQ[\[Nu], \[Mu], x] == -Divide[1,2]*Pi*Sin[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 1][\[Nu], \[Mu], x]+Divide[1,2]*Pi*Cos[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 2][\[Nu], \[Mu], x]	Failure	Failure	Failed [300 / 300] Result: -3.819326549-.1470472359*I+(5.421288855+1.025621334*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000) Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = 1/2*3^(1/2)+1/2*I} Result: -3.819326549-.1470472359*I+(-.33478055e-1+.324815778*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000) Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Error
14.3.E13	${\displaystyle{\displaystyle w_{1}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}\left(1-x^{2}\right)^{-\mu/2}\mathbf{F}\left(-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2};\tfrac{1}{2};x^{2}\right)}}$ w_{1}(\nu,\mu,x) = \frac{2^{\mu}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}}\left(1-x^{2}\right)^{-\mu/2}\hyperOlverF@{-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu}{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}}{\tfrac{1}{2}}{x^{2}}	${\displaystyle{\displaystyle\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+1)>0,|(x^{2})|<1}}$	w[1](nu , mu , x) = ((2)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu + 1))*(1 - (x)^(2))^(- mu/2)* hypergeom([-(1)/(2)*nu -(1)/(2)*mu, (1)/(2)*nu -(1)/(2)*mu +(1)/(2)], [(1)/(2)], (x)^(2))/GAMMA((1)/(2))	Subscript[w, 1][\[Nu], \[Mu], x] == Divide[(2)^\[Mu]* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1]]*(1 - (x)^(2))^(- \[Mu]/2)* Hypergeometric2F1Regularized[-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu], Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2], Divide[1,2], (x)^(2)]	Failure	Failure	Failed [300 / 300] Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.6893070382-.1737378889*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[1] = 1/2*3^(1/2)+1/2*I} Result: (-.5000000000+.8660254040*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.6893070382-.1737378889*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[1] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Error
14.3.E14	${\displaystyle{\displaystyle w_{2}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1-x^{2}\right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right)}}$ w_{2}(\nu,\mu,x) = \frac{2^{\mu}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}}x\left(1-x^{2}\right)^{-\mu/2}\hyperOlverF@{\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu}{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1}{\tfrac{3}{2}}{x^{2}}	${\displaystyle{\displaystyle\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2})>0,|(x^{2})|<1}}$	w[2](nu , mu , x) = ((2)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))*x*(1 - (x)^(2))^(- mu/2)* hypergeom([(1)/(2)-(1)/(2)*nu -(1)/(2)*mu, (1)/(2)*nu -(1)/(2)*mu + 1], [(3)/(2)], (x)^(2))/GAMMA((3)/(2))	Subscript[w, 2][\[Nu], \[Mu], x] == Divide[(2)^\[Mu]* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]*x*(1 - (x)^(2))^(- \[Mu]/2)* Hypergeometric2F1Regularized[Divide[1,2]-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu], Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1, Divide[3,2], (x)^(2)]	Failure	Failure	Failed [300 / 300] Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.4687612945-.2577588545*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[2] = 1/2*3^(1/2)+1/2*I} Result: (-.5000000000+.8660254040*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.4687612945-.2577588545*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Error
14.3.E15	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \assLegendreP[-\mu]{\nu}@{x} = 2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\hyperOlverF@{\mu-\nu}{\nu+\mu+1}{\mu+1}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, - mu, x) = (2)^(- mu)*((x)^(2)- 1)^(mu/2)* hypergeom([mu - nu, nu + mu + 1], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)	LegendreP[\[Nu], - \[Mu], 3, x] == (2)^(- \[Mu])*((x)^(2)- 1)^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Mu]- \[Nu], \[Nu]+ \[Mu]+ 1, \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Failed [27 / 300] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Successful [Tested: 300]
14.3.E16	${\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\pi^{1/2}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{\nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right)}}$ \cos@{\nu\pi}\assLegendreP[-\mu]{\nu}@{x} = \frac{2^{\nu}\pi^{1/2}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\EulerGamma@{\nu+\mu+1}}\hyperOlverF@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu}{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}{\tfrac{1}{2}-\nu}{\frac{1}{x^{2}}}-\frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}\EulerGamma@{\mu-\nu}x^{\nu+\mu+1}}\hyperOlverF@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1}{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}}{\nu+\tfrac{3}{2}}{\frac{1}{x^{2}}}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\mu-\nu)>0}}$	cos(nu*Pi)*LegendreP(nu, - mu, x) = ((2)^(nu)* (Pi)^(1/2)* (x)^(nu - mu)*((x)^(2)- 1)^(mu/2))/(GAMMA(nu + mu + 1))*hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*mu -(1)/(2)*nu +(1)/(2)], [(1)/(2)- nu], (1)/((x)^(2)))/GAMMA((1)/(2)- nu)-((Pi)^(1/2)*((x)^(2)- 1)^(mu/2))/((2)^(nu + 1)* GAMMA(mu - nu)*(x)^(nu + mu + 1))*hypergeom([(1)/(2)*nu +(1)/(2)*mu + 1, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [nu +(3)/(2)], (1)/((x)^(2)))/GAMMA(nu +(3)/(2))	Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], 3, x] == Divide[(2)^\[Nu]* (Pi)^(1/2)* (x)^(\[Nu]- \[Mu])*((x)^(2)- 1)^(\[Mu]/2),Gamma[\[Nu]+ \[Mu]+ 1]]*Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], Divide[1,2]- \[Nu], Divide[1,(x)^(2)]]-Divide[(Pi)^(1/2)*((x)^(2)- 1)^(\[Mu]/2),(2)^(\[Nu]+ 1)* Gamma[\[Mu]- \[Nu]]*(x)^(\[Nu]+ \[Mu]+ 1)]*Hypergeometric2F1Regularized[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Divide[1,(x)^(2)]]	Failure	Failure	Failed [14 / 58] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -3/2, x = 3/2} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -3/2, x = 2} ... skip entries to safe data	Successful [Tested: 64]
14.3.E17	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\pi\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}}\left(\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}-\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}\right)}}$ \assLegendreP[-\mu]{\nu}@{x} = \frac{\pi\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}}\left(\frac{\hyperOlverF@{\frac{1}{2}\mu-\frac{1}{2}\nu}{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}{\frac{1}{2}}{x^{2}}}{\EulerGamma@{\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}-\frac{x\hyperOlverF@{\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}}{\frac{1}{2}\nu+\frac{1}{2}\mu+1}{\frac{3}{2}}{x^{2}}}{\EulerGamma@{\frac{1}{2}\mu-\frac{1}{2}\nu}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}\right)	${\displaystyle{\displaystyle\Re(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,\Re(\frac{1}{2}\mu-\frac{1}{2}\nu)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,|(x^{2})|<1}}$	LegendreP(nu, - mu, x) = (Pi*((x)^(2)- 1)^(mu/2))/((2)^(mu))*((hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [(1)/(2)], (x)^(2))/GAMMA((1)/(2)))/(GAMMA((1)/(2)*mu -(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))-(x*hypergeom([(1)/(2)*mu -(1)/(2)*nu +(1)/(2), (1)/(2)*nu +(1)/(2)*mu + 1], [(3)/(2)], (x)^(2))/GAMMA((3)/(2)))/(GAMMA((1)/(2)*mu -(1)/(2)*nu)*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2))))	LegendreP[\[Nu], - \[Mu], 3, x] == Divide[Pi*((x)^(2)- 1)^(\[Mu]/2),(2)^\[Mu]]*(Divide[Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], Divide[1,2], (x)^(2)],Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]]-Divide[x*Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[3,2], (x)^(2)],Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]]])	Failure	Failure	Successful [Tested: 29]	Successful [Tested: 32]
14.3.E18	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\mu+1;1-\frac{1}{x^{2}}\right)}}$ \assLegendreP[-\mu]{\nu}@{x} = 2^{-\mu}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}\hyperOlverF@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu}{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}{\mu+1}{1-\frac{1}{x^{2}}}		LegendreP(nu, - mu, x) = (2)^(- mu)* (x)^(nu - mu)*((x)^(2)- 1)^(mu/2)* hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*mu -(1)/(2)*nu +(1)/(2)], [mu + 1], 1 -(1)/((x)^(2)))/GAMMA(mu + 1)	LegendreP[\[Nu], - \[Mu], 3, x] == (2)^(- \[Mu])* (x)^(\[Nu]- \[Mu])*((x)^(2)- 1)^(\[Mu]/2)* Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], \[Mu]+ 1, 1 -Divide[1,(x)^(2)]]	Failure	Failure	Failed [18 / 200] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 2} ... skip entries to safe data	Successful [Tested: 200]
14.3.E19	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\Gamma\left(\nu+1\right)(x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\mathbf{F}\left(\nu+1,\nu+\mu+1;2\nu+2;\frac{2}{1-x}\right)}}$ \assLegendreOlverQ[\mu]{\nu}@{x} = \frac{2^{\nu}\EulerGamma@{\nu+1}(x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\hyperOlverF@{\nu+1}{\nu+\mu+1}{2\nu+2}{\frac{2}{1-x}}	${\displaystyle{\displaystyle\Re(\nu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((2)^(nu)* GAMMA(nu + 1)*(x + 1)^(mu/2))/((x - 1)^((mu/2)+ nu + 1))*hypergeom([nu + 1, nu + mu + 1], [2*nu + 2], (2)/(1 - x))/GAMMA(2*nu + 2)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(2)^\[Nu]* Gamma[\[Nu]+ 1]*(x + 1)^(\[Mu]/2),(x - 1)^((\[Mu]/2)+ \[Nu]+ 1)]*Hypergeometric2F1Regularized[\[Nu]+ 1, \[Nu]+ \[Mu]+ 1, 2*\[Nu]+ 2, Divide[2,1 - x]]	Failure	Failure	Error	Skip - No test values generated
14.3.E20	${\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{(x+1)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)(x-1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{(x-1)^{\mu/2}}{\Gamma\left(\nu-\mu+1\right)(x+1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$ \frac{2\sin@{\mu\pi}}{\pi}\assLegendreOlverQ[\mu]{\nu}@{x} = \frac{(x+1)^{\mu/2}}{\EulerGamma@{\nu+\mu+1}(x-1)^{\mu/2}}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}-\frac{(x-1)^{\mu/2}}{\EulerGamma@{\nu-\mu+1}(x+1)^{\mu/2}}\hyperOlverF@{\nu+1}{-\nu}{\mu+1}{\tfrac{1}{2}-\tfrac{1}{2}x}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	(2*sin(mu*Pi))/(Pi)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((x + 1)^(mu/2))/(GAMMA(nu + mu + 1)*(x - 1)^(mu/2))*hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)-((x - 1)^(mu/2))/(GAMMA(nu - mu + 1)*(x + 1)^(mu/2))*hypergeom([nu + 1, - nu], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)	Divide[2*Sin[\[Mu]*Pi],Pi]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(x + 1)^(\[Mu]/2),Gamma[\[Nu]+ \[Mu]+ 1]*(x - 1)^(\[Mu]/2)]*Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]-Divide[(x - 1)^(\[Mu]/2),Gamma[\[Nu]- \[Mu]+ 1]*(x + 1)^(\[Mu]/2)]*Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]	Failure	Successful	Failed [12 / 120] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 3/2, x = 3/2} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = -2, nu = 3/2, x = 1/2} ... skip entries to safe data	Successful [Tested: 135]
