14.9

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
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