Results of Legendre and Related Functions II

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.12.E1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{1/2}(\sin\theta)^{\mu}}{\pi^{1/2}\Gamma\left(\frac{1}{2}-\mu\right)}\int_{0}^{\theta}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)t\right)}{(\cos t-\cos\theta)^{\mu+(1/2)}}\mathrm{d}t}}$ \FerrersP[\mu]{\nu}@{\cos@@{\theta}} = \frac{2^{1/2}(\sin@@{\theta})^{\mu}}{\pi^{1/2}\EulerGamma@{\frac{1}{2}-\mu}}\int_{0}^{\theta}\frac{\cos@{\left(\nu+\frac{1}{2}\right)t}}{(\cos@@{t}-\cos@@{\theta})^{\mu+(1/2)}}\diff{t}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi,\Re(\frac{1}{2}-\mu)>0}}$	LegendreP(nu, mu, cos(theta)) = ((2)^(1/2)*(sin(theta))^(mu))/((Pi)^(1/2)* GAMMA((1)/(2)- mu))*int((cos((nu +(1)/(2))*t))/((cos(t)- cos(theta))^(mu +(1/2))), t = 0..theta)	LegendreP[\[Nu], \[Mu], Cos[\[Theta]]] == Divide[(2)^(1/2)*(Sin[\[Theta]])^\[Mu],(Pi)^(1/2)* Gamma[Divide[1,2]- \[Mu]]]*Integrate[Divide[Cos[(\[Nu]+Divide[1,2])*t],(Cos[t]- Cos[\[Theta]])^(\[Mu]+(1/2))], {t, 0, \[Theta]}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
14.12.E2	${\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\left(1-x^{2}\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{x}^{1}\mathsf{P}_{\nu}\left(t\right)(t-x)^{\mu-1}\mathrm{d}t}}$ \FerrersP[-\mu]{\nu}@{x} = \frac{\left(1-x^{2}\right)^{-\mu/2}}{\EulerGamma@{\mu}}\int_{x}^{1}\FerrersP[]{\nu}@{t}(t-x)^{\mu-1}\diff{t}	${\displaystyle{\displaystyle\Re\mu>0,\Re(\mu)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, - mu, x) = ((1 - (x)^(2))^(- mu/2))/(GAMMA(mu))*int(LegendreP(nu, t)*(t - x)^(mu - 1), t = x..1)	LegendreP[\[Nu], - \[Mu], x] == Divide[(1 - (x)^(2))^(- \[Mu]/2),Gamma[\[Mu]]]*Integrate[LegendreP[\[Nu], t]*(t - x)^(\[Mu]- 1), {t, x, 1}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Skipped - Because timed out
14.12.E3	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}\mathrm{d}t+\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta\cosh t)^{\nu+\mu+1}}\mathrm{d}t\right)}}$ \FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \frac{\pi^{1/2}\EulerGamma@{\nu+\mu+1}(\sin@@{\theta})^{\mu}}{2^{\mu+1}\EulerGamma@{\mu+\frac{1}{2}}\EulerGamma@{\nu-\mu+1}}\*\left(\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\mu}}{(\cos@@{\theta}+i\sin@@{\theta}\cosh@@{t})^{\nu+\mu+1}}\diff{t}+\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\mu}}{(\cos@@{\theta}-i\sin@@{\theta}\cosh@@{t})^{\nu+\mu+1}}\diff{t}\right)	${\displaystyle{\displaystyle 0<\theta,\theta<\pi,\Re\mu>-\tfrac{1}{2},\Re\nu+\mu>-1,\Re\nu-\mu>-1,\Re(\nu+\mu+1)>0,\Re(\mu+\frac{1}{2})>0,\Re(\nu-\mu+1)>0}}$	LegendreQ(nu, mu, cos(theta)) = ((Pi)^(1/2)* GAMMA(nu + mu + 1)*(sin(theta))^(mu))/((2)^(mu + 1)* GAMMA(mu +(1)/(2))*GAMMA(nu - mu + 1))*(int(((sinh(t))^(2*mu))/((cos(theta)+ I*sin(theta)*cosh(t))^(nu + mu + 1)), t = 0..infinity)+ int(((sinh(t))^(2*mu))/((cos(theta)- I*sin(theta)*cosh(t))^(nu + mu + 1)), t = 0..infinity))	LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == Divide[(Pi)^(1/2)* Gamma[\[Nu]+ \[Mu]+ 1]*(Sin[\[Theta]])^\[Mu],(2)^(\[Mu]+ 1)* Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]- \[Mu]+ 1]]*(Integrate[Divide[(Sinh[t])^(2*\[Mu]),(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]+ Integrate[Divide[(Sinh[t])^(2*\[Mu]),(Cos[\[Theta]]- I*Sin[\[Theta]]*Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None])	Error	Aborted	-	Skipped - Because timed out