14.3.E21	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(1-x^{2}\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right)}}$ \FerrersP[\mu]{\nu}@{x} = \frac{2^{\mu}\EulerGamma@{1-2\mu}\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\EulerGamma@{1-\mu}\left(1-x^{2}\right)^{\mu/2}}\ultrasphpoly{\frac{1}{2}-\mu}{\nu+\mu}@{x}	${\displaystyle{\displaystyle\Re(1-2\mu)>0,\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,\Re(1-\mu)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x) = ((2)^(mu)* GAMMA(1 - 2*mu)*GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*GAMMA(1 - mu)*(1 - (x)^(2))^(mu/2))*GegenbauerC(nu + mu, (1)/(2)- mu, x)	LegendreP[\[Nu], \[Mu], x] == Divide[(2)^\[Mu]* Gamma[1 - 2*\[Mu]]*Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*Gamma[1 - \[Mu]]*(1 - (x)^(2))^(\[Mu]/2)]*GegenbauerC[\[Nu]+ \[Mu], Divide[1,2]- \[Mu], x]	Failure	Failure	Successful [Tested: 60]	Successful [Tested: 69]
14.3.E22	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(x^{2}-1\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right)}}$ \assLegendreP[\mu]{\nu}@{x} = \frac{2^{\mu}\EulerGamma@{1-2\mu}\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\EulerGamma@{1-\mu}\left(x^{2}-1\right)^{\mu/2}}\ultrasphpoly{\frac{1}{2}-\mu}{\nu+\mu}@{x}	${\displaystyle{\displaystyle\Re(1-2\mu)>0,\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,\Re(1-\mu)>0}}$	LegendreP(nu, mu, x) = ((2)^(mu)* GAMMA(1 - 2*mu)*GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*GAMMA(1 - mu)*((x)^(2)- 1)^(mu/2))*GegenbauerC(nu + mu, (1)/(2)- mu, x)	LegendreP[\[Nu], \[Mu], 3, x] == Divide[(2)^\[Mu]* Gamma[1 - 2*\[Mu]]*Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*Gamma[1 - \[Mu]]*((x)^(2)- 1)^(\[Mu]/2)]*GegenbauerC[\[Nu]+ \[Mu], Divide[1,2]- \[Mu], x]	Failure	Failure	Successful [Tested: 60]	Successful [Tested: 69]
14.3.E23	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{\mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i}(2\nu+1)}\left(\operatorname{arcsinh}\left((\tfrac{1}{2}x-\tfrac{1}{2})^{\ifrac{1}{2}}\right)\right)}}$ \assLegendreP[\mu]{\nu}@{x} = \frac{1}{\EulerGamma@{1-\mu}}\left(\frac{x+1}{x-1}\right)^{\mu/2}\Jacobiphi{-\mu}{\mu}{-\iunit(2\nu+1)}@{\asinh@{(\tfrac{1}{2}x-\tfrac{1}{2})^{\ifrac{1}{2}}}}	${\displaystyle{\displaystyle\Re(1-\mu)>0}}$	LegendreP(nu, mu, x) = (1)/(GAMMA(1 - mu))*((x + 1)/(x - 1))^(mu/2)* hypergeom([((- mu)+(mu)+1-I*(- I*(2*nu + 1)))/2, ((- mu)+(mu)+1+I*(- I*(2*nu + 1)))], [(- mu)+1], -sinh(arcsinh(((1)/(2)*x -(1)/(2))^((1)/(2))))^2)	Error	Failure	Missing Macro Error	Failed [240 / 240] Result: -.318116688-.9248307299*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: -5.010614457+.9472052439*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	-
14.5.E1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(0\right)=\frac{2^{\mu}\pi^{1/2}}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\nu-\frac{1}{2}\mu\right)}}}$ \FerrersP[\mu]{\nu}@{0} = \frac{2^{\mu}\pi^{1/2}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}-\frac{1}{2}\nu-\frac{1}{2}\mu}}	${\displaystyle{\displaystyle\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+1)>0,\Re(\frac{1}{2}-\frac{1}{2}\nu-\frac{1}{2}\mu)>0}}$	LegendreP(nu, mu, 0) = ((2)^(mu)* (Pi)^(1/2))/(GAMMA((1)/(2)*nu -(1)/(2)*mu + 1)*GAMMA((1)/(2)-(1)/(2)*nu -(1)/(2)*mu))	LegendreP[\[Nu], \[Mu], 0] == Divide[(2)^\[Mu]* (Pi)^(1/2),Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]]]	Successful	Failure	-	Successful [Tested: 54]
14.5.E3	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(0\right)=-\frac{2^{\mu-1}\pi^{1/2}\sin\left(\frac{1}{2}(\nu+\mu)\pi\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}}}$ \FerrersQ[\mu]{\nu}@{0} = -\frac{2^{\mu-1}\pi^{1/2}\sin@{\frac{1}{2}(\nu+\mu)\pi}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}}	${\displaystyle{\displaystyle\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+1)>0,\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}}$	LegendreQ(nu, mu, 0) = -((2)^(mu - 1)* (Pi)^(1/2)* sin((1)/(2)*(nu + mu)*Pi)*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu + 1))	LegendreQ[\[Nu], \[Mu], 0] == -Divide[(2)^(\[Mu]- 1)* (Pi)^(1/2)* Sin[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1]]	Successful	Failure	-	Successful [Tested: 45]
14.5.E5	${\displaystyle{\displaystyle\mathsf{P}_{0}\left(x\right)=P_{0}\left(x\right)}}$ \FerrersP[]{0}@{x} = \assLegendreP[]{0}@{x}		LegendreP(0, x) = LegendreP(0, x)	LegendreP[0, x] == LegendreP[0, 0, 3, x]	Successful	Successful	-	Successful [Tested: 3]
14.5.E5	${\displaystyle{\displaystyle P_{0}\left(x\right)=1}}$ \assLegendreP[]{0}@{x} = 1		LegendreP(0, x) = 1	LegendreP[0, 0, 3, x] == 1	Successful	Successful	-	Successful [Tested: 3]
14.5.E6	${\displaystyle{\displaystyle\mathsf{P}_{1}\left(x\right)=P_{1}\left(x\right)}}$ \FerrersP[]{1}@{x} = \assLegendreP[]{1}@{x}		LegendreP(1, x) = LegendreP(1, x)	LegendreP[1, x] == LegendreP[1, 0, 3, x]	Successful	Successful	-	Successful [Tested: 3]
14.5.E6	${\displaystyle{\displaystyle P_{1}\left(x\right)=x}}$ \assLegendreP[]{1}@{x} = x		LegendreP(1, x) = x	LegendreP[1, 0, 3, x] == x	Successful	Successful	-	Successful [Tested: 3]
14.5.E7	${\displaystyle{\displaystyle\mathsf{Q}_{0}\left(x\right)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)}}$ \FerrersQ[]{0}@{x} = \frac{1}{2}\ln@{\frac{1+x}{1-x}}		LegendreQ(0, x) = (1)/(2)*ln((1 + x)/(1 - x))	LegendreQ[0, x] == Divide[1,2]*Log[Divide[1 + x,1 - x]]	Failure	Failure	Failed [2 / 3] Result: .2e-9-3.141592654*I Test Values: {x = 3/2} Result: -.2e-9-3.141592654*I Test Values: {x = 2}	Failed [2 / 3] Result: Complex[1.1102230246251565*^-16, -3.141592653589793] Test Values: {Rule[x, 1.5]} Result: Complex[0.0, -3.141592653589793] Test Values: {Rule[x, 2]}
14.5.E8	${\displaystyle{\displaystyle\mathsf{Q}_{1}\left(x\right)=\frac{x}{2}\ln\left(\frac{1+x}{1-x}\right)-1}}$ \FerrersQ[]{1}@{x} = \frac{x}{2}\ln@{\frac{1+x}{1-x}}-1		LegendreQ(1, x) = (x)/(2)*ln((1 + x)/(1 - x))- 1	LegendreQ[1, x] == Divide[x,2]*Log[Divide[1 + x,1 - x]]- 1	Failure	Failure	Failed [2 / 3] Result: .3e-9-4.712388980*I Test Values: {x = 3/2} Result: 0.-6.283185308*I Test Values: {x = 2}	Failed [2 / 3] Result: Complex[2.220446049250313*^-16, -4.71238898038469] Test Values: {Rule[x, 1.5]} Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[x, 2]}
14.5.E9	${\displaystyle{\displaystyle\boldsymbol{Q}_{0}\left(x\right)=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)}}$ \assLegendreOlverQ[]{0}@{x} = \frac{1}{2}\ln@{\frac{x+1}{x-1}}		LegendreQ(0,x)/GAMMA(0+1) = (1)/(2)*ln((x + 1)/(x - 1))	Exp[-(0) Pi I] LegendreQ[0, 2, 3, x]/Gamma[0 + 3] == Divide[1,2]*Log[Divide[x + 1,x - 1]]	Failure	Failure	Failed [1 / 3] Result: -.2e-9-3.141592654*I Test Values: {x = 1/2}	Failed [3 / 3] Result: Complex[0.3952810437829498, -2.9391523179536476*^-16] Test Values: {Rule[x, 1.5]} Result: Complex[-1.2159728110007215, -1.5707963267948966] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
14.5.E10	${\displaystyle{\displaystyle\boldsymbol{Q}_{1}\left(x\right)=\frac{x}{2}\ln\left(\frac{x+1}{x-1}\right)-1}}$ \assLegendreOlverQ[]{1}@{x} = \frac{x}{2}\ln@{\frac{x+1}{x-1}}-1		LegendreQ(1,x)/GAMMA(1+1) = (x)/(2)*ln((x + 1)/(x - 1))- 1	Exp[-(1) Pi I] LegendreQ[1, 2, 3, x]/Gamma[1 + 3] == Divide[x,2]*Log[Divide[x + 1,x - 1]]- 1	Failure	Failure	Failed [1 / 3] Result: 0.-1.570796327*I Test Values: {x = 1/2}	Failed [3 / 3] Result: Complex[-0.47374510099224176, 6.531449595452549*^-17] Test Values: {Rule[x, 1.5]} Result: Complex[1.1697913722774167, -0.7853981633974483] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
14.5.E11	${\displaystyle{\displaystyle\mathsf{P}^{1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\cos\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)}}$ \FerrersP[1/2]{\nu}@{\cos@@{\theta}} = \left(\frac{2}{\pi\sin@@{\theta}}\right)^{1/2}\cos@{\left(\nu+\tfrac{1}{2}\right)\theta}		LegendreP(nu, 1/2, cos(theta)) = ((2)/(Pi*sin(theta)))^(1/2)* cos((nu +(1)/(2))*theta)	LegendreP[\[Nu], 1/2, Cos[\[Theta]]] == (Divide[2,Pi*Sin[\[Theta]]])^(1/2)* Cos[(\[Nu]+Divide[1,2])*\[Theta]]	Failure	Failure	Failed [50 / 100] Result: -.7596743150+.9986452891*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)} Result: -.3969265290-1.700808098*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [50 / 100] Result: Complex[-0.7596743150203076, 0.9986452891592468] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.5932078691227823, 0.7119534787783219] Test Values: {Rule[θ, 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
14.5.E12	${\displaystyle{\displaystyle\mathsf{P}^{-1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\frac{\sin\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}}}}$ \FerrersP[-1/2]{\nu}@{\cos@@{\theta}} = \left(\frac{2}{\pi\sin@@{\theta}}\right)^{1/2}\frac{\sin@{\left(\nu+\frac{1}{2}\right)\theta}}{\nu+\frac{1}{2}}		LegendreP(nu, - 1/2, cos(theta)) = ((2)/(Pi*sin(theta)))^(1/2)*(sin((nu +(1)/(2))*theta))/(nu +(1)/(2))	LegendreP[\[Nu], - 1/2, Cos[\[Theta]]] == (Divide[2,Pi*Sin[\[Theta]]])^(1/2)*Divide[Sin[(\[Nu]+Divide[1,2])*\[Theta]],\[Nu]+Divide[1,2]]	Failure	Failure	Failed [55 / 100] Result: .5392263657-.8901760048*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)} Result: .9027151592+.9035040024*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [55 / 100] Result: Indeterminate Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]} Result: Complex[0.5392263655684584, -0.8901760046482097] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.5.E13	${\displaystyle{\displaystyle\mathsf{Q}^{1/2}_{\nu}\left(\cos\theta\right)=-\left(\frac{\pi}{2\sin\theta}\right)^{1/2}\sin\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)}}$ \FerrersQ[1/2]{\nu}@{\cos@@{\theta}} = -\left(\frac{\pi}{2\sin@@{\theta}}\right)^{1/2}\sin@{\left(\nu+\tfrac{1}{2}\right)\theta}		LegendreQ(nu, 1/2, cos(theta)) = -((Pi)/(2*sin(theta)))^(1/2)* sin((nu +(1)/(2))*theta)	LegendreQ[\[Nu], 1/2, Cos[\[Theta]]] == -(Divide[Pi,2*Sin[\[Theta]]])^(1/2)* Sin[(\[Nu]+Divide[1,2])*\[Theta]]	Failure	Failure	Failed [25 / 50] Result: -1.856186326+1.486585706*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)} Result: -1.227388580-2.647682452*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [25 / 50] Result: Complex[-1.8561863256089288, 1.4865857054438434] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.690848965325271, 2.3698178156702956] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
