14.17

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.17.E1	${\displaystyle{\displaystyle{\int\left(1-x^{2}\right)^{-\mu/2}\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathrm{d}x}={-\left(1-x^{2}\right)^{-(\mu-1)/2}\mathsf{P}^{\mu-1}_{\nu}\left(x\right)}}}$ {\int\left(1-x^{2}\right)^{-\mu/2}\FerrersP[\mu]{\nu}@{x}\diff{x}} = {-\left(1-x^{2}\right)^{-(\mu-1)/2}\FerrersP[\mu-1]{\nu}@{x}}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	int((1 - (x)^(2))^(- mu/2)* LegendreP(nu, mu, x), x) = -(1 - (x)^(2))^(-(mu - 1)/2)* LegendreP(nu, mu - 1, x)	Integrate[(1 - (x)^(2))^(- \[Mu]/2)* LegendreP[\[Nu], \[Mu], x], x, GenerateConditions->None] == -(1 - (x)^(2))^(-(\[Mu]- 1)/2)* LegendreP[\[Nu], \[Mu]- 1, x]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[3.8842606727900413, 5.104372500552582], Integrate[Complex[-4.747850387868644, -1.1425414738949808], 1.5, Rule[GenerateConditions, None]]] 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: Plus[Complex[3.976584990156878, 2.3595388807039552], Integrate[Complex[-2.482845880898655, 4.683216982349827], 1.5, Rule[GenerateConditions, None]]] 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
14.17.E2	${\displaystyle{\displaystyle\int\left(1-x^{2}\right)^{\mu/2}\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{\left(1-x^{2}\right)^{(\mu+1)/2}}{(\nu-\mu)(\nu+\mu+1)}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)}}$ \int\left(1-x^{2}\right)^{\mu/2}\FerrersP[\mu]{\nu}@{x}\diff{x} = \frac{\left(1-x^{2}\right)^{(\mu+1)/2}}{(\nu-\mu)(\nu+\mu+1)}\FerrersP[\mu+1]{\nu}@{x}	${\displaystyle{\displaystyle\mu\neq\nu,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	int((1 - (x)^(2))^(mu/2)* LegendreP(nu, mu, x), x) = ((1 - (x)^(2))^((mu + 1)/2))/((nu - mu)*(nu + mu + 1))*LegendreP(nu, mu + 1, x)	Integrate[(1 - (x)^(2))^(\[Mu]/2)* LegendreP[\[Nu], \[Mu], x], x, GenerateConditions->None] == Divide[(1 - (x)^(2))^((\[Mu]+ 1)/2),(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)]*LegendreP[\[Nu], \[Mu]+ 1, x]	Error	Failure	-	Failed [270 / 270] Result: Plus[Complex[-0.5646480599960819, 1.3746025553854266], Integrate[Complex[0.23690790481776922, -1.3156471186304795], 1.5, Rule[GenerateConditions, None]]] 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]]]} Result: Plus[Complex[-0.228607897264037, 1.5189132046928975], Integrate[Complex[0.8670522613344679, -2.293703747689092], 1.5, Rule[GenerateConditions, None]]] 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
14.17.E3	${\displaystyle{\displaystyle\int x\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{1}{2\nu(\nu+1)}\left((\mu^{2}-(\nu+1)(\nu+x^{2}))\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)+(\nu+1)(\nu-\mu+1)x(\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)+\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right))-(\nu-\mu+1)^{2}\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)\right)}}$ \int