14.12.E4	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{2^{1/2}\Gamma\left(\mu+\frac{1}{2}\right)\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\*\int_{0}^{\infty}\frac{\cosh\left(\left(\nu+\frac{1}{2}\right)t\right)}{(x+\cosh t)^{\mu+(1/2)}}\mathrm{d}t}}$ \assLegendreP[-\mu]{\nu}@{x} = \frac{2^{1/2}\EulerGamma@{\mu+\frac{1}{2}}\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\EulerGamma@{\nu+\mu+1}\EulerGamma@{\mu-\nu}}\*\int_{0}^{\infty}\frac{\cosh@{\left(\nu+\frac{1}{2}\right)t}}{(x+\cosh@@{t})^{\mu+(1/2)}}\diff{t}	${\displaystyle{\displaystyle\Re\left(\mu-\nu\right)>0,\Re(\mu+\frac{1}{2})>0,\Re(\nu+\mu+1)>0,\Re(\mu-\nu)>0}}$	LegendreP(nu, - mu, x) = ((2)^(1/2)* GAMMA(mu +(1)/(2))*((x)^(2)- 1)^(mu/2))/((Pi)^(1/2)* GAMMA(nu + mu + 1)*GAMMA(mu - nu))* int((cosh((nu +(1)/(2))*t))/((x + cosh(t))^(mu +(1/2))), t = 0..infinity)	LegendreP[\[Nu], - \[Mu], 3, x] == Divide[(2)^(1/2)* Gamma[\[Mu]+Divide[1,2]]*((x)^(2)- 1)^(\[Mu]/2),(Pi)^(1/2)* Gamma[\[Nu]+ \[Mu]+ 1]*Gamma[\[Mu]- \[Nu]]]* Integrate[Divide[Cosh[(\[Nu]+Divide[1,2])*t],(x + Cosh[t])^(\[Mu]+(1/2))], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.12.E5	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\mathrm{d}t}}$ \assLegendreP[-\mu]{\nu}@{x} = \frac{\left(x^{2}-1\right)^{-\mu/2}}{\EulerGamma@{\mu}}\int_{1}^{x}\LegendrepolyP{\nu}@{t}(x-t)^{\mu-1}\diff{t}	${\displaystyle{\displaystyle\Re\mu>0,\Re(\mu)>0}}$	LegendreP(nu, - mu, x) = (((x)^(2)- 1)^(- mu/2))/(GAMMA(mu))*int(LegendreP(nu, t)*(x - t)^(mu - 1), t = 1..x)	LegendreP[\[Nu], - \[Mu], 3, x] == Divide[((x)^(2)- 1)^(- \[Mu]/2),Gamma[\[Mu]]]*Integrate[LegendreP[\[Nu], t]*(x - t)^(\[Mu]- 1), {t, 1, x}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Skipped - Because timed out
14.12.E6	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\mathrm{d}t}}$ \assLegendreOlverQ[\mu]{\nu}@{x} = \frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}\EulerGamma@{\mu+\frac{1}{2}}\EulerGamma@{\nu-\mu+1}}\*\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\mu}}{\left(x+(x^{2}-1)^{1/2}\cosh@@{t}\right)^{\nu+\mu+1}}\diff{t}	${\displaystyle{\displaystyle\Re\left(\nu+1\right)>\Re\mu,\Re\mu>-\tfrac{1}{2},\Re(\mu+\frac{1}{2})>0,\Re(\nu-\mu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((Pi)^(1/2)*((x)^(2)- 1)^(mu/2))/((2)^(mu)* GAMMA(mu +(1)/(2))*GAMMA(nu - mu + 1))* int(((sinh(t))^(2*mu))/((x +((x)^(2)- 1)^(1/2)* cosh(t))^(nu + mu + 1)), t = 0..infinity)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(Pi)^(1/2)*((x)^(2)- 1)^(\[Mu]/2),(2)^\[Mu]* Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]- \[Mu]+ 1]]* Integrate[Divide[(Sinh[t])^(2*\[Mu]),(x +((x)^(2)- 1)^(1/2)* Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