14.5.E14	${\displaystyle{\displaystyle\mathsf{Q}^{-1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{\pi}{2\sin\theta}\right)^{1/2}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}}}}$ \FerrersQ[-1/2]{\nu}@{\cos@@{\theta}} = \left(\frac{\pi}{2\sin@@{\theta}}\right)^{1/2}\frac{\cos@{\left(\nu+\frac{1}{2}\right)\theta}}{\nu+\frac{1}{2}}		LegendreQ(nu, - 1/2, cos(theta)) = ((Pi)/(2*sin(theta)))^(1/2)*(cos((nu +(1)/(2))*theta))/(nu +(1)/(2))	LegendreQ[\[Nu], - 1/2, Cos[\[Theta]]] == (Divide[Pi,2*Sin[\[Theta]]])^(1/2)*Divide[Cos[(\[Nu]+Divide[1,2])*\[Theta]],\[Nu]+Divide[1,2]]	Failure	Failure	Error	Failed [25 / 50] Result: Complex[-0.3996810371463801, 1.2946383468829223] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.41345894273326, 2.4734286705879205] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
14.5.E15	${\displaystyle{\displaystyle P^{1/2}_{\nu}\left(\cosh\xi\right)=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\cosh\left(\left(\nu+\tfrac{1}{2}\right)\xi\right)}}$ \assLegendreP[1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{2}{\pi\sinh@@{\xi}}\right)^{1/2}\cosh@{\left(\nu+\tfrac{1}{2}\right)\xi}		LegendreP(nu, 1/2, cosh(xi)) = ((2)/(Pi*sinh(xi)))^(1/2)* cosh((nu +(1)/(2))*xi)	LegendreP[\[Nu], 1/2, 3, Cosh[\[Xi]]] == (Divide[2,Pi*Sinh[\[Xi]]])^(1/2)* Cosh[(\[Nu]+Divide[1,2])*\[Xi]]	Failure	Failure	Failed [100 / 100] Result: -.5866633690+.3419889424*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I} Result: .9326102256+.153785626*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [50 / 100] Result: Complex[1.483322380543576, 0.9219835006286831] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.2433197156086089, -0.16897799632039867] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
14.5.E16	${\displaystyle{\displaystyle P^{-1/2}_{\nu}\left(\cosh\xi\right)=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\frac{\sinh\left(\left(\nu+\frac{1}{2}\right)\xi\right)}{\nu+\frac{1}{2}}}}$ \assLegendreP[-1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{2}{\pi\sinh@@{\xi}}\right)^{1/2}\frac{\sinh@{\left(\nu+\frac{1}{2}\right)\xi}}{\nu+\frac{1}{2}}		LegendreP(nu, - 1/2, cosh(xi)) = ((2)/(Pi*sinh(xi)))^(1/2)*(sinh((nu +(1)/(2))*xi))/(nu +(1)/(2))	LegendreP[\[Nu], - 1/2, 3, Cosh[\[Xi]]] == (Divide[2,Pi*Sinh[\[Xi]]])^(1/2)*Divide[Sinh[(\[Nu]+Divide[1,2])*\[Xi]],\[Nu]+Divide[1,2]]	Failure	Failure	Failed [100 / 100] Result: .852516959e-1-.5567654394*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I} Result: .2647935712-.6384793854*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [55 / 100] Result: Complex[5.577974291320897*^-4, -1.2771898182050043] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.2481588696482635, 1.0107401090243302] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
14.5.E17	${\displaystyle{\displaystyle\boldsymbol{Q}^{+1/2}_{\nu}\left(\cosh\xi\right)=\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}\frac{\exp\left(-\left(\nu+\frac{1}{2}\right)\xi\right)}{\Gamma\left(\nu+\frac{3}{2}\right)}}}$ \assLegendreOlverQ[+ 1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{\pi}{2\sinh@@{\xi}}\right)^{1/2}\frac{\exp@{-\left(\nu+\frac{1}{2}\right)\xi}}{\EulerGamma@{\nu+\frac{3}{2}}}	${\displaystyle{\displaystyle\Re(\nu+\frac{3}{2})>0}}$	exp(-(+ 1/2)*Pi*I)*LegendreQ(nu,+ 1/2,cosh(xi))/GAMMA(nu++ 1/2+1) = ((Pi)/(2*sinh(xi)))^(1/2)*(exp(-(nu +(1)/(2))*xi))/(GAMMA(nu +(3)/(2)))	Exp[-(+ 1/2) Pi I] LegendreQ[\[Nu], + 1/2, 3, Cosh[\[Xi]]]/Gamma[\[Nu] + + 1/2 + 1] == (Divide[Pi,2*Sinh[\[Xi]]])^(1/2)*Divide[Exp[-(\[Nu]+Divide[1,2])*\[Xi]],Gamma[\[Nu]+Divide[3,2]]]	Error	Failure	-	Failed [40 / 80] Result: Complex[2.271329177520301, 3.117315294925537] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.110539983099107, -2.8061475441370582] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
14.5.E17	${\displaystyle{\displaystyle\boldsymbol{Q}^{-1/2}_{\nu}\left(\cosh\xi\right)=\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}\frac{\exp\left(-\left(\nu+\frac{1}{2}\right)\xi\right)}{\Gamma\left(\nu+\frac{3}{2}\right)}}}$ \assLegendreOlverQ[- 1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{\pi}{2\sinh@@{\xi}}\right)^{1/2}\frac{\exp@{-\left(\nu+\frac{1}{2}\right)\xi}}{\EulerGamma@{\nu+\frac{3}{2}}}	${\displaystyle{\displaystyle\Re(\nu+\frac{3}{2})>0}}$	exp(-(- 1/2)*Pi*I)*LegendreQ(nu,- 1/2,cosh(xi))/GAMMA(nu+- 1/2+1) = ((Pi)/(2*sinh(xi)))^(1/2)*(exp(-(nu +(1)/(2))*xi))/(GAMMA(nu +(3)/(2)))	Exp[-(- 1/2) Pi I] LegendreQ[\[Nu], - 1/2, 3, Cosh[\[Xi]]]/Gamma[\[Nu] + - 1/2 + 1] == (Divide[Pi,2*Sinh[\[Xi]]])^(1/2)*Divide[Exp[-(\[Nu]+Divide[1,2])*\[Xi]],Gamma[\[Nu]+Divide[3,2]]]	Error	Failure	-	Failed [45 / 80] Result: Complex[2.271329177520301, 3.1173152949255365] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.1105399830991072, -2.806147544137058] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
14.5.E18	${\displaystyle{\displaystyle\mathsf{P}^{-\nu}_{\nu}\left(\cos\theta\right)=\frac{(\sin\theta)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}}}$ \FerrersP[-\nu]{\nu}@{\cos@@{\theta}} = \frac{(\sin@@{\theta})^{\nu}}{2^{\nu}\EulerGamma@{\nu+1}}	${\displaystyle{\displaystyle\Re(\nu+1)>0}}$	LegendreP(nu, - nu, cos(theta)) = ((sin(theta))^(nu))/((2)^(nu)* GAMMA(nu + 1))	LegendreP[\[Nu], - \[Nu], Cos[\[Theta]]] == Divide[(Sin[\[Theta]])^\[Nu],(2)^\[Nu]* Gamma[\[Nu]+ 1]]	Failure	Failure	Failed [35 / 80] Result: .2949209281-1.238111915*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)} Result: 2.775912070+.3102767417*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [35 / 80] Result: Complex[0.29492092804949727, -1.2381119148256148] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.772257638440087, 3.7251537153578904] Test Values: {Rule[θ, 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
14.5.E19	${\displaystyle{\displaystyle P^{-\nu}_{\nu}\left(\cosh\xi\right)=\frac{(\sinh\xi)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}}}$ \assLegendreP[-\nu]{\nu}@{\cosh@@{\xi}} = \frac{(\sinh@@{\xi})^{\nu}}{2^{\nu}\EulerGamma@{\nu+1}}	${\displaystyle{\displaystyle\Re(\nu+1)>0}}$	LegendreP(nu, - nu, cosh(xi)) = ((sinh(xi))^(nu))/((2)^(nu)* GAMMA(nu + 1))	LegendreP[\[Nu], - \[Nu], 3, Cosh[\[Xi]]] == Divide[(Sinh[\[Xi]])^\[Nu],(2)^\[Nu]* Gamma[\[Nu]+ 1]]	Failure	Failure	Failed [35 / 80] Result: -.1260431913-1.267273114*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)} Result: 2.520491622+1.199838208*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [35 / 80] Result: Complex[-0.12604319089926652, -1.2672731138072273] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[2.5204916224127887, 1.1998382094597244] Test Values: {Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
14.5.E20	${\displaystyle{\displaystyle\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}\left(2E\left(\sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(\sin\left(\tfrac{1}{2}\theta\right)\right)\right)}}$ \FerrersP[]{\frac{1}{2}}@{\cos@@{\theta}} = \frac{2}{\pi}\left(2\compellintEk@{\sin@{\tfrac{1}{2}\theta}}-\compellintKk@{\sin@{\tfrac{1}{2}\theta}}\right)		LegendreP((1)/(2), cos(theta)) = (2)/(Pi)*(2*EllipticE(sin((1)/(2)*theta))- EllipticK(sin((1)/(2)*theta)))	LegendreP[Divide[1,2], Cos[\[Theta]]] == Divide[2,Pi]*(2*EllipticE[(Sin[Divide[1,2]*\[Theta]])^2]- EllipticK[(Sin[Divide[1,2]*\[Theta]])^2])	Failure	Failure	Successful [Tested: 10]	Successful [Tested: 10]
14.5.E21	${\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}K\left(\sin\left(\tfrac{1}{2}\theta\right)\right)}}$ \FerrersP[]{-\frac{1}{2}}@{\cos@@{\theta}} = \frac{2}{\pi}\compellintKk@{\sin@{\tfrac{1}{2}\theta}}		LegendreP(-(1)/(2), cos(theta)) = (2)/(Pi)*EllipticK(sin((1)/(2)*theta))	LegendreP[-Divide[1,2], Cos[\[Theta]]] == Divide[2,Pi]*EllipticK[(Sin[Divide[1,2]*\[Theta]])^2]	Failure	Successful	Successful [Tested: 10]	Successful [Tested: 10]
14.5.E22	${\displaystyle{\displaystyle\mathsf{Q}_{\frac{1}{2}}\left(\cos\theta\right)=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right)-2E\left(\cos\left(\tfrac{1}{2}\theta\right)\right)}}$ \FerrersQ[]{\frac{1}{2}}@{\cos@@{\theta}} = \compellintKk@{\cos@{\tfrac{1}{2}\theta}}-2\compellintEk@{\cos@{\tfrac{1}{2}\theta}}		LegendreQ((1)/(2), cos(theta)) = EllipticK(cos((1)/(2)*theta))- 2*EllipticE(cos((1)/(2)*theta))	LegendreQ[Divide[1,2], Cos[\[Theta]]] == EllipticK[(Cos[Divide[1,2]*\[Theta]])^2]- 2*EllipticE[(Cos[Divide[1,2]*\[Theta]])^2]	Failure	Failure	Successful [Tested: 10]	Successful [Tested: 10]
14.5.E23	${\displaystyle{\displaystyle\mathsf{Q}_{-\frac{1}{2}}\left(\cos\theta\right)=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right)}}$ \FerrersQ[]{-\frac{1}{2}}@{\cos@@{\theta}} = \compellintKk@{\cos@{\tfrac{1}{2}\theta}}		LegendreQ(-(1)/(2), cos(theta)) = EllipticK(cos((1)/(2)*theta))	LegendreQ[-Divide[1,2], Cos[\[Theta]]] == EllipticK[(Cos[Divide[1,2]*\[Theta]])^2]	Failure	Failure	Successful [Tested: 10]	Successful [Tested: 10]
14.5.E24	${\displaystyle{\displaystyle P_{\frac{1}{2}}\left(\cosh\xi\right)=\frac{2}{\pi}e^{\xi/2}E\left(\left(1-e^{-2\xi}\right)^{1/2}\right)}}$ \assLegendreP[]{\frac{1}{2}}@{\cosh@@{\xi}} = \frac{2}{\pi}e^{\xi/2}\compellintEk@{\left(1-e^{-2\xi}\right)^{1/2}}		LegendreP((1)/(2), cosh(xi)) = (2)/(Pi)*exp(xi/2)*EllipticE((1 - exp(- 2*xi))^(1/2))	LegendreP[Divide[1,2], 0, 3, Cosh[\[Xi]]] == Divide[2,Pi]*Exp[\[Xi]/2]*EllipticE[((1 - Exp[- 2*\[Xi]])^(1/2))^2]	Failure	Failure	Successful [Tested: 10]	Successful [Tested: 10]
14.5.E25	${\displaystyle{\displaystyle P_{-\frac{1}{2}}\left(\cosh\xi\right)=\frac{2}{\pi\cosh\left(\frac{1}{2}\xi\right)}K\left(\tanh\left(\tfrac{1}{2}\xi\right)\right)}}$ \assLegendreP[]{-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{2}{\pi\cosh@{\frac{1}{2}\xi}}\compellintKk@{\tanh@{\tfrac{1}{2}\xi}}		LegendreP(-(1)/(2), cosh(xi)) = (2)/(Pi*cosh((1)/(2)*xi))*EllipticK(tanh((1)/(2)*xi))	LegendreP[-Divide[1,2], 0, 3, Cosh[\[Xi]]] == Divide[2,Pi*Cosh[Divide[1,2]*\[Xi]]]*EllipticK[(Tanh[Divide[1,2]*\[Xi]])^2]	Failure	Failure	Successful [Tested: 10]	Successful [Tested: 10]