x\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}\diff{x} = \frac{1}{2\nu(\nu+1)}\left((\mu^{2}-(\nu+1)(\nu+x^{2}))\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}+(\nu+1)(\nu-\mu+1)x(\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x}+\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x})-(\nu-\mu+1)^{2}\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu+1}@{x}\right)	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,\Re((\nu+1)+\mu+1)>0,\Re(\nu-\mu+1)>0,\Re((\nu+1)-\mu+1)>0}}$	int(x*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x), x) = (1)/(2*nu*(nu + 1))*(((mu)^(2)-(nu + 1)*(nu + (x)^(2)))*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x)+(nu + 1)*(nu - mu + 1)*x*(LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x)+ LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x))-(nu - mu + 1)^(2)* LegendreP(nu + 1, mu, x)*LegendreQ(nu + 1, mu, x))	Integrate[x*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x], x, GenerateConditions->None] == Divide[1,2*\[Nu]*(\[Nu]+ 1)]*((\[Mu]^(2)-(\[Nu]+ 1)*(\[Nu]+ (x)^(2)))*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x]+(\[Nu]+ 1)*(\[Nu]- \[Mu]+ 1)*x*(LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x]+ LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu], \[Mu], x])-(\[Nu]- \[Mu]+ 1)^(2)* LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x])	Error	Aborted	-	Skip - No test values generated
14.17.E4	${\displaystyle{\displaystyle\int\frac{x}{\left(1-x^{2}\right)^{3/2}}\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{1/2}}\left((1-2\mu^{2}+2\nu(\nu+1))\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)+(2\nu+1)(\mu-\nu-1)x(\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)+\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right))+2(\mu-\nu-1)^{2}\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)\right)}}$ \int\frac{x}{\left(1-x^{2}\right)^{3/2}}\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}\diff{x} = \frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{1/2}}\left((1-2\mu^{2}+2\nu(\nu+1))\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}+(2\nu+1)(\mu-\nu-1)x(\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x}+\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x})+2(\mu-\nu-1)^{2}\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu+1}@{x}\right)	${\displaystyle{\displaystyle\mu\neq+\frac{1}{2},\mu\neq-\frac{1}{2},|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,\Re((\nu+1)+\mu+1)>0,\Re(\nu-\mu+1)>0,\Re((\nu+1)-\mu+1)>0}}$	int((x)/((1 - (x)^(2))^(3/2))*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x), x) = (1)/((1 - 4*(mu)^(2))*(1 - (x)^(2))^(1/2))*((1 - 2*(mu)^(2)+ 2*nu*(nu + 1))*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x)+(2*nu + 1)*(mu - nu - 1)*x*(LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x)+ LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x))+ 2*(mu - nu - 1)^(2)* LegendreP(nu + 1, mu, x)*LegendreQ(nu + 1, mu, x))	Integrate[Divide[x,(1 - (x)^(2))^(3/2)]*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x], x, GenerateConditions->None] == Divide[1,(1 - 4*\[Mu]^(2))*(1 - (x)^(2))^(1/2)]*((1 - 2*\[Mu]^(2)+ 2*\[Nu]*(\[Nu]+ 1))*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x]+(2*\[Nu]+ 1)*(\[Mu]- \[Nu]- 1)*x*(LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x]+ LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu], \[Mu], x])+ 2*(\[Mu]- \[Nu]- 1)^(2)* LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x])	Failure	Aborted	Error	Failed [99 / 99] Result: Plus[Complex[-15.417707085194902, 19.940158970813897], Integrate[Complex[-9.988309927179525, -1.2041271824131927], 1.5, Rule[GenerateConditions, None]]] 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: Plus[Complex[17.198725078389664, -1.5826141510664629], Integrate[Complex[20.92420958974465, 36.064324396521705], 1.5, Rule[GenerateConditions, None]]] 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
14.17.E5	${\displaystyle{\displaystyle\int_{0}^{1}x^{\sigma}\left(1-x^{2}\right)^{\mu/2}\mathsf{P}^{-\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{\Gamma\left(\frac{1}{2}\sigma+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\sigma+1\right)}{2^{\mu+1}\Gamma\left(\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}\right)}}}$ \int_{0}^{1}x^{\sigma}\left(1-x^{2}\right)^{\mu/2}\FerrersP[-\mu]{\nu}@{x}\diff{x} = \frac{\EulerGamma@{\frac{1}{2}\sigma+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\sigma+1}}{2^{\mu+1}\EulerGamma@{\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}}}	${\displaystyle{\displaystyle\Re\sigma>-1,\Re\mu>-1,\Re(\frac{1}{2}\sigma+\frac{1}{2})>0,\Re(\frac{1}{2}\sigma+1)>0,\Re(\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,\Re(\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2})>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	int((x)^(sigma)*(1 - (x)^(2))^(mu/2)* LegendreP(nu, - mu, x), x = 0..1) = (GAMMA((1)/(2)*sigma +(1)/(2))*GAMMA((1)/(2)*sigma + 1))/((2)^(mu + 1)* GAMMA((1)/(2)*sigma -(1)/(2)*nu +(1)/(2)*mu + 1)*GAMMA((1)/(2)*sigma +(1)/(2)*nu +(1)/(2)*mu +(3)/(2)))	Integrate[(x)^\[Sigma]*(1 - (x)^(2))^(\[Mu]/2)* LegendreP[\[Nu], - \[Mu], x], {x, 0, 1}, GenerateConditions->None] == Divide[Gamma[Divide[1,2]*\[Sigma]+Divide[1,2]]*Gamma[Divide[1,2]*\[Sigma]+ 1],(2)^(\[Mu]+ 1)* Gamma[Divide[1,2]*\[Sigma]-Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]*\[Sigma]+Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[3,2]]]	Failure	Failure	Manual Skip!	Skipped - Because timed out
14.17.E6	${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}^{m}_{l}\left(x\right)\mathsf{P}^{m}_{n}\left(x\right)\mathrm{d}x=\frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}\delta_{l,n}}}$ \int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersP[m]{n}@{x}\diff{x} = \frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}\Kroneckerdelta{l}{n}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	int(LegendreP(l, m, x)*LegendreP(n, m, x), x = - 1..1) = (factorial(n + m))/(factorial(n - m)*(n +(1)/(2)))*KroneckerDelta[l, n]	Integrate[LegendreP[l, m, x]*LegendreP[n, m, x], {x, - 1, 1}, GenerateConditions->None] == Divide[(n + m)!,(n - m)!*(n +Divide[1,2])]*KroneckerDelta[l, n]	Aborted	Failure	Successful [Tested: 27]	Successful [Tested: 27]