14.12.E7	${\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{{\left(\nu+1\right)_{m}}}{\pi}\*\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi\right)^{\nu}\cos\left(m\phi\right)\mathrm{d}\phi}}$ \assLegendreP[m]{\nu}@{x} = \frac{\Pochhammersym{\nu+1}{m}}{\pi}\*\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos@@{\phi}\right)^{\nu}\cos@{m\phi}\diff{\phi}		LegendreP(nu, m, x) = (pochhammer(nu + 1, m))/(Pi)* int((x +((x)^(2)- 1)^(1/2)* cos(phi))^(nu)* cos(m*phi), phi = 0..Pi)	LegendreP[\[Nu], m, 3, x] == Divide[Pochhammer[\[Nu]+ 1, m],Pi]* Integrate[(x +((x)^(2)- 1)^(1/2)* Cos[\[Phi]])^\[Nu]* Cos[m*\[Phi]], {\[Phi], 0, Pi}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Successful [Tested: 90]
14.12.E8	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{2^{m}m!(n+m)!\left(x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi\right)^{n-m}(\sin\phi)^{2m}\mathrm{d}\phi}}$ \assLegendreP[m]{n}@{x} = \frac{2^{m}m!(n+m)!\left(x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos@@{\phi}\right)^{n-m}(\sin@@{\phi})^{2m}\diff{\phi}	${\displaystyle{\displaystyle n\geq m}}$	LegendreP(n, m, x) = ((2)^(m)* factorial(m)*factorial(n + m)*((x)^(2)- 1)^(m/2))/(factorial(2*m)*factorial(n - m)*Pi)*int((x +((x)^(2)- 1)^(1/2)* cos(phi))^(n - m)*(sin(phi))^(2*m), phi = 0..Pi)	LegendreP[n, m, 3, x] == Divide[(2)^(m)* (m)!*(n + m)!*((x)^(2)- 1)^(m/2),(2*m)!*(n - m)!*Pi]*Integrate[(x +((x)^(2)- 1)^(1/2)* Cos[\[Phi]])^(n - m)*(Sin[\[Phi]])^(2*m), {\[Phi], 0, Pi}, GenerateConditions->None]	Error	Aborted	-	Successful [Tested: 18]
14.12.E9	${\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{u}\left(x-\left(x^{2}-1\right)^{1/2}\cosh t\right)^{n}\cosh\left(mt\right)\mathrm{d}t}}$ \assLegendreOlverQ[m]{n}@{x} = \frac{1}{n!}\int_{0}^{u}\left(x-\left(x^{2}-1\right)^{1/2}\cosh@@{t}\right)^{n}\cosh@{mt}\diff{t}		exp(-(m)*Pi*I)*LegendreQ(n,m,x)/GAMMA(n+m+1) = (1)/(factorial(n))*int((x -((x)^(2)- 1)^(1/2)* cosh(t))^(n)* cosh(m*t), t = 0..u)	Exp[-(m) Pi I] LegendreQ[n, m, 3, x]/Gamma[n + m + 1] == Divide[1,(n)!]*Integrate[(x -((x)^(2)- 1)^(1/2)* Cosh[t])^(n)* Cosh[m*t], {t, 0, u}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.12.E10	${\displaystyle{\displaystyle u=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)}}$ u = \frac{1}{2}\ln@{\frac{x+1}{x-1}}		u = (1)/(2)*ln((x + 1)/(x - 1))	u == Divide[1,2]*Log[Divide[x + 1,x - 1]]	Failure	Failure	Failed [30 / 30] Result: .613064480e-1+.5000000000*I Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: .3167192595-1.070796327*I Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[0.06130644756738857, 0.49999999999999994] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]} Result: Complex[0.3167192594503838, -1.0707963267948966] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]} ... skip entries to safe data