14.5.E26	${\displaystyle{\displaystyle\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}\cosh\xi\operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(\tfrac{1}{2}\xi\right)E\left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right)}}$ \assLegendreOlverQ[]{\frac{1}{2}}@{\cosh@@{\xi}} = 2\pi^{-1/2}\cosh@@{\xi}\sech@{\tfrac{1}{2}\xi}\compellintKk@{\sech@{\tfrac{1}{2}\xi}}-4\pi^{-1/2}\cosh@{\tfrac{1}{2}\xi}\compellintEk@{\sech@{\tfrac{1}{2}\xi}}		LegendreQ((1)/(2),cosh(xi))/GAMMA((1)/(2)+1) = 2*(Pi)^(- 1/2)* cosh(xi)*sech((1)/(2)*xi)*EllipticK(sech((1)/(2)*xi))- 4*(Pi)^(- 1/2)* cosh((1)/(2)*xi)*EllipticE(sech((1)/(2)*xi))	Exp[-(Divide[1,2]) Pi I] LegendreQ[Divide[1,2], 2, 3, Cosh[\[Xi]]]/Gamma[Divide[1,2] + 3] == 2*(Pi)^(- 1/2)* Cosh[\[Xi]]*Sech[Divide[1,2]*\[Xi]]*EllipticK[(Sech[Divide[1,2]*\[Xi]])^2]- 4*(Pi)^(- 1/2)* Cosh[Divide[1,2]*\[Xi]]*EllipticE[(Sech[Divide[1,2]*\[Xi]])^2]	Failure	Failure	Successful [Tested: 10]	Failed [10 / 10] Result: Complex[-0.8843996963296057, 0.10723567454157107] Test Values: {Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.4538488510851968, -0.4630204881028235] Test Values: {Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.5.E27	${\displaystyle{\displaystyle\boldsymbol{Q}_{-\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}e^{-\xi/2}K\left(e^{-\xi}\right)}}$ \assLegendreOlverQ[]{-\frac{1}{2}}@{\cosh@@{\xi}} = 2\pi^{-1/2}e^{-\xi/2}\compellintKk@{e^{-\xi}}		LegendreQ(-(1)/(2),cosh(xi))/GAMMA(-(1)/(2)+1) = 2*(Pi)^(- 1/2)* exp(- xi/2)*EllipticK(exp(- xi))	Exp[-(-Divide[1,2]) Pi I] LegendreQ[-Divide[1,2], 2, 3, Cosh[\[Xi]]]/Gamma[-Divide[1,2] + 3] == 2*(Pi)^(- 1/2)* Exp[- \[Xi]/2]*EllipticK[(Exp[- \[Xi]])^2]	Failure	Failure	Failed [5 / 10] Result: -.101404509+1.824239856*I Test Values: {xi = -1/2+1/2*I*3^(1/2)} Result: -.90465021e-1-1.714290815*I Test Values: {xi = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [10 / 10] Result: Complex[0.16749403535362406, 1.47562407248214] Test Values: {Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.5106529782887232, 0.796583020821415] Test Values: {Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.5.E28	${\displaystyle{\displaystyle\mathsf{P}_{2}\left(x\right)=P_{2}\left(x\right)}}$ \FerrersP[]{2}@{x} = \assLegendreP[]{2}@{x}		LegendreP(2, x) = LegendreP(2, x)	LegendreP[2, x] == LegendreP[2, 0, 3, x]	Successful	Successful	-	Successful [Tested: 3]
14.5.E28	${\displaystyle{\displaystyle P_{2}\left(x\right)=\frac{3x^{2}-1}{2}}}$ \assLegendreP[]{2}@{x} = \frac{3x^{2}-1}{2}		LegendreP(2, x) = (3*(x)^(2)- 1)/(2)	LegendreP[2, 0, 3, x] == Divide[3*(x)^(2)- 1,2]	Successful	Successful	-	Successful [Tested: 3]
14.5.E29	${\displaystyle{\displaystyle\mathsf{Q}_{2}\left(x\right)=\frac{3x^{2}-1}{4}\ln\left(\frac{1+x}{1-x}\right)-\frac{3}{2}x}}$ \FerrersQ[]{2}@{x} = \frac{3x^{2}-1}{4}\ln@{\frac{1+x}{1-x}}-\frac{3}{2}x		LegendreQ(2, x) = (3*(x)^(2)- 1)/(4)*ln((1 + x)/(1 - x))-(3)/(2)*x	LegendreQ[2, x] == Divide[3*(x)^(2)- 1,4]*Log[Divide[1 + x,1 - x]]-Divide[3,2]*x	Failure	Failure	Failed [2 / 3] Result: .1e-8-9.032078880*I Test Values: {x = 3/2} Result: -.1e-8-17.27875960*I Test Values: {x = 2}	Failed [2 / 3] Result: Complex[0.0, -9.032078879070655] Test Values: {Rule[x, 1.5]} Result: Complex[0.0, -17.27875959474386] Test Values: {Rule[x, 2]}
14.5.E30	${\displaystyle{\displaystyle\boldsymbol{Q}_{2}\left(x\right)=\frac{3x^{2}-1}{8}\ln\left(\frac{x+1}{x-1}\right)-\frac{3}{4}x}}$ \assLegendreOlverQ[]{2}@{x} = \frac{3x^{2}-1}{8}\ln@{\frac{x+1}{x-1}}-\frac{3}{4}x		LegendreQ(2,x)/GAMMA(2+1) = (3*(x)^(2)- 1)/(8)*ln((x + 1)/(x - 1))-(3)/(4)*x	Exp[-(2) Pi I] LegendreQ[2, 2, 3, x]/Gamma[2 + 3] == Divide[3*(x)^(2)- 1,8]*Log[Divide[x + 1,x - 1]]-Divide[3,4]*x	Failure	Failure	Failed [1 / 3] Result: 0.+.1963495409*I Test Values: {x = 1/2}	Failed [2 / 3] Result: Complex[0.006453837346904523, -9.365446450684121*^-18] Test Values: {Rule[x, 1.5]} Result: Complex[0.23977862743400533, 0.2454369260617026] Test Values: {Rule[x, 0.5]}
14.6.E1	${\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{P}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$ \FerrersP[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersP[]{\nu}@{x}}{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreP(nu, x), [x$(m)])	LegendreP[\[Nu], m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreP[\[Nu], x], {x, m}]	Failure	Failure	Failed [3 / 90] Result: Float(undefined)+Float(undefined)*I Test Values: {nu = -2, x = 3/2, m = 1} Result: Float(undefined)+Float(undefined)*I Test Values: {nu = -2, x = 1/2, m = 1} ... skip entries to safe data	Successful [Tested: 90]
14.6.E2	${\displaystyle{\displaystyle\mathsf{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$ \FerrersQ[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersQ[]{\nu}@{x}}{x}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu+m+1)>0,\Re(\nu-\mu+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(nu, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreQ(nu, x), [x$(m)])	LegendreQ[\[Nu], m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreQ[\[Nu], x], {x, m}]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
14.6.E3	${\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}P_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$ \assLegendreP[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreP[]{\nu}@{x}}{x}		LegendreP(nu, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreP(nu, x), [x$(m)])	LegendreP[\[Nu], m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreP[\[Nu], 0, 3, x], {x, m}]	Failure	Failure	Failed [3 / 90] Result: Float(undefined)+Float(undefined)*I Test Values: {nu = -2, x = 3/2, m = 1} Result: Float(undefined)+Float(undefined)*I Test Values: {nu = -2, x = 1/2, m = 1} ... skip entries to safe data	Successful [Tested: 90]
14.6.E4	${\displaystyle{\displaystyle Q^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}Q_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$ \assLegendreQ[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreQ[]{\nu}@{x}}{x}		LegendreQ(nu, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreQ(nu, x), [x$(m)])	LegendreQ[\[Nu], m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreQ[\[Nu], 0, 3, x], {x, m}]	Failure	Failure	Error	Failed [75 / 90] Result: Plus[Complex[-0.4598393885300628, 0.18181080125096066], Times[-1.118033988749895, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Equal[[1], Times[-1, Power[Plus[-1, Power[1.5, 2]], -1], Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[1.5, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[1.6909557968522604, -0.413901027514361], Times[-2.5, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Equal[[1], Times[-1, Power[Plus[-1, Power[1.5, 2]], -1], Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[1.5, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 0, 3, 1.5]]]]]}]][2.0]]], {Rule[m, 2], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.6.E5	${\displaystyle{\displaystyle{\left(\nu+1\right)_{m}}\boldsymbol{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\boldsymbol{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$ \Pochhammersym{\nu+1}{m}\assLegendreOlverQ[m]{\nu}@{x} = (-1)^{m}\left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreOlverQ[]{\nu}@{x}}{x}		pochhammer(nu + 1, m)*exp(-(m)*Pi*I)*LegendreQ(nu,m,x)/GAMMA(nu+m+1) = (- 1)^(m)*((x)^(2)- 1)^(m/2)* diff(LegendreQ(nu,x)/GAMMA(nu+1), [x$(m)])	Pochhammer[\[Nu]+ 1, m]*Exp[-(m) Pi I] LegendreQ[\[Nu], m, 3, x]/Gamma[\[Nu] + m + 1] == (- 1)^(m)*((x)^(2)- 1)^(m/2)* D[Exp[-(\[Nu]) Pi I] LegendreQ[\[Nu], 2, 3, x]/Gamma[\[Nu] + 3], {x, m}]	Failure	Failure	Error	Failed [90 / 90] Result: Plus[Complex[0.482758812955306, -0.29762130115013324], Times[Complex[-1.0778621920495528, 0.20681719187113978], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, 1.5, Plus[3, Times[5, ], Times[2, Power[, 2]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[Plus[-12, Times[-8, ], Times[-2, Power[, 2]], Times[24, Power[1.5, 2]], Times[24, , Power[1.5, 2]], Times[6, Power[, 2], Power[1.5, 2]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[-1, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Times[-1, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[2, ]]], Times[2, P<syntaxhighlight lang=mathematica>Result: Plus[Complex[1.8263637314445087, -0.806860371328253], Times[Complex[2.4101731317997332, -0.4624572999394857], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, 1.5, Plus[3, Times[5, ], Times[2, Power[, 2]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[Plus[-12, Times[-8, ], Times[-2, Power[, 2]], Times[24, Power[1.5, 2]], Times[24, , Power[1.5, 2]], Times[6, Power[, 2], Power[1.5, 2]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[-1, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Times[-1, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[2, ]]], Times[2, Plus[3, ], Plus[5, Times[2, ]], Plus[-1, 1.5], 1.5, Plus[1, 1.5], [Plus[3, ]]], Times[Plus[3, ], Plus[4, ], Power[Plus[-1, 1.5], 2], Power[Plus[1, 1.5], 2], [Plus[4, ]]]], 0], Equal[[0], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Equal[[1], Times[Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[-1, 1.5, Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[Plus[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]]]]], Equal[[2], Times[Rational[1, 2], Power[Plus[-1, Power[1.5, 2]], -2], Plus[Times[4, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[2, Power[1.5, 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[3, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[2, 1.5, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-2, 1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]]]]], Equal[[3], Times[Rational[-1, 6], Power[Plus[-1, Power[1.5, 2]], -3], Plus[Times[30, 1.5, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[6, Power[1.5, 3], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[11, Power[1.5, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-6, 1.5, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[6, Power[1.5, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-1, 1.5, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[Power[1.5, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[6, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[6, Power[1.5, 2], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-7, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-5, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-1, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]]]]]}]][2.0]]], {Rule[m, 2], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.7.E1	${\displaystyle{\displaystyle\mathsf{P}^{0}_{n}\left(x\right)=\mathsf{P}_{n}\left(x\right)}}$ \FerrersP[0]{n}@{x} = \FerrersP[]{n}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, 0, x) = LegendreP(n, x)	LegendreP[n, 0, x] == LegendreP[n, x]	Successful	Successful	-	Successful [Tested: 3]