14.17.E7	${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}^{m}_{l}\left(x\right)\mathsf{P}^{-m}_{n}\left(x\right)\mathrm{d}x=\frac{(-1)^{m}}{l+\frac{1}{2}}\delta_{l,n}}}$ \int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersP[-m]{n}@{x}\diff{x} = \frac{(-1)^{m}}{l+\frac{1}{2}}\Kroneckerdelta{l}{n}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	int(LegendreP(l, m, x)*LegendreP(n, - m, x), x = - 1..1) = ((- 1)^(m))/(l +(1)/(2))*KroneckerDelta[l, n]	Integrate[LegendreP[l, m, x]*LegendreP[n, - m, x], {x, - 1, 1}, GenerateConditions->None] == Divide[(- 1)^(m),l +Divide[1,2]]*KroneckerDelta[l, n]	Aborted	Failure	Failed [7 / 27] Result: -.6666666667 Test Values: {l = 1, m = 2, n = 1} Result: .6666666667 Test Values: {l = 1, m = 3, n = 1} ... skip entries to safe data	Failed [7 / 27] Result: -0.6666666666666666 Test Values: {Rule[l, 1], Rule[m, 2], Rule[n, 1]} Result: 0.6666666666666666 Test Values: {Rule[l, 1], Rule[m, 3], Rule[n, 1]} ... skip entries to safe data
14.17.E8	${\displaystyle{\displaystyle\int_{-1}^{1}\frac{\mathsf{P}^{l}_{n}\left(x\right)\mathsf{P}^{m}_{n}\left(x\right)}{1-x^{2}}\mathrm{d}x=\frac{(n+m)!}{(n-m)!m}\delta_{l,m}}}$ \int_{-1}^{1}\frac{\FerrersP[l]{n}@{x}\FerrersP[m]{n}@{x}}{1-x^{2}}\diff{x} = \frac{(n+m)!}{(n-m)!m}\Kroneckerdelta{l}{m}	${\displaystyle{\displaystyle m>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	int((LegendreP(n, l, x)*LegendreP(n, m, x))/(1 - (x)^(2)), x = - 1..1) = (factorial(n + m))/(factorial(n - m)*m)*KroneckerDelta[l, m]	Integrate[Divide[LegendreP[n, l, x]*LegendreP[n, m, x],1 - (x)^(2)], {x, - 1, 1}, GenerateConditions->None] == Divide[(n + m)!,(n - m)!*m]*KroneckerDelta[l, m]	Failure	Aborted	Skipped - Because timed out	Successful [Tested: 27]
14.17.E9	${\displaystyle{\displaystyle\int_{-1}^{1}\frac{\mathsf{P}^{l}_{n}\left(x\right)\mathsf{P}^{-m}_{n}\left(x\right)}{1-x^{2}}\mathrm{d}x=\frac{(-1)^{l}}{l}\delta_{l,m}}}$ \int_{-1}^{1}\frac{\FerrersP[l]{n}@{x}\FerrersP[-m]{n}@{x}}{1-x^{2}}\diff{x} = \frac{(-1)^{l}}{l}\Kroneckerdelta{l}{m}	${\displaystyle{\displaystyle l>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	int((LegendreP(n, l, x)*LegendreP(n, - m, x))/(1 - (x)^(2)), x = - 1..1) = ((- 1)^(l))/(l)*KroneckerDelta[l, m]	Integrate[Divide[LegendreP[n, l, x]*LegendreP[n, - m, x],1 - (x)^(2)], {x, - 1, 1}, GenerateConditions->None] == Divide[(- 1)^(l),l]*KroneckerDelta[l, m]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.17.E10	${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf{P}_{\lambda}\left(x\right)\mathrm{d}x=\frac{2\left(2\sin\left(\nu\pi\right)\sin\left(\lambda\pi\right)\left(\psi\left(\nu+1\right)-\psi\left(\lambda+1\right)\right)+\pi\sin\left((\lambda-\nu)\pi\right)\right)}{\pi^{2}(\lambda-\nu)(\lambda+\nu+1)}}}$ \int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersP[]{\lambda}@{x}\diff{x} = \frac{2\left(2\sin@{\nu\pi}\sin@{\lambda\pi}\left(\digamma@{\nu+1}-\digamma@{\lambda+1}\right)+\pi\sin@{(\lambda-\nu)\pi}\right)}{\pi^{2}(\lambda-\nu)(\lambda+\nu+1)}	${\displaystyle{\displaystyle\lambda\neq\nu}}$	int(LegendreP(nu, x)*LegendreP(lambda, x), x = - 1..1) = (2*(2*sin(nu*Pi)*sin(lambda*Pi)*(Psi(nu + 1)- Psi(lambda + 1))+ Pi*sin((lambda - nu)*Pi)))/((Pi)^(2)*(lambda - nu)*(lambda + nu + 1))	Integrate[LegendreP[\[Nu], x]*LegendreP[\[Lambda], x], {x, - 1, 1}, GenerateConditions->None] == Divide[2*(2*Sin[\[Nu]*Pi]*Sin[\[Lambda]*Pi]*(PolyGamma[\[Nu]+ 1]- PolyGamma[\[Lambda]+ 1])+ Pi*Sin[(\[Lambda]- \[Nu])*Pi]),(Pi)^(2)*(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]	Error	Aborted	-	Skipped - Because timed out
14.17.E11	${\displaystyle{\displaystyle\int_{-1}^{1}\left(\mathsf{P}_{\nu}\left(x\right)\right)^{2}\mathrm{d}x=\frac{\pi^{2}-2{\sin^{2}}\left(\nu\pi\right)\psi'\left(\nu+1\right)}{\pi^{2}\left(\nu+\frac{1}{2}\right)}}}$ \int_{-1}^{1}\left(\FerrersP[]{\nu}@{x}\right)^{2}\diff{x} = \frac{\pi^{2}-2\sin^{2}@{\nu\pi}\digamma'@{\nu+1}}{\pi^{2}\left(\nu+\frac{1}{2}\right)}	${\displaystyle{\displaystyle\nu\neq-\frac{1}{2}}}$	int((LegendreP(nu, x))^(2), x = - 1..1) = ((Pi)^(2)- 2*(sin(nu*Pi))^(2)* subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/((Pi)^(2)*(nu +(1)/(2)))	Integrate[(LegendreP[\[Nu], x])^(2), {x, - 1, 1}, GenerateConditions->None] == Divide[(Pi)^(2)- 2*(Sin[\[Nu]*Pi])^(2)* (D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),(Pi)^(2)*(\[Nu]+Divide[1,2])]	Failure	Aborted	Failed [1 / 9] Result: Float(infinity)+Float(infinity)*I Test Values: {nu = -2}	Skipped - Because timed out
14.17.E12	${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{Q}_{\nu}\left(x\right)\mathsf{Q}_{\lambda}\left(x\right)\mathrm{d}x=\frac{\left((\psi\left(\nu+1\right)-\psi\left(\lambda+1\right))(1+\cos\left(\nu\pi\right)\cos\left(\lambda\pi\right))+\frac{1}{2}\pi\sin\left((\lambda-\nu)\pi\right)\right)}{(\lambda-\nu)(\lambda+\nu+1)}}}$ \int_{-1}^{1}\FerrersQ[]{\nu}@{x}\FerrersQ[]{\lambda}@{x}\diff{x} = \frac{\left((\digamma@{\nu+1}-\digamma@{\lambda+1})(1+\cos@{\nu\pi}\cos@{\lambda\pi})+\frac{1}{2}\pi\sin@{(\lambda-\nu)\pi}\right)}{(\lambda-\nu)(\lambda+\nu+1)}	${\displaystyle{\displaystyle\lambda\neq\nu}}$	int(LegendreQ(nu, x)*LegendreQ(lambda, x), x = - 1..1) = ((Psi(nu + 1)- Psi(lambda + 1))*(1 + cos(nu*Pi)*cos(lambda*Pi))+(1)/(2)*Pi*sin((lambda - nu)*Pi))/((lambda - nu)*(lambda + nu + 1))	Integrate[LegendreQ[\[Nu], x]*LegendreQ[\[Lambda], x], {x, - 1, 1}, GenerateConditions->None] == Divide[(PolyGamma[\[Nu]+ 1]- PolyGamma[\[Lambda]+ 1])*(1 + Cos[\[Nu]*Pi]*Cos[\[Lambda]*Pi])+Divide[1,2]*Pi*Sin[(\[Lambda]- \[Nu])*Pi],(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.17.E13	${\displaystyle{\displaystyle\int_{-1}^{1}\left(\mathsf{Q}_{\nu}\left(x\right)\right)^{2}\mathrm{d}x=\frac{\pi^{2}-2\left(1+{\cos^{2}}\left(\nu\pi\right)\right)\psi'\left(\nu+1\right)}{2(2\nu+1)}}}$ \int_{-1}^{1}\left(\FerrersQ[]{\nu}@{x}\right)^{2}\diff{x} = \frac{\pi^{2}-2\left(1+\cos^{2}@{\nu\pi}\right)\digamma'@{\nu+1}}{2(2\nu+1)}	${\displaystyle{\displaystyle\nu\neq-\frac{1}{2}}}$	int((LegendreQ(nu, x))^(2), x = - 1..1) = ((Pi)^(2)- 2*(1 + (cos(nu*Pi))^(2))*subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/(2*(2*nu + 1))	Integrate[(LegendreQ[\[Nu], x])^(2), {x, - 1, 1}, GenerateConditions->None] == Divide[(Pi)^(2)- 2*(1 + (Cos[\[Nu]*Pi])^(2))*(D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),2*(2*\[Nu]+ 1)]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.17.E14	${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf{Q}_{\lambda}\left(x\right)\mathrm{d}x=\frac{2\sin\left(\nu\pi\right)\cos\left(\lambda\pi\right)\left(\psi\left(\nu+1\right)-\psi\left(\lambda+1\right)\right)+\pi\cos\left((\lambda-\nu)\pi\right)-\pi}{\pi(\lambda-\nu)(\lambda+\nu+1)}}}$ \int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersQ[]{\lambda}@{x}\diff{x} = \frac{2\sin@{\nu\pi}\cos@{\lambda\pi}\left(\digamma@{\nu+1}-\digamma@{\lambda+1}\right)+\pi\cos@{(\lambda-\nu)\pi}-\pi}{\pi(\lambda-\nu)(\lambda+\nu+1)}	${\displaystyle{\displaystyle\Re\lambda>0,\Re\nu>0,\lambda\neq\nu}}$	int(LegendreP(nu, x)*LegendreQ(lambda, x), x = - 1..1) = (2*sin(nu*Pi)*cos(lambda*Pi)*(Psi(nu + 1)- Psi(lambda + 1))+ Pi*cos((lambda - nu)*Pi)- Pi)/(Pi*(lambda - nu)*(lambda + nu + 1))	Integrate[LegendreP[\[Nu], x]*LegendreQ[\[Lambda], x], {x, - 1, 1}, GenerateConditions->None] == Divide[2*Sin[\[Nu]*Pi]*Cos[\[Lambda]*Pi]*(PolyGamma[\[Nu]+ 1]- PolyGamma[\[Lambda]+ 1])+ Pi*Cos[(\[Lambda]- \[Nu])*Pi]- Pi,Pi*(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]	Error	Aborted	-	Skipped - Because timed out
14.17.E15	${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf{Q}_{\nu}\left(x\right)\mathrm{d}x=-\frac{\sin\left(2\nu\pi\right)\psi'\left(\nu+1\right)}{\pi(2\nu+1)}}}$ \int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersQ[]{\nu}@{x}\diff{x} = -\frac{\sin@{2\nu\pi}\digamma'@{\nu+1}}{\pi(2\nu+1)}	${\displaystyle{\displaystyle\Re\nu>0}}$	int(LegendreP(nu, x)*LegendreQ(nu, x), x = - 1..1) = -(sin(2*nu*Pi)*subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/(Pi*(2*nu + 1))	Integrate[LegendreP[\[Nu], x]*LegendreQ[\[Nu], x], {x, - 1, 1}, GenerateConditions->None] == -Divide[Sin[2*\[Nu]*Pi]*(D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),Pi*(2*\[Nu]+ 1)]	Error	Aborted	-	Skipped - Because timed out
14.17.E16	${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}^{m}_{l}\left(x\right)\mathsf{Q}^{m}_{n}\left(x\right)\mathrm{d}x=\frac{\left(1-(-1)^{l+n}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!}}}$ \int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersQ[m]{n}@{x}\diff{x} = \frac{\left(1-(-1)^{l+n}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!}	${\displaystyle{\displaystyle l\neq n,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(n+\mu+1)>0,\Re(\nu+m+1)>0,\Re(n-\mu+1)>0,\Re(\nu-m+1)>0}}$	int(LegendreP(l, m, x)*LegendreQ(n, m, x), x = - 1..1) = ((1 -(- 1)^(l + n))*factorial(l + m))/((l - n)*(l + n + 1)*factorial(l - m))	Integrate[LegendreP[l, m, x]*LegendreQ[n, m, x], {x, - 1, 1}, GenerateConditions->None] == Divide[(1 -(- 1)^(l + n))*(l + m)!,(l - n)*(l + n + 1)*(l - m)!]	Aborted	Failure	Error	Skipped - Because timed out