14.12.E11	${\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n+1}n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n+m+1}}\mathrm{d}t}}$ \assLegendreOlverQ[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n+1}n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n+m+1}}\diff{t}		exp(-(m)*Pi*I)*LegendreQ(n,m,x)/GAMMA(n+m+1) = (((x)^(2)- 1)^(m/2))/((2)^(n + 1)* factorial(n))*int(((1 - (t)^(2))^(n))/((x - t)^(n + m + 1)), t = - 1..1)	Exp[-(m) Pi I] LegendreQ[n, m, 3, x]/Gamma[n + m + 1] == Divide[((x)^(2)- 1)^(m/2),(2)^(n + 1)* (n)!]*Integrate[Divide[(1 - (t)^(2))^(n),(x - t)^(n + m + 1)], {t, - 1, 1}, GenerateConditions->None]	Failure	Failure	Failed [9 / 27] Result: -.6801747617+Float(undefined)*I Test Values: {x = 1/2, m = 1, n = 1} Result: -.3400873809-Float(infinity)*I Test Values: {x = 1/2, m = 1, n = 2} ... skip entries to safe data	Successful [Tested: 27]
14.12.E12	${\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{(n-m)!}P^{m}_{n}\left(x\right)\int_{x}^{\infty}\frac{\mathrm{d}t}{\left(t^{2}-1\right)\left(\displaystyle P^{m}_{n}\left(t\right)\right)^{2}}}}$ \assLegendreOlverQ[m]{n}@{x} = \frac{1}{(n-m)!}\assLegendreP[m]{n}@{x}\int_{x}^{\infty}\frac{\diff{t}}{\left(t^{2}-1\right)\left(\displaystyle\assLegendreP[m]{n}@{t}\right)^{2}}	${\displaystyle{\displaystyle n\geq m}}$	exp(-(m)*Pi*I)*LegendreQ(n,m,x)/GAMMA(n+m+1) = (1)/(factorial(n - m))*LegendreP(n, m, x)*int((1)/(((t)^(2)- 1)*(LegendreP(n, m, t))^(2)), t = x..infinity)	Exp[-(m) Pi I] LegendreQ[n, m, 3, x]/Gamma[n + m + 1] == Divide[1,(n - m)!]*LegendreP[n, m, 3, x]*Integrate[Divide[1,((t)^(2)- 1)*(LegendreP[n, m, 3, t])^(2)], {t, x, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [6 / 18] Result: -.6801747617-Float(infinity)*I Test Values: {x = 1/2, m = 1, n = 1} Result: -.3400873809-Float(infinity)*I Test Values: {x = 1/2, m = 1, n = 2} ... skip entries to safe data	Skipped - Because timed out
14.12.E13	${\displaystyle{\displaystyle\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}\int_{-1}^{1}\frac{P_{n}\left(t\right)}{x-t}\mathrm{d}t}}$ \assLegendreOlverQ[]{n}@{x} = \frac{1}{2(n!)}\int_{-1}^{1}\frac{\LegendrepolyP{n}@{t}}{x-t}\diff{t}		LegendreQ(n,x)/GAMMA(n+1) = (1)/(2*(factorial(n)))*int((LegendreP(n, t))/(x - t), t = - 1..1)	Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3] == Divide[1,2*((n)!)]*Integrate[Divide[LegendreP[n, t],x - t], {t, - 1, 1}, GenerateConditions->None]	Failure	Aborted	Failed [3 / 9] Result: Float(undefined)-.7853981634*I Test Values: {x = 1/2, n = 1} Result: Float(undefined)+.9817477045e-1*I Test Values: {x = 1/2, n = 2} ... skip entries to safe data	Skipped - Because timed out
14.12.E14	${\displaystyle{\displaystyle\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{\infty}\frac{\mathrm{d}t}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{n+1}}}}$ \assLegendreOlverQ[]{n}@{x} = \frac{1}{n!}\int_{0}^{\infty}\frac{\diff{t}}{\left(x+(x^{2}-1)^{1/2}\cosh@@{t}\right)^{n+1}}		LegendreQ(n,x)/GAMMA(n+1) = (1)/(factorial(n))*int((1)/((x +((x)^(2)- 1)^(1/2)* cosh(t))^(n + 1)), t = 0..infinity)	Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3] == Divide[1,(n)!]*Integrate[Divide[1,(x +((x)^(2)- 1)^(1/2)* Cosh[t])^(n + 1)], {t, 0, Infinity}, GenerateConditions->None]	Aborted	Aborted	Successful [Tested: 9]	Skipped - Because timed out