14.7.E1	${\displaystyle{\displaystyle\mathsf{P}_{n}\left(x\right)=P^{0}_{n}\left(x\right)}}$ \FerrersP[]{n}@{x} = \assLegendreP[0]{n}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, x) = LegendreP(n, 0, x)	LegendreP[n, x] == LegendreP[n, 0, 3, x]	Successful	Successful	-	Successful [Tested: 3]
14.7.E1	${\displaystyle{\displaystyle P^{0}_{n}\left(x\right)=P_{n}\left(x\right)}}$ \assLegendreP[0]{n}@{x} = \LegendrepolyP{n}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, 0, x) = LegendreP(n, x)	LegendreP[n, 0, 3, x] == LegendreP[n, x]	Successful	Successful	-	Successful [Tested: 3]
14.7.E2	${\displaystyle{\displaystyle\mathsf{Q}^{0}_{n}\left(x\right)=\mathsf{Q}_{n}\left(x\right)}}$ \FerrersQ[0]{n}@{x} = \FerrersQ[]{n}@{x}	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+0+1)>0,\Re(n-\mu+1)>0,\Re(\nu-0+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(n, 0, x) = LegendreQ(n, x)	LegendreQ[n, 0, x] == LegendreQ[n, x]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 9]
14.7.E2	${\displaystyle{\displaystyle\mathsf{Q}_{n}\left(x\right)=\frac{1}{2}P_{n}\left(x\right)\ln\left(\frac{1+x}{1-x}\right)-W_{n-1}(x)}}$ \FerrersQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{1+x}{1-x}}-W_{n-1}(x)	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+0+1)>0,\Re(n-\mu+1)>0,\Re(\nu-0+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(n, x) = (1)/(2)*LegendreP(n, x)*ln((1 + x)/(1 - x))- W[n - 1](x)	LegendreQ[n, x] == Divide[1,2]*LegendreP[n, x]*Log[Divide[1 + x,1 - x]]- Subscript[W, n - 1][x]	Failure	Failure	Failed [88 / 90] Result: .2990381063-3.962388980*I Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1} Result: -.950961893-8.282078880*I Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [88 / 90] Result: Complex[0.299038105676658, -3.9623889803846897] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.9509618943233424, -8.282078879070655] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.7.E3	${\displaystyle{\displaystyle W_{n-1}(x)=\sum_{s=0}^{n-1}\frac{(n+s)!(\psi\left(n+1\right)-\psi\left(s+1\right))}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}}}$ W_{n-1}(x) = \sum_{s=0}^{n-1}\frac{(n+s)!(\digamma@{n+1}-\digamma@{s+1})}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}		W[n - 1](x) = sum((factorial(n + s)*(Psi(n + 1)- Psi(s + 1)))/((2)^(s)*factorial(n - s)*(factorial(s))^(2))*(x - 1)^(s), s = 0..n - 1)	Subscript[W, n - 1][x] == Sum[Divide[(n + s)!*(PolyGamma[n + 1]- PolyGamma[s + 1]),(2)^(s)*(n - s)!*((s)!)^(2)]*(x - 1)^(s), {s, 0, n - 1}, GenerateConditions->None]	Failure	Failure	Failed [85 / 90] Result: .2990381061+.7500000000*I Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1} Result: -.950961893+.7500000000*I Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [88 / 90] Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[0.5, Plus[-0.845568670196934, Times[2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Plus[1, , 1], Plus[2, , 1], Power[Plus[-1, 1.5], 2], []], Times[Plus[-1, Times[-1, ], 1], Plus[2, , 1], Plus[-1, 1.5], Plus[6, Times[11, ], Times[5, Power[, 2]], Times[-1, 1], Times[-1, Power[1, 2]], Times[-1, , 1.5], Times[-1, Power[, 2], 1.5], Times[1, 1.5], Times[Power[1, 2], 1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-22, Times[-37, ], Times[-21, Power[, 2]], Times[-4, Power[, 3]], Times[3, 1], Times[2, , 1], Times[3, Power[1, 2]], Times[2, , Power[1, 2]], Times[6, 1.5], Times[13, , 1.5], Times[9, Power[, 2], 1.5], Times[2, Power[, 3], 1.5], Times[-3, 1, 1.5], Times[-2, , 1, 1.5], Times[-3, Power[1, 2], 1.5], Times[-2, , Power[1, 2], 1.5]], [Plus[2, ]]], Times[4, Plus[1, ], Power[Plus[2, ], 3], [Pl<syntaxhighlight lang=mathematica>Result: Plus[Complex[1.299038105676658, 0.7499999999999999], Times[0.0625, Plus[-36.91137340393869, Times[16.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Plus[1, , 2], Plus[2, , 2], Power[Plus[-1, 1.5], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[2, , 2], Plus[-1, 1.5], Plus[6, Times[11, ], Times[5, Power[, 2]], Times[-1, 2], Times[-1, Power[2, 2]], Times[-1, , 1.5], Times[-1, Power[, 2], 1.5], Times[2, 1.5], Times[Power[2, 2], 1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-22, Times[-37, ], Times[-21, Power[, 2]], Times[-4, Power[, 3]], Times[3, 2], Times[2, , 2], Times[3, Power[2, 2]], Times[2, , Power[2, 2]], Times[6, 1.5], Times[13, , 1.5], Times[9, Power[, 2], 1.5], Times[2, Power[, 3], 1.5], Times[-3, 2, 1.5], Times[-2, , 2, 1.5], Times[-3, Power[2, 2], 1.5], Times[-2, , Power[2, 2], 1.5]], [Plus[2, ]]], Times[4, Plus[1, ], Power[Plus[2, ], 3], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[-1, EulerGamma]], Equal[[2], Plus[Times[-1, EulerGamma], Times[Rational[1, 2], Plus[1, Times[-1, EulerGamma]], 2, Plus[1, 2], Plus[-1, 1.5]]]]}]][2.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.7.E4	${\displaystyle{\displaystyle W_{n-1}(x)=\sum_{k=1}^{n}\frac{1}{k}P_{k-1}\left(x\right)P_{n-k}\left(x\right)}}$ W_{n-1}(x) = \sum_{k=1}^{n}\frac{1}{k}\LegendrepolyP{k-1}@{x}\LegendrepolyP{n-k}@{x}		W[n - 1](x) = sum((1)/(k)*LegendreP(k - 1, x)*LegendreP(n - k, x), k = 1..n)	Subscript[W, n - 1][x] == Sum[Divide[1,k]*LegendreP[k - 1, x]*LegendreP[n - k, x], {k, 1, n}, GenerateConditions->None]	Failure	Failure	Failed [85 / 90] Result: .299038106+.7500000000*I Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 1} Result: -.950961894+.7500000000*I Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Skipped - Because timed out
14.7#Ex1	${\displaystyle{\displaystyle W_{0}(x)=1}}$ W_{0}(x) = 1		W[0](x) = 1	Subscript[W, 0][x] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
14.7#Ex2	${\displaystyle{\displaystyle W_{1}(x)=\tfrac{3}{2}x}}$ W_{1}(x) = \tfrac{3}{2}x		W[1](x) = (3)/(2)*x	Subscript[W, 1][x] == Divide[3,2]*x	Skipped - no semantic math	Skipped - no semantic math	-	-
14.7#Ex3	${\displaystyle{\displaystyle W_{2}(x)=\tfrac{5}{2}x^{2}-\tfrac{2}{3}}}$ W_{2}(x) = \tfrac{5}{2}x^{2}-\tfrac{2}{3}		W[2](x) = (5)/(2)*(x)^(2)-(2)/(3)	Subscript[W, 2][x] == Divide[5,2]*(x)^(2)-Divide[2,3]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.7.E6	${\displaystyle{\displaystyle Q^{0}_{n}\left(x\right)=Q_{n}\left(x\right)}}$ \assLegendreQ[0]{n}@{x} = \assLegendreQ[]{n}@{x}		LegendreQ(n, 0, x) = LegendreQ(n, x)	LegendreQ[n, 0, 3, x] == LegendreQ[n, 0, 3, x]	Successful	Successful	-	Successful [Tested: 9]
14.7.E6	${\displaystyle{\displaystyle Q_{n}\left(x\right)=n!\boldsymbol{Q}^{0}_{n}\left(x\right)}}$ \assLegendreQ[]{n}@{x} = n!\assLegendreOlverQ[0]{n}@{x}		LegendreQ(n, x) = factorial(n)*exp(-(0)*Pi*I)*LegendreQ(n,0,x)/GAMMA(n+0+1)	LegendreQ[n, 0, 3, x] == (n)!*Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1]	Successful	Successful	-	Successful [Tested: 9]
14.7.E6	${\displaystyle{\displaystyle n!\boldsymbol{Q}^{0}_{n}\left(x\right)=n!\boldsymbol{Q}_{n}\left(x\right)}}$ n!\assLegendreOlverQ[0]{n}@{x} = n!\assLegendreOlverQ[]{n}@{x}		factorial(n)*exp(-(0)*Pi*I)*LegendreQ(n,0,x)/GAMMA(n+0+1) = factorial(n)*LegendreQ(n,x)/GAMMA(n+1)	(n)!*Exp[-(0) Pi I] LegendreQ[n, 0, 3, x]/Gamma[n + 0 + 1] == (n)!*Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3]	Successful	Failure	-	Failed [9 / 9] Result: Complex[0.47374510099224165, -6.531449595452549*^-17] Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: Complex[-0.012907674693808963, 1.8730892901368242*^-17] Test Values: {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E7	${\displaystyle{\displaystyle Q_{n}\left(x\right)=\frac{1}{2}P_{n}\left(x\right)\ln\left(\frac{x+1}{x-1}\right)-W_{n-1}(x)}}$ \assLegendreQ[]{n}@{x} = \frac{1}{2}\LegendrepolyP{n}@{x}\ln@{\frac{x+1}{x-1}}-W_{n-1}(x)		LegendreQ(n, x) = (1)/(2)*LegendreP(n, x)*ln((x + 1)/(x - 1))- W[n - 1](x)	LegendreQ[n, 0, 3, x] == Divide[1,2]*LegendreP[n, x]*Log[Divide[x + 1,x - 1]]- Subscript[W, n - 1][x]	Failure	Failure	Failed [30 / 30] Result: -3.659295226+.7500000000*I Test Values: {x = 3/2, W[n-1] = 1/2*3^(1/2)+1/2*I, n = 3} Result: -5.708333332+1.299038106*I Test Values: {x = 3/2, W[n-1] = -1/2+1/2*I*3^(1/2), n = 3} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-3.659295227656675, 0.7499999999999999] Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-5.708333333333333, 1.299038105676658] Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[Subscript[W, Plus[-1, n]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.7.E8	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{P}_{n}\left(x\right)}}$ \FerrersP[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersP[]{n}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreP(n, x), [x$(m)])	LegendreP[n, m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreP[n, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
14.7.E9	${\displaystyle{\displaystyle\mathsf{Q}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{Q}_{n}\left(x\right)}}$ \FerrersQ[m]{n}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{}{x}\FerrersQ[]{n}@{x}	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+m+1)>0,\Re(n-\mu+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(n, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreQ(n, x), [x$(m)])	LegendreQ[n, m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreQ[n, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
14.7.E10	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(1-x^{2}\right)^{n}}}$ \FerrersP[m]{n}@{x} = (-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(1-x^{2}\right)^{n}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, m, x) = (- 1)^(m + n)*((1 - (x)^(2))^(m/2))/((2)^(n)* factorial(n))*diff((1 - (x)^(2))^(n), [x$(m + n)])	LegendreP[n, m, x] == (- 1)^(m + n)*Divide[(1 - (x)^(2))^(m/2),(2)^(n)* (n)!]*D[(1 - (x)^(2))^(n), {x, m + n}]	Failure	Failure	Failed [18 / 27] Result: Float(undefined)+Float(undefined)*I Test Values: {x = 3/2, m = 1, n = 1} Result: Float(undefined)+Float(undefined)*I Test Values: {x = 3/2, m = 1, n = 2} ... skip entries to safe data	Failed [27 / 27] Result: Plus[Complex[0.0, -1.118033988749895], Times[Complex[0.0, -0.5590169943749475], D[-1.25 Test Values: {1.5, 2.0}]]], {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]} Result: Plus[Complex[0.0, -5.031152949374526], Times[Complex[0.0, 0.13975424859373686], D[1.5625 Test Values: {1.5, 3.0}]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E11	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}P_{n}\left(x\right)}}$ \assLegendreP[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\LegendrepolyP{n}@{x}		LegendreP(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreP(n, x), [x$(m)])	LegendreP[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreP[n, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