14.17.E17	${\displaystyle{\displaystyle\int_{0}^{\pi}\mathsf{Q}_{l}\left(\cos\theta\right)\mathsf{P}_{m}\left(\cos\theta\right)\mathsf{P}_{n}\left(\cos\theta\right)\sin\theta\mathrm{d}\theta=0}}$ \int_{0}^{\pi}\FerrersQ[]{l}@{\cos@@{\theta}}\FerrersP[]{m}@{\cos@@{\theta}}\FerrersP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta} = 0	${\displaystyle{\displaystyle|m-n|<l,l<m+n}}$	int(LegendreQ(l, cos(theta))*LegendreP(m, cos(theta))*LegendreP(n, cos(theta))*sin(theta), theta = 0..Pi) = 0	Integrate[LegendreQ[l, Cos[\[Theta]]]*LegendreP[m, Cos[\[Theta]]]*LegendreP[n, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == 0	Aborted	Aborted	Error	Skipped - Because timed out
14.17.E18	${\displaystyle{\displaystyle\int_{1}^{\infty}P_{\nu}\left(x\right)Q_{\lambda}\left(x\right)\mathrm{d}x=\frac{1}{(\lambda-\nu)(\nu+\lambda+1)}}}$ \int_{1}^{\infty}\assLegendreP[]{\nu}@{x}\assLegendreQ[]{\lambda}@{x}\diff{x} = \frac{1}{(\lambda-\nu)(\nu+\lambda+1)}	${\displaystyle{\displaystyle\Re\lambda>\Re\nu,\Re\nu>0}}$	int(LegendreP(nu, x)*LegendreQ(lambda, x), x = 1..infinity) = (1)/((lambda - nu)*(nu + lambda + 1))	Integrate[LegendreP[\[Nu], 0, 3, x]*LegendreQ[\[Lambda], 0, 3, x], {x, 1, Infinity}, GenerateConditions->None] == Divide[1,(\[Lambda]- \[Nu])*(\[Nu]+ \[Lambda]+ 1)]	Error	Failure	-	Skipped - Because timed out
14.17.E19	${\displaystyle{\displaystyle\int_{1}^{\infty}Q_{\nu}\left(x\right)Q_{\lambda}\left(x\right)\mathrm{d}x=\frac{\psi\left(\lambda+1\right)-\psi\left(\nu+1\right)}{(\lambda-\nu)(\lambda+\nu+1)}}}$ \int_{1}^{\infty}\assLegendreQ[]{\nu}@{x}\assLegendreQ[]{\lambda}@{x}\diff{x} = \frac{\digamma@{\lambda+1}-\digamma@{\nu+1}}{(\lambda-\nu)(\lambda+\nu+1)}	${\displaystyle{\displaystyle\Re\left(\lambda+\nu\right)>-1,\lambda\neq\nu}}$	int(LegendreQ(nu, x)*LegendreQ(lambda, x), x = 1..infinity) = (Psi(lambda + 1)- Psi(nu + 1))/((lambda - nu)*(lambda + nu + 1))	Integrate[LegendreQ[\[Nu], 0, 3, x]*LegendreQ[\[Lambda], 0, 3, x], {x, 1, Infinity}, GenerateConditions->None] == Divide[PolyGamma[\[Lambda]+ 1]- PolyGamma[\[Nu]+ 1],(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.17.E20	${\displaystyle{\displaystyle\int_{1}^{\infty}(Q_{\nu}\left(x\right))^{2}\mathrm{d}x=\frac{\psi'\left(\nu+1\right)}{2\nu+1}}}$ \int_{1}^{\infty}(\assLegendreQ[]{\nu}@{x})^{2}\diff{x} = \frac{\digamma'@{\nu+1}}{2\nu+1}	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}$	int((LegendreQ(nu, x))^(2), x = 1..infinity) = (subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/(2*nu + 1)	Integrate[(LegendreQ[\[Nu], 0, 3, x])^(2), {x, 1, Infinity}, GenerateConditions->None] == Divide[D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1,2*\[Nu]+ 1]	Error	Failure	-	Successful [Tested: 5]