14.13#Ex1	${\displaystyle{\displaystyle+\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)+\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}\Gamma\left(\nu+\mu+1\right)(2\sin\theta)^{\mu}e^{+(\nu+\mu+1)i\theta}\*\mathbf{F}\left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{+2i\theta}\right)}}$ +\frac{1}{2}\pi i\FerrersP[\mu]{\nu}@{\cos@@{\theta}}+\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{\frac{1}{2}}\EulerGamma@{\nu+\mu+1}(2\sin@@{\theta})^{\mu}e^{+(\nu+\mu+1)i\theta}\*\hyperOlverF@{\nu+\mu+1}{\mu+\frac{1}{2}}{\nu+\frac{3}{2}}{e^{+ 2i\theta}}		+(1)/(2)*Pi*I*LegendreP(nu, mu, cos(theta))+ LegendreQ(nu, mu, cos(theta)) = (Pi)^((1)/(2))* GAMMA(nu + mu + 1)*(2*sin(theta))^(mu)* exp(+(nu + mu + 1)*I*theta)* hypergeom([nu + mu + 1, mu +(1)/(2)], [nu +(3)/(2)], exp(+ 2*I*theta))/GAMMA(nu +(3)/(2))	+Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], Cos[\[Theta]]]+ LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == (Pi)^(Divide[1,2])* Gamma[\[Nu]+ \[Mu]+ 1]*(2*Sin[\[Theta]])^\[Mu]* Exp[+(\[Nu]+ \[Mu]+ 1)*I*\[Theta]]* Hypergeometric2F1Regularized[\[Nu]+ \[Mu]+ 1, \[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Exp[+ 2*I*\[Theta]]]	Failure	Failure	Skipped - Because timed out	Failed [113 / 300] Result: Indeterminate Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]} Result: Indeterminate Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]} ... skip entries to safe data
14.13#Ex1	${\displaystyle{\displaystyle-\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)+\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}\Gamma\left(\nu+\mu+1\right)(2\sin\theta)^{\mu}e^{-(\nu+\mu+1)i\theta}\*\mathbf{F}\left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{-2i\theta}\right)}}$ -\frac{1}{2}\pi i\FerrersP[\mu]{\nu}@{\cos@@{\theta}}+\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{\frac{1}{2}}\EulerGamma@{\nu+\mu+1}(2\sin@@{\theta})^{\mu}e^{-(\nu+\mu+1)i\theta}\*\hyperOlverF@{\nu+\mu+1}{\mu+\frac{1}{2}}{\nu+\frac{3}{2}}{e^{- 2i\theta}}		-(1)/(2)*Pi*I*LegendreP(nu, mu, cos(theta))+ LegendreQ(nu, mu, cos(theta)) = (Pi)^((1)/(2))* GAMMA(nu + mu + 1)*(2*sin(theta))^(mu)* exp(-(nu + mu + 1)*I*theta)* hypergeom([nu + mu + 1, mu +(1)/(2)], [nu +(3)/(2)], exp(- 2*I*theta))/GAMMA(nu +(3)/(2))	-Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], Cos[\[Theta]]]+ LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == (Pi)^(Divide[1,2])* Gamma[\[Nu]+ \[Mu]+ 1]*(2*Sin[\[Theta]])^\[Mu]* Exp[-(\[Nu]+ \[Mu]+ 1)*I*\[Theta]]* Hypergeometric2F1Regularized[\[Nu]+ \[Mu]+ 1, \[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Exp[- 2*I*\[Theta]]]	Failure	Failure	Skipped - Because timed out	Failed [113 / 300] Result: Indeterminate Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]} Result: Indeterminate Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]} ... skip entries to safe data