14.7.E12	${\displaystyle{\displaystyle Q^{m}_{n}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}Q_{n}\left(x\right)}}$ \assLegendreQ[m]{n}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{}{x}\assLegendreQ[]{n}@{x}		LegendreQ(n, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreQ(n, x), [x$(m)])	LegendreQ[n, m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreQ[n, 0, 3, x], {x, m}]	Failure	Failure	Successful [Tested: 27]	Failed [18 / 27] Result: Plus[Complex[-0.4419376420578732, 5.412175187689032*^-17], Times[-1.118033988749895, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[Times[-1, ], 1], Plus[1, , 1], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[1, 0, 3, 1.5]], Equal[[1], Times[-1, Plus[1, 1], Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[1.5, LegendreQ[1, 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, 1], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]} Result: Plus[Complex[-0.1998650072605977, 2.447640414032535*^-17], Times[-1.118033988749895, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[Times[-1, ], 2], Plus[1, , 2], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[2, 0, 3, 1.5]], Equal[[1], Times[-1, Plus[1, 2], Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[1.5, LegendreQ[2, 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, 2], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E13	${\displaystyle{\displaystyle P_{n}\left(x\right)=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{2}-1\right)^{n}}}$ \LegendrepolyP{n}@{x} = \frac{1}{2^{n}n!}\deriv[n]{}{x}\left(x^{2}-1\right)^{n}		LegendreP(n, x) = (1)/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(n)])	LegendreP[n, x] == Divide[1,(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, n}]	Failure	Failure	Error	Failed [6 / 9] Result: Plus[1.5, Times[-0.5, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-2, 1]], []], Times[-2, Plus[-1, Times[-1, ], 1], 1.5, [Plus[1, ]]], Times[Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], Power[Plus[-1, Power[1.5, 2]], 1]], Equal[[1], Times[2, 1, 1.5, Power[Plus[-1, Power[1.5, 2]], Plus[-1, 1]]]]}]][1.0]]], {Rule[n, 1], Rule[x, 1.5]} Result: Plus[2.875, Times[-0.25, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[, Times[-2, 2]], []], Times[-2, Plus[-1, Times[-1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], Power[Plus[-1, Power[1.5, 2]], 2]], Equal[[1], Times[2, 2, 1.5, Power[Plus[-1, Power[1.5, 2]], Plus[-1, 2]]]]}]][2.0]]], {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E14	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(x^{2}-1\right)^{n}}}$ \assLegendreP[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\deriv[m+n]{}{x}\left(x^{2}-1\right)^{n}		LegendreP(n, m, x) = (((x)^(2)- 1)^(m/2))/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(m + n)])	LegendreP[n, m, 3, x] == Divide[((x)^(2)- 1)^(m/2),(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, m + n}]	Failure	Failure	Failed [18 / 27] Result: Float(undefined)+Float(undefined)*I Test Values: {x = 3/2, m = 1, n = 1} Result: Float(undefined)+Float(undefined)*I Test Values: {x = 3/2, m = 1, n = 2} ... skip entries to safe data	Failed [27 / 27] Result: Plus[1.118033988749895, Times[-0.5590169943749475, D[1.25 Test Values: {1.5, 2.0}]]], {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]} Result: Plus[5.031152949374526, Times[-0.13975424859373686, D[1.5625 Test Values: {1.5, 3.0}]]], {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
14.7.E15	${\displaystyle{\displaystyle P^{m}_{m}\left(x\right)=\frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}}}$ \assLegendreP[m]{m}@{x} = \frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}		LegendreP(m, m, x) = (factorial(2*m))/((2)^(m)* factorial(m))*((x)^(2)- 1)^(m/2)	LegendreP[m, m, 3, x] == Divide[(2*m)!,(2)^(m)* (m)!]*((x)^(2)- 1)^(m/2)	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
14.7.E16	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=P^{m}_{n}\left(x\right)}}$ \FerrersP[m]{n}@{x} = \assLegendreP[m]{n}@{x}	${\displaystyle{\displaystyle m>n,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, m, x) = LegendreP(n, m, x)	LegendreP[n, m, x] == LegendreP[n, m, 3, x]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 9]
14.7.E16	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=0}}$ \assLegendreP[m]{n}@{x} = 0	${\displaystyle{\displaystyle m>n,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, m, x) = 0	LegendreP[n, m, 3, x] == 0	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
14.7.E17	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(-x\right)=(-1)^{n-m}\mathsf{P}^{m}_{n}\left(x\right)}}$ \FerrersP[m]{n}@{-x} = (-1)^{n-m}\FerrersP[m]{n}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, m, - x) = (- 1)^(n - m)* LegendreP(n, m, x)	LegendreP[n, m, - x] == (- 1)^(n - m)* LegendreP[n, m, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
14.7.E18	${\displaystyle{\displaystyle\mathsf{Q}^{+m}_{n}\left(-x\right)=(-1)^{n-m-1}\mathsf{Q}^{+m}_{n}\left(x\right)}}$ \FerrersQ[+ m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[+ m]{n}@{x}		LegendreQ(n, + m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, + m, x)	LegendreQ[n, + m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, + m, x]	Failure	Failure	Error	Successful [Tested: 9]
14.7.E18	${\displaystyle{\displaystyle\mathsf{Q}^{-m}_{n}\left(-x\right)=(-1)^{n-m-1}\mathsf{Q}^{-m}_{n}\left(x\right)}}$ \FerrersQ[- m]{n}@{-x} = (-1)^{n-m-1}\FerrersQ[- m]{n}@{x}	${\displaystyle{\displaystyle\Re(n+\mu+1)>0,\Re(\nu+(-m)+1)>0,\Re(n-\mu+1)>0,\Re(\nu-(-m)+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(n, - m, - x) = (- 1)^(n - m - 1)* LegendreQ(n, - m, x)	LegendreQ[n, - m, - x] == (- 1)^(n - m - 1)* LegendreQ[n, - m, x]	Failure	Failure	Error	Failed [3 / 9] Result: Indeterminate Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 0.5]} Result: Indeterminate Test Values: {Rule[m, 3], Rule[n, 1], Rule[x, 0.5]} ... skip entries to safe data
14.7.E19	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{n}=\left(1-2xh+h^{2}\right)^{-1/2}}}$ \sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{n} = \left(1-2xh+h^{2}\right)^{-1/2}		sum(LegendreP(n, x)*(h)^(n), n = 0..infinity) = (1 - 2*x*h + (h)^(2))^(- 1/2)	Sum[LegendreP[n, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*h + (h)^(2))^(- 1/2)	Failure	Successful	Error	Successful [Tested: 30]
14.7.E20	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{Q}_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}\right)}}$ \sum_{n=0}^{\infty}\FerrersQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}}		sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity) = (1)/((1 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/2))/((1 - (x)^(2))^(1/2)))	Sum[LegendreQ[n, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2*x*h + (h)^(2))^(1/2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/2),(1 - (x)^(2))^(1/2)]]	Failure	Failure	Manual Skip!	Skipped - Because timed out
14.7.E21	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{-n-1}=\left(1-2xh+h^{2}\right)^{-1/2}}}$ \sum_{n=0}^{\infty}\FerrersP[]{n}@{x}h^{-n-1} = \left(1-2xh+h^{2}\right)^{-1/2}		sum(LegendreP(n, x)*(h)^(- n - 1), n = 0..infinity) = (1 - 2*x*h + (h)^(2))^(- 1/2)	Sum[LegendreP[n, x]*(h)^(- n - 1), {n, 0, Infinity}, GenerateConditions->None] == (1 - 2*x*h + (h)^(2))^(- 1/2)	Failure	Failure	Error	Failed [20 / 30] Result: Complex[-0.45970084338098294, -1.7156269037800917] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]} Result: Complex[-0.3437237693334403, -1.2827945709214845] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 2]} ... skip entries to safe data
14.7.E22	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}Q_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}\right)}}$ \sum_{n=0}^{\infty}\assLegendreQ[]{n}@{x}h^{n} = \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln@{\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}}		sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity) = (1)/((1 - 2*x*h + (h)^(2))^(1/2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/2))/(((x)^(2)- 1)^(1/2)))	Sum[LegendreQ[n, 0, 3, x]*(h)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - 2*x*h + (h)^(2))^(1/2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/2),((x)^(2)- 1)^(1/2)]]	Failure	Failure	Successful [Tested: 30]	Skipped - Because timed out
14.9.E1	${\displaystyle{\displaystyle\frac{\pi\sin\left(\mu\pi\right)}{2\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)+\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right)}}$ \frac{\pi\sin@{\mu\pi}}{2\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{x} = -\frac{1}{\EulerGamma@{\nu+\mu+1}}\FerrersQ[\mu]{\nu}@{x}+\frac{\cos@{\mu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersQ[-\mu]{\nu}@{x}	${\displaystyle{\displaystyle\Re(\nu-\mu+1)>0,\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+(-\mu)+1)>0,\Re(\nu-(-\mu)+1)>0}}$	(Pi*sin(mu*Pi))/(2*GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x) = -(1)/(GAMMA(nu + mu + 1))*LegendreQ(nu, mu, x)+(cos(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreQ(nu, - mu, x)	Divide[Pi*Sin[\[Mu]*Pi],2*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x] == -Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreQ[\[Nu], \[Mu], x]+Divide[Cos[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreQ[\[Nu], - \[Mu], x]	Successful	Successful	-	Successful [Tested: 135]
14.9.E2	${\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)}}$ \frac{2\sin@{\mu\pi}}{\pi\EulerGamma@{\nu-\mu+1}}\FerrersQ[-\mu]{\nu}@{x} = \frac{1}{\EulerGamma@{\nu+\mu+1}}\FerrersP[\mu]{\nu}@{x}-\frac{\cos@{\mu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{x}	${\displaystyle{\displaystyle\Re(\nu-\mu+1)>0,\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+(-\mu)+1)>0,\Re(\nu-(-\mu)+1)>0}}$	(2*sin(mu*Pi))/(Pi*GAMMA(nu - mu + 1))*LegendreQ(nu, - mu, x) = (1)/(GAMMA(nu + mu + 1))*LegendreP(nu, mu, x)-(cos(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x)	Divide[2*Sin[\[Mu]*Pi],Pi*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreQ[\[Nu], - \[Mu], x] == Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreP[\[Nu], \[Mu], x]-Divide[Cos[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x]	Successful	Successful	-	Successful [Tested: 135]