14.13.E1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{\mu+1}(\sin\theta)^{\mu}}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu+k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*\sin\left((\nu+\mu+2k+1)\theta\right)}}$ \FerrersP[\mu]{\nu}@{\cos@@{\theta}} = \frac{2^{\mu+1}(\sin@@{\theta})^{\mu}}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\EulerGamma@{\nu+\mu+k+1}}{\EulerGamma@{\nu+k+\frac{3}{2}}}\frac{\Pochhammersym{\mu+\frac{1}{2}}{k}}{k!}\*\sin@{(\nu+\mu+2k+1)\theta}	${\displaystyle{\displaystyle\Re(\nu+\mu+k+1)>0,\Re(\nu+k+\frac{3}{2})>0}}$	LegendreP(nu, mu, cos(theta)) = ((2)^(mu + 1)*(sin(theta))^(mu))/((Pi)^(1/2))* sum((GAMMA(nu + mu + k + 1))/(GAMMA(nu + k +(3)/(2)))*(pochhammer(mu +(1)/(2), k))/(factorial(k))* sin((nu + mu + 2*k + 1)*theta), k = 0..infinity)	LegendreP[\[Nu], \[Mu], Cos[\[Theta]]] == Divide[(2)^(\[Mu]+ 1)*(Sin[\[Theta]])^\[Mu],(Pi)^(1/2)]* Sum[Divide[Gamma[\[Nu]+ \[Mu]+ k + 1],Gamma[\[Nu]+ k +Divide[3,2]]]*Divide[Pochhammer[\[Mu]+Divide[1,2], k],(k)!]* Sin[(\[Nu]+ \[Mu]+ 2*k + 1)*\[Theta]], {k, 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Skipped - Because timed out	Failed [127 / 300] Result: Indeterminate Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -1.5]} Result: Indeterminate Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -1.5]} ... skip entries to safe data
14.13.E2	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{1/2}2^{\mu}(\sin\theta)^{\mu}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu+k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*\cos\left((\nu+\mu+2k+1)\theta\right)}}$ \FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{1/2}2^{\mu}(\sin@@{\theta})^{\mu}\*\sum_{k=0}^{\infty}\frac{\EulerGamma@{\nu+\mu+k+1}}{\EulerGamma@{\nu+k+\frac{3}{2}}}\frac{\Pochhammersym{\mu+\frac{1}{2}}{k}}{k!}\*\cos@{(\nu+\mu+2k+1)\theta}	${\displaystyle{\displaystyle\Re(\nu+\mu+k+1)>0,\Re(\nu+k+\frac{3}{2})>0}}$	LegendreQ(nu, mu, cos(theta)) = (Pi)^(1/2)* (2)^(mu)*(sin(theta))^(mu)* sum((GAMMA(nu + mu + k + 1))/(GAMMA(nu + k +(3)/(2)))*(pochhammer(mu +(1)/(2), k))/(factorial(k))* cos((nu + mu + 2*k + 1)*theta), k = 0..infinity)	LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == (Pi)^(1/2)* (2)^\[Mu]*(Sin[\[Theta]])^\[Mu]* Sum[Divide[Gamma[\[Nu]+ \[Mu]+ k + 1],Gamma[\[Nu]+ k +Divide[3,2]]]*Divide[Pochhammer[\[Mu]+Divide[1,2], k],(k)!]* Cos[(\[Nu]+ \[Mu]+ 2*k + 1)*\[Theta]], {k, 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Skipped - Because timed out	Failed [153 / 300] Result: Complex[-0.9838922770586165, -0.844402487080167] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.06813222813420483, 1.1810252600164224] Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 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
14.14#Ex1	${\displaystyle{\displaystyle x_{k}=\tfrac{1}{4}(\nu-\mu-k+1)(\nu+\mu+k)\left(x^{2}-1\right)}}$ x_{k} = \tfrac{1}{4}(\nu-\mu-k+1)(\nu+\mu+k)\left(x^{2}-1\right)		x[k] = (1)/(4)*(nu - mu - k + 1)*(nu + mu + k)*((x)^(2)- 1)	Subscript[x, k] == Divide[1,4]*(\[Nu]- \[Mu]- k + 1)*(\[Nu]+ \[Mu]+ k)*((x)^(2)- 1)	Skipped - no semantic math	Skipped - no semantic math	-	-
14.14#Ex2	${\displaystyle{\displaystyle y_{k}=(\mu+k)x}}$ y_{k} = (\mu+k)x		y[k] = (mu + k)*x	Subscript[y, k] == (\[Mu]+ k)*x	Skipped - no semantic math	Skipped - no semantic math	-	-
14.14#Ex3	${\displaystyle{\displaystyle x_{k}=(\nu+\mu+k)(\nu-\mu+k)}}$ x_{k} = (\nu+\mu+k)(\nu-\mu+k)		x[k] = (nu + mu + k)*(nu - mu + k)	Subscript[x, k] == (\[Nu]+ \[Mu]+ k)*(\[Nu]- \[Mu]+ k)	Skipped - no semantic math	Skipped - no semantic math	-	-
14.14#Ex4	${\displaystyle{\displaystyle y_{k}=(2\nu+2k+1)x}}$ y_{k} = (2\nu+2k+1)x		y[k] = (2*nu + 2*k + 1)*x	Subscript[y, k] == (2*\[Nu]+ 2*k + 1)*x	Skipped - no semantic math	Skipped - no semantic math	-	-
14.15.E6	${\displaystyle{\displaystyle p=\frac{x}{\left(\alpha^{2}x^{2}+1-\alpha^{2}\right)^{1/2}}}}$ p = \frac{x}{\left(\alpha^{2}x^{2}+1-\alpha^{2}\right)^{1/2}}		p = (x)/(((alpha)^(2)* (x)^(2)+ 1 - (alpha)^(2))^(1/2))	p == Divide[x,(\[Alpha]^(2)* (x)^(2)+ 1 - \[Alpha]^(2))^(1/2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.15.E7	${\displaystyle{\displaystyle\rho=\frac{1}{2}\ln\left(\frac{1+p}{1-p}\right)+\frac{1}{2}\alpha\ln\left(\frac{1-\alpha p}{1+\alpha p}\right)}}$ \rho = \frac{1}{2}\ln@{\frac{1+p}{1-p}}+\frac{1}{2}\alpha\ln@{\frac{1-\alpha p}{1+\alpha p}}		rho = (1)/(2)*ln((1 + p)/(1 - p))+(1)/(2)*alpha*ln((1 - alpha*p)/(1 + alpha*p))	\[Rho] == Divide[1,2]*Log[Divide[1 + p,1 - p]]+Divide[1,2]*\[Alpha]*Log[Divide[1 - \[Alpha]*p,1 + \[Alpha]*p]]	Failure	Failure	Failed [300 / 300] Result: 1.030274093+1.413752788*I Test Values: {alpha = 3/2, p = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I} Result: -.3357513108+1.779778192*I Test Values: {alpha = 3/2, p = 1/2*3^(1/2)+1/2*I, rho = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.030274092896748, 1.4137527888462516] Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.3357513108876905, 1.7797781926306904] Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.15.E10	${\displaystyle{\displaystyle\alpha\ln\left(\left(\alpha^{2}+\eta^{2}\right)^{1/2}+\alpha\right)-\alpha\ln\eta-\left(\alpha^{2}+\eta^{2}\right)^{1/2}=\frac{1}{2}\ln\left(\frac{\left(1+\alpha^{2}\right)x^{2}+1-\alpha^{2}-2x\left(\alpha^{2}x^{2}-\alpha^{2}+1\right)^{1/2}}{\left(x^{2}-1\right)\left(1-\alpha^{2}\right)}\right)+\frac{1}{2}\alpha\ln\left(\frac{\alpha^{2}\left(2x^{2}-1\right)+1+2\alpha x\left(\alpha^{2}x^{2}-\alpha^{2}+1\right)^{1/2}}{1-\alpha^{2}}\right)}}$ \alpha\ln@{\left(\alpha^{2}+\eta^{2}\right)^{1/2}+\alpha}-\alpha\ln@@{\eta}-\left(\alpha^{2}+\eta^{2}\right)^{1/2} = \frac{1}{2}\ln@{\frac{\left(1+\alpha^{2}\right)x^{2}+1-\alpha^{2}-2x\left(\alpha^{2}x^{2}-\alpha^{2}+1\right)^{1/2}}{\left(x^{2}-1\right)\left(1-\alpha^{2}\right)}}+\frac{1}{2}\alpha\ln@{\frac{\alpha^{2}\left(2x^{2}-1\right)+1+2\alpha