14.9.E3	${\displaystyle{\displaystyle\mathsf{P}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\mathsf{P}^{m}_{\nu}\left(x\right)}}$ \FerrersP[-m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\FerrersP[m]{\nu}@{x}	${\displaystyle{\displaystyle\Re(\nu-m+1)>0,\Re(\nu+m+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, - m, x) = (- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreP(nu, m, x)	LegendreP[\[Nu], - m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreP[\[Nu], m, x]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
14.9.E4	${\displaystyle{\displaystyle\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right)}}$ \FerrersQ[-m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\FerrersQ[m]{\nu}@{x}	${\displaystyle{\displaystyle\nu\neq m-1,\Re(\nu-m+1)>0,\Re(\nu+m+1)>0,\Re(\nu+\mu+1)>0,\Re(\nu+(-m)+1)>0,\Re(\nu-\mu+1)>0,\Re(\nu-(-m)+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(nu, - m, x) = (- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreQ(nu, m, x)	LegendreQ[\[Nu], - m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreQ[\[Nu], m, x]	Failure	Failure	Error	Successful [Tested: 21]
14.9#Ex1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$ \FerrersP[\mu]{-\nu-1}@{x} = \FerrersP[\mu]{\nu}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(- nu - 1, mu, x) = LegendreP(nu, mu, x)	LegendreP[- \[Nu]- 1, \[Mu], x] == LegendreP[\[Nu], \[Mu], x]	Successful	Failure	-	Successful [Tested: 300]
14.9#Ex2	${\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{-\mu}_{\nu}\left(x\right)}}$ \FerrersP[-\mu]{-\nu-1}@{x} = \FerrersP[-\mu]{\nu}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(- nu - 1, - mu, x) = LegendreP(nu, - mu, x)	LegendreP[- \[Nu]- 1, - \[Mu], x] == LegendreP[\[Nu], - \[Mu], x]	Successful	Failure	-	Successful [Tested: 300]
14.9.E6	${\displaystyle{\displaystyle\pi\cos\left(\nu\pi\right)\cos\left(\mu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)=\sin\left((\nu+\mu)\pi\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)}}$ \pi\cos@{\nu\pi}\cos@{\mu\pi}\FerrersP[\mu]{\nu}@{x} = \sin@{(\nu+\mu)\pi}\FerrersQ[\mu]{\nu}@{x}-\sin@{(\nu-\mu)\pi}\FerrersQ[\mu]{-\nu-1}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,\Re((-\nu-1)+\mu+1)>0,\Re(\nu-\mu+1)>0,\Re((-\nu-1)-\mu+1)>0}}$	Pi*cos(nu*Pi)*cos(mu*Pi)*LegendreP(nu, mu, x) = sin((nu + mu)*Pi)*LegendreQ(nu, mu, x)- sin((nu - mu)*Pi)*LegendreQ(- nu - 1, mu, x)	Pi*Cos[\[Nu]*Pi]*Cos[\[Mu]*Pi]*LegendreP[\[Nu], \[Mu], x] == Sin[(\[Nu]+ \[Mu])*Pi]*LegendreQ[\[Nu], \[Mu], x]- Sin[(\[Nu]- \[Mu])*Pi]*LegendreQ[- \[Nu]- 1, \[Mu], x]	Successful	Failure	-	Successful [Tested: 3]
14.9.E7	${\displaystyle{\displaystyle\frac{\sin\left((\nu-\mu)\pi\right)}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\frac{\sin\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(-x\right)}}$ \frac{\sin@{(\nu-\mu)\pi}}{\EulerGamma@{\nu+\mu+1}}\FerrersP[\mu]{\nu}@{x} = \frac{\sin@{\nu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{x}-\frac{\sin@{\mu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{-x}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1}}$	(sin((nu - mu)*Pi))/(GAMMA(nu + mu + 1))*LegendreP(nu, mu, x) = (sin(nu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x)-(sin(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, - x)	Divide[Sin[(\[Nu]- \[Mu])*Pi],Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreP[\[Nu], \[Mu], x] == Divide[Sin[\[Nu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x]-Divide[Sin[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], - x]	Failure	Failure	Successful [Tested: 40]	Successful [Tested: 45]
14.9.E8	${\displaystyle{\displaystyle\tfrac{1}{2}\pi\sin\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\cos\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)-\mathsf{Q}^{-\mu}_{\nu}\left(-x\right)}}$ \tfrac{1}{2}\pi\sin@{(\nu-\mu)\pi}\FerrersP[-\mu]{\nu}@{x} = -\cos@{(\nu-\mu)\pi}\FerrersQ[-\mu]{\nu}@{x}-\FerrersQ[-\mu]{\nu}@{-x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,\Re(\nu+(-\mu)+1)>0,\Re(\nu-\mu+1)>0,\Re(\nu-(-\mu)+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1}}$	(1)/(2)*Pi*sin((nu - mu)*Pi)*LegendreP(nu, - mu, x) = - cos((nu - mu)*Pi)*LegendreQ(nu, - mu, x)- LegendreQ(nu, - mu, - x)	Divide[1,2]*Pi*Sin[(\[Nu]- \[Mu])*Pi]*LegendreP[\[Nu], - \[Mu], x] == - Cos[(\[Nu]- \[Mu])*Pi]*LegendreQ[\[Nu], - \[Mu], x]- LegendreQ[\[Nu], - \[Mu], - x]	Failure	Failure	Error	Successful [Tested: 45]
14.9.E9	${\displaystyle{\displaystyle\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right)}}$ \frac{2}{\EulerGamma@{\nu+\mu+1}\EulerGamma@{\mu-\nu}}\FerrersQ[\mu]{\nu}@{x} = -\cos@{\nu\pi}\FerrersP[-\mu]{\nu}@{x}+\cos@{\mu\pi}\FerrersP[-\mu]{\nu}@{-x}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\mu-\nu)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1,\Re(\nu-\mu+1)>0}}$	(2)/(GAMMA(nu + mu + 1)*GAMMA(mu - nu))*LegendreQ(nu, mu, x) = - cos(nu*Pi)*LegendreP(nu, - mu, x)+ cos(mu*Pi)*LegendreP(nu, - mu, - x)	Divide[2,Gamma[\[Nu]+ \[Mu]+ 1]*Gamma[\[Mu]- \[Nu]]]*LegendreQ[\[Nu], \[Mu], x] == - Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], x]+ Cos[\[Mu]*Pi]*LegendreP[\[Nu], - \[Mu], - x]	Failure	Failure	Successful [Tested: 4]	Successful [Tested: 8]
14.9.E10	${\displaystyle{\displaystyle(2/\pi)\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\cos\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\mathsf{P}^{-\mu}_{\nu}\left(-x\right)}}$ (2/\pi)\sin@{(\nu-\mu)\pi}\FerrersQ[-\mu]{\nu}@{x} = \cos@{(\nu-\mu)\pi}\FerrersP[-\mu]{\nu}@{x}-\FerrersP[-\mu]{\nu}@{-x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1,\Re(\nu+\mu+1)>0,\Re(\nu+(-\mu)+1)>0,\Re(\nu-\mu+1)>0,\Re(\nu-(-\mu)+1)>0}}$	(2/Pi)*sin((nu - mu)*Pi)*LegendreQ(nu, - mu, x) = cos((nu - mu)*Pi)*LegendreP(nu, - mu, x)- LegendreP(nu, - mu, - x)	(2/Pi)*Sin[(\[Nu]- \[Mu])*Pi]*LegendreQ[\[Nu], - \[Mu], x] == Cos[(\[Nu]- \[Mu])*Pi]*LegendreP[\[Nu], - \[Mu], x]- LegendreP[\[Nu], - \[Mu], - x]	Failure	Failure	Error	Successful [Tested: 45]
14.9#Ex3	${\displaystyle{\displaystyle P^{-\mu}_{-\nu-1}\left(x\right)=P^{-\mu}_{\nu}\left(x\right)}}$ \assLegendreP[-\mu]{-\nu-1}@{x} = \assLegendreP[-\mu]{\nu}@{x}		LegendreP(- nu - 1, - mu, x) = LegendreP(nu, - mu, x)	LegendreP[- \[Nu]- 1, - \[Mu], 3, x] == LegendreP[\[Nu], - \[Mu], 3, x]	Successful	Successful	-	Successful [Tested: 300]
14.9#Ex4	${\displaystyle{\displaystyle P^{\mu}_{-\nu-1}\left(x\right)=P^{\mu}_{\nu}\left(x\right)}}$ \assLegendreP[\mu]{-\nu-1}@{x} = \assLegendreP[\mu]{\nu}@{x}		LegendreP(- nu - 1, mu, x) = LegendreP(nu, mu, x)	LegendreP[- \[Nu]- 1, \[Mu], 3, x] == LegendreP[\[Nu], \[Mu], 3, x]	Successful	Successful	-	Successful [Tested: 300]
14.9.E12	${\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=-\frac{\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\mu-\nu\right)}+\frac{\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}}}$ \cos@{\nu\pi}\assLegendreP[-\mu]{\nu}@{x} = -\frac{\assLegendreOlverQ[\mu]{\nu}@{x}}{\EulerGamma@{\mu-\nu}}+\frac{\assLegendreOlverQ[\mu]{-\nu-1}@{x}}{\EulerGamma@{\nu+\mu+1}}	${\displaystyle{\displaystyle\Re(\mu-\nu)>0,\Re(\nu+\mu+1)>0}}$	cos(nu*Pi)*LegendreP(nu, - mu, x) = -(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1))/(GAMMA(mu - nu))+(exp(-(mu)*Pi*I)*LegendreQ(- nu - 1,mu,x)/GAMMA(- nu - 1+mu+1))/(GAMMA(nu + mu + 1))	Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], 3, x] == -Divide[Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1],Gamma[\[Mu]- \[Nu]]]+Divide[Exp[-(\[Mu]) Pi I] LegendreQ[- \[Nu]- 1, \[Mu], 3, x]/Gamma[- \[Nu]- 1 + \[Mu] + 1],Gamma[\[Nu]+ \[Mu]+ 1]]	Failure	Failure	Failed [36 / 87] Result: -9.22033570+3.98641277*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -1/2+1/2*I*3^(1/2), x = 3/2} Result: 4.85982369+35.02749311*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -1/2+1/2*I*3^(1/2), x = 1/2} ... skip entries to safe data	Successful [Tested: 96]
14.9.E13	${\displaystyle{\displaystyle P^{-m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}P^{m}_{\nu}\left(x\right)}}$ \assLegendreP[-m]{\nu}@{x} = \frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\assLegendreP[m]{\nu}@{x}	${\displaystyle{\displaystyle\nu\neq m-1,\Re(\nu-m+1)>0,\Re(\nu+m+1)>0}}$	LegendreP(nu, - m, x) = (GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreP(nu, m, x)	LegendreP[\[Nu], - m, 3, x] == Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreP[\[Nu], m, 3, x]	Failure	Failure	Failed [15 / 21] Result: -.1566814731+1.035406980*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, m = 1} Result: .9394863529-.1899097116*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2, m = 1} ... skip entries to safe data	Successful [Tested: 21]
14.9.E14	${\displaystyle{\displaystyle\boldsymbol{Q}^{-\mu}_{\nu}\left(x\right)=\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)}}$ \assLegendreOlverQ[-\mu]{\nu}@{x} = \assLegendreOlverQ[\mu]{\nu}@{x}		exp(-(- mu)*Pi*I)*LegendreQ(nu,- mu,x)/GAMMA(nu+- mu+1) = exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1)	Exp[-(- \[Mu]) Pi I] LegendreQ[\[Nu], - \[Mu], 3, x]/Gamma[\[Nu] + - \[Mu] + 1] == Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1]	Error	Successful	-	Failed [36 / 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
14.9.E15	${\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{P^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}-\frac{P^{-\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu-\mu+1\right)}}}$ \frac{2\sin@{\mu\pi}}{\pi}\assLegendreOlverQ[\mu]{\nu}@{x} = \frac{\assLegendreP[\mu]{\nu}@{x}}{\EulerGamma@{\nu+\mu+1}}-\frac{\assLegendreP[-\mu]{\nu}@{x}}{\EulerGamma@{\nu-\mu+1}}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}}$	(2*sin(mu*Pi))/(Pi)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = (LegendreP(nu, mu, x))/(GAMMA(nu + mu + 1))-(LegendreP(nu, - mu, x))/(GAMMA(nu - mu + 1))	Divide[2*Sin[\[Mu]*Pi],Pi]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[LegendreP[\[Nu], \[Mu], 3, x],Gamma[\[Nu]+ \[Mu]+ 1]]-Divide[LegendreP[\[Nu], - \[Mu], 3, x],Gamma[\[Nu]- \[Mu]+ 1]]	Failure	Successful	Failed [108 / 120] Result: 3.058402749-19.69019192*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: .1602155595-16.40144782*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	Successful [Tested: 135]
14.9.E16	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(\tfrac{1}{2}\pi\right)^{1/2}\left(x^{2}-1\right)^{-1/4}\*P^{-\nu-(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1\right)^{-1/2}\right)}}$ \assLegendreOlverQ[\mu]{\nu}@{x} = \left(\tfrac{1}{2}\pi\right)^{1/2}\left(x^{2}-1\right)^{-1/4}\*\assLegendreP[-\nu-(1/2)]{-\mu-(1/2)}@{x\left(x^{2}-1\right)^{-1/2}}		exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((1)/(2)*Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* LegendreP(- mu -(1/2), - nu -(1/2), x*((x)^(2)- 1)^(- 1/2))	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == (Divide[1,2]*Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* LegendreP[- \[Mu]-(1/2), - \[Nu]-(1/2), 3, x*((x)^(2)- 1)^(- 1/2)]	Failure	Failure	Failed [292 / 300] Result: 13.31105553-5.485346831*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: 8.925040493-5.300266523*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 [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
14.9.E17	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=(2/\pi)^{1/2}\left(x^{2}-1\right)^{-1/4}\*\boldsymbol{Q}^{\nu+(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1\right)^{-1/2}\right)}}$ \assLegendreP[\mu]{\nu}@{x} = (2/\pi)^{1/2}\left(x^{2}-1\right)^{-1/4}\*\assLegendreOlverQ[\nu+(1/2)]{-\mu-(1/2)}@{x\left(x^{2}-1\right)^{-1/2}}		LegendreP(nu, mu, x) = (2/Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* exp(-(nu +(1/2))*Pi*I)*LegendreQ(- mu -(1/2),nu +(1/2),x*((x)^(2)- 1)^(- 1/2))/GAMMA(- mu -(1/2)+nu +(1/2)+1)	LegendreP[\[Nu], \[Mu], 3, x] == (2/Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* Exp[-(\[Nu]+(1/2)) Pi I] LegendreQ[- \[Mu]-(1/2), \[Nu]+(1/2), 3, x*((x)^(2)- 1)^(- 1/2)]/Gamma[- \[Mu]-(1/2) + \[Nu]+(1/2) + 1]	Failure	Failure	Failed [297 / 300] Result: 15.05963282-19.56004465*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: 2.964591568-6.756538622*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 [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
14.10.E1	${\displaystyle{\displaystyle{\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}$ {\FerrersP[\mu+2]{\nu}@{x}+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}\FerrersP[\mu+1]{\nu}@{x}}+(\nu-\mu)(\nu+\mu+1)\FerrersP[\mu]{\nu}@{x} = 0	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu + 2, x)+ 2*(mu + 1)*x*(1 - (x)^(2))^(- 1/2)* LegendreP(nu, mu + 1, x)+(nu - mu)*(nu + mu + 1)*LegendreP(nu, mu, x) = 0	LegendreP[\[Nu], \[Mu]+ 2, x]+ 2*(\[Mu]+ 1)*x*(1 - (x)^(2))^(- 1/2)* LegendreP[\[Nu], \[Mu]+ 1, x]+(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)*LegendreP[\[Nu], \[Mu], x] == 0	Failure	Successful	Successful [Tested: 300]	Successful [Tested: 300]
14.10.E2	${\displaystyle{\displaystyle{\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}$ {\left(1-x^{2}\right)^{1/2}\FerrersP[\mu+1]{\nu}@{x}-(\nu-\mu+1)\FerrersP[\mu]{\nu+1}@{x}}+(\nu+\mu+1)x\FerrersP[\mu]{\nu}@{x} = 0	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	(1 - (x)^(2))^(1/2)* LegendreP(nu, mu + 1, x)-(nu - mu + 1)*LegendreP(nu + 1, mu, x)+(nu + mu + 1)*x*LegendreP(nu, mu, x) = 0	(1 - (x)^(2))^(1/2)* LegendreP[\[Nu], \[Mu]+ 1, x]-(\[Nu]- \[Mu]+ 1)*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ \[Mu]+ 1)*x*LegendreP[\[Nu], \[Mu], x] == 0	Failure	Successful	Successful [Tested: 300]	Successful [Tested: 300]
14.10.E3	${\displaystyle{\displaystyle{(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}$ {(\nu-\mu+2)\FerrersP[\mu]{\nu+2}@{x}-(2\nu+3)x\FerrersP[\mu]{\nu+1}@{x}}+(\nu+\mu+1)\FerrersP[\mu]{\nu}@{x} = 0	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	(nu - mu + 2)*LegendreP(nu + 2, mu, x)-(2*nu + 3)*x*LegendreP(nu + 1, mu, x)+(nu + mu + 1)*LegendreP(nu, mu, x) = 0	(\[Nu]- \[Mu]+ 2)*LegendreP[\[Nu]+ 2, \[Mu], x]-(2*\[Nu]+ 3)*x*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ \[Mu]+ 1)*LegendreP[\[Nu], \[Mu], x] == 0	Successful	Successful	-	Successful [Tested: 300]
14.10.E4	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x\mathsf{P}^{\mu}_{\nu}\left(x\right)}}}$ \left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = {(\mu-\nu-1)\FerrersP[\mu]{\nu+1}@{x}+(\nu+1)x\FerrersP[\mu]{\nu}@{x}}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	(1 - (x)^(2))*diff(LegendreP(nu, mu, x), x) = (mu - nu - 1)*LegendreP(nu + 1, mu, x)+(nu + 1)*x*LegendreP(nu, mu, x)	(1 - (x)^(2))*D[LegendreP[\[Nu], \[Mu], x], x] == (\[Mu]- \[Nu]- 1)*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ 1)*x*LegendreP[\[Nu], \[Mu], x]	Successful	Successful	-	Successful [Tested: 300]
14.10.E5	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(x\right)-\nu x\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$ \left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = (\nu+\mu)\FerrersP[\mu]{\nu-1}@{x}-\nu x\FerrersP[\mu]{\nu}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	(1 - (x)^(2))*diff(LegendreP(nu, mu, x), x) = (nu + mu)*LegendreP(nu - 1, mu, x)- nu*x*LegendreP(nu, mu, x)	(1 - (x)^(2))*D[LegendreP[\[Nu], \[Mu], x], x] == (\[Nu]+ \[Mu])*LegendreP[\[Nu]- 1, \[Mu], x]- \[Nu]*x*LegendreP[\[Nu], \[Mu], x]	Successful	Successful	-	Successful [Tested: 300]
14.10.E6	${\displaystyle{\displaystyle{P^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}P^{\mu+1}_{\nu}\left(x\right)}-(\nu-\mu)(\nu+\mu+1)P^{\mu}_{\nu}\left(x\right)=0}}$ {\assLegendreP[\mu+2]{\nu}@{x}+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}\assLegendreP[\mu+1]{\nu}@{x}}-(\nu-\mu)(\nu+\mu+1)\assLegendreP[\mu]{\nu}@{x} = 0		LegendreP(nu, mu + 2, x)+ 2*(mu + 1)*x*((x)^(2)- 1)^(- 1/2)* LegendreP(nu, mu + 1, x)-(nu - mu)*(nu + mu + 1)*LegendreP(nu, mu, x) = 0	LegendreP[\[Nu], \[Mu]+ 2, 3, x]+ 2*(\[Mu]+ 1)*x*((x)^(2)- 1)^(- 1/2)* LegendreP[\[Nu], \[Mu]+ 1, 3, x]-(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)*LegendreP[\[Nu], \[Mu], 3, x] == 0	Failure	Failure	Successful [Tested: 300]	Successful [Tested: 300]
14.10.E7	${\displaystyle{\displaystyle{\left(x^{2}-1\right)^{1/2}P^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)P^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)xP^{\mu}_{\nu}\left(x\right)=0}}$ {\left(x^{2}-1\right)^{1/2}\assLegendreP[\mu+1]{\nu}@{x}-(\nu-\mu+1)\assLegendreP[\mu]{\nu+1}@{x}}+(\nu+\mu+1)x\assLegendreP[\mu]{\nu}@{x} = 0		((x)^(2)- 1)^(1/2)* LegendreP(nu, mu + 1, x)-(nu - mu + 1)*LegendreP(nu + 1, mu, x)+(nu + mu + 1)*x*LegendreP(nu, mu, x) = 0	((x)^(2)- 1)^(1/2)* LegendreP[\[Nu], \[Mu]+ 1, 3, x]-(\[Nu]- \[Mu]+ 1)*LegendreP[\[Nu]+ 1, \[Mu], 3, x]+(\[Nu]+ \[Mu]+ 1)*x*LegendreP[\[Nu], \[Mu], 3, x] == 0	Failure	Failure	Successful [Tested: 300]	Successful [Tested: 300]
14.11.E1	${\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\pi\cot\left(\nu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{1}{\pi}\mathsf{A}_{\nu}^{\mu}(x)}}$ \pderiv{}{\nu}\FerrersP[\mu]{\nu}@{x} = \pi\cot@{\nu\pi}\FerrersP[\mu]{\nu}@{x}-\frac{1}{\pi}\mathsf{A}_{\nu}^{\mu}(x)	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	diff(LegendreP(nu, mu, x), nu) = Pi*cot(nu*Pi)*LegendreP(nu, mu, x)-(1)/(Pi)*(A[nu])^(mu)(x)	D[LegendreP[\[Nu], \[Mu], x], \[Nu]] == Pi*Cot[\[Nu]*Pi]*LegendreP[\[Nu], \[Mu], x]-Divide[1,Pi]*(Subscript[A, \[Nu]])^\[Mu][x]	Failure	Failure	Failed [300 / 300] Result: 11.90824559-1.654502830*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = 1/2*3^(1/2)+1/2*I} Result: 11.53757926-1.652858974*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[11.90824558684297, -1.654502826549051] 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]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[11.53757925943594, -1.6528589711511499] 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]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.11.E2	${\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}\pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{\pi\sin\left(\mu\pi\right)}{\sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}(-x)}}$ \pderiv{}{\nu}\FerrersQ[\mu]{\nu}@{x} = -\tfrac{1}{2}\pi^{2}\FerrersP[\mu]{\nu}@{x}+\frac{\pi\sin@{\mu\pi}}{\sin@{\nu\pi}\sin@{(\nu+\mu)\pi}}\FerrersQ[\mu]{\nu}@{x}-\tfrac{1}{2}\cot@{(\nu+\mu)\pi}\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc@{(\nu+\mu)\pi}\mathsf{A}_{\nu}^{\mu}(-x)	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}}$	diff(LegendreQ(nu, mu, x), nu) = -(1)/(2)*(Pi)^(2)* LegendreP(nu, mu, x)+(Pi*sin(mu*Pi))/(sin(nu*Pi)*sin((nu + mu)*Pi))*LegendreQ(nu, mu, x)-(1)/(2)*cot((nu + mu)*Pi)*(A[nu])^(mu)(x)+(1)/(2)*csc((nu + mu)*Pi)*(A[nu])^(mu)(- x)	D[LegendreQ[\[Nu], \[Mu], x], \[Nu]] == -Divide[1,2]*(Pi)^(2)* LegendreP[\[Nu], \[Mu], x]+Divide[Pi*Sin[\[Mu]*Pi],Sin[\[Nu]*Pi]*Sin[(\[Nu]+ \[Mu])*Pi]]*LegendreQ[\[Nu], \[Mu], x]-Divide[1,2]*Cot[(\[Nu]+ \[Mu])*Pi]*(Subscript[A, \[Nu]])^\[Mu][x]+Divide[1,2]*Csc[(\[Nu]+ \[Mu])*Pi]*(Subscript[A, \[Nu]])^\[Mu][- x]	Failure	Failure	Failed [300 / 300] Result: -2.639260453-18.83790600*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = 1/2*3^(1/2)+1/2*I} Result: -2.596785248-18.22264548*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-2.639260449912798, -18.837906001053177] 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]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.5967852433828247, -18.222645474383306] 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]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.11.E3	${\displaystyle{\displaystyle\mathsf{A}_{\nu}^{\mu}(x)=\sin\left(\nu\pi\right)\left(\frac{1+x}{1-x}\right)^{\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}\Gamma\left(k-\nu\right)\Gamma\left(k+\nu+1\right)}{k!\Gamma\left(k-\mu+1\right)}\*\left(\psi\left(k+\nu+1\right)-\psi\left(k-\nu\right)\right)}}$ \mathsf{A}_{\nu}^{\mu}(x) = \sin@{\nu\pi}\left(\frac{1+x}{1-x}\right)^{\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}\EulerGamma@{k-\nu}\EulerGamma@{k+\nu+1}}{k!\EulerGamma@{k-\mu+1}}\*\left(\digamma@{k+\nu+1}-\digamma@{k-\nu}\right)	${\displaystyle{\displaystyle\Re(k-\nu)>0,\Re(k+\nu+1)>0,\Re(k-\mu+1)>0}}$	(A[nu])^(mu)(x) = sin(nu*Pi)*((1 + x)/(1 - x))^(mu/2)* sum((((1)/(2)-(1)/(2)*x)^(k)* GAMMA(k - nu)*GAMMA(k + nu + 1))/(factorial(k)*GAMMA(k - mu + 1))*(Psi(k + nu + 1)- Psi(k - nu)), k = 0..infinity)	(Subscript[A, \[Nu]])^\[Mu][x] == Sin[\[Nu]*Pi]*(Divide[1 + x,1 - x])^(\[Mu]/2)* Sum[Divide[(Divide[1,2]-Divide[1,2]*x)^(k)* Gamma[k - \[Nu]]*Gamma[k + \[Nu]+ 1],(k)!*Gamma[k - \[Mu]+ 1]]*(PolyGamma[k + \[Nu]+ 1]- PolyGamma[k - \[Nu]]), {k, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out