x\left(\alpha^{2}x^{2}-\alpha^{2}+1\right)^{1/2}}{1-\alpha^{2}}}		alpha*ln(((alpha)^(2)+ (eta)^(2))^(1/2)+ alpha)- alpha*ln(eta)-((alpha)^(2)+ (eta)^(2))^(1/2) = (1)/(2)*ln(((1 + (alpha)^(2))*(x)^(2)+ 1 - (alpha)^(2)- 2*x*((alpha)^(2)* (x)^(2)- (alpha)^(2)+ 1)^(1/2))/(((x)^(2)- 1)*(1 - (alpha)^(2))))+(1)/(2)*alpha*ln(((alpha)^(2)*(2*(x)^(2)- 1)+ 1 + 2*alpha*x*((alpha)^(2)* (x)^(2)- (alpha)^(2)+ 1)^(1/2))/(1 - (alpha)^(2)))	\[Alpha]*Log[(\[Alpha]^(2)+ \[Eta]^(2))^(1/2)+ \[Alpha]]- \[Alpha]*Log[\[Eta]]-(\[Alpha]^(2)+ \[Eta]^(2))^(1/2) == Divide[1,2]*Log[Divide[(1 + \[Alpha]^(2))*(x)^(2)+ 1 - \[Alpha]^(2)- 2*x*(\[Alpha]^(2)* (x)^(2)- \[Alpha]^(2)+ 1)^(1/2),((x)^(2)- 1)*(1 - \[Alpha]^(2))]]+Divide[1,2]*\[Alpha]*Log[Divide[\[Alpha]^(2)*(2*(x)^(2)- 1)+ 1 + 2*\[Alpha]*x*(\[Alpha]^(2)* (x)^(2)- \[Alpha]^(2)+ 1)^(1/2),1 - \[Alpha]^(2)]]	Failure	Failure	Failed [90 / 90] Result: -.909045744-4.848897315*I Test Values: {alpha = 3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: .6116511952e-1+1.209222406*I Test Values: {alpha = 3/2, eta = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[-0.9090457411289452, -4.848897314881391] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.7450466678010295, -6.916529733960363] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.15.E20	${\displaystyle{\displaystyle\beta=e^{\mu}\left(\frac{\nu-\mu+\frac{1}{2}}{\nu+\mu+\frac{1}{2}}\right)^{(\nu/2)+(1/4)}\left(\left(\nu+\tfrac{1}{2}\right)^{2}-\mu^{2}\right)^{-\mu/2}}}$ \beta = e^{\mu}\left(\frac{\nu-\mu+\frac{1}{2}}{\nu+\mu+\frac{1}{2}}\right)^{(\nu/2)+(1/4)}\left(\left(\nu+\tfrac{1}{2}\right)^{2}-\mu^{2}\right)^{-\mu/2}		beta = exp(mu)*((nu - mu +(1)/(2))/(nu + mu +(1)/(2)))^((nu/2)+(1/4))*((nu +(1)/(2))^(2)- (mu)^(2))^(- mu/2)	\[Beta] == Exp[\[Mu]]*(Divide[\[Nu]- \[Mu]+Divide[1,2],\[Nu]+ \[Mu]+Divide[1,2]])^((\[Nu]/2)+(1/4))*((\[Nu]+Divide[1,2])^(2)- \[Mu]^(2))^(- \[Mu]/2)	Skipped - no semantic math	Skipped - no semantic math	-	-
14.15.E21	${\displaystyle{\displaystyle\left(y-\alpha^{2}\right)^{1/2}-\alpha\operatorname{arctan}\left(\frac{\left(y-\alpha^{2}\right)^{1/2}}{\alpha}\right)=\operatorname{arccos}\left(\frac{x}{\left(1-\alpha^{2}\right)^{1/2}}\right)-\frac{\alpha}{2}\operatorname{arccos}\left(\frac{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}}{\left(1-\alpha^{2}\right)\left(1-x^{2}\right)}\right)}}$ \left(y-\alpha^{2}\right)^{1/2}-\alpha\atan@{\frac{\left(y-\alpha^{2}\right)^{1/2}}{\alpha}} = \acos@{\frac{x}{\left(1-\alpha^{2}\right)^{1/2}}}-\frac{\alpha}{2}\acos@{\frac{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}}{\left(1-\alpha^{2}\right)\left(1-x^{2}\right)}}	${\displaystyle{\displaystyle x\leq\left(1-\alpha^{2}\right)^{1/2},y\geq\alpha^{2}}}$	(y - (alpha)^(2))^(1/2)- alpha*arctan(((y - (alpha)^(2))^(1/2))/(alpha)) = arccos((x)/((1 - (alpha)^(2))^(1/2)))-(alpha)/(2)*arccos(((1 + (alpha)^(2))*(x)^(2)- 1 + (alpha)^(2))/((1 - (alpha)^(2))*(1 - (x)^(2))))	(y - \[Alpha]^(2))^(1/2)- \[Alpha]*ArcTan[Divide[(y - \[Alpha]^(2))^(1/2),\[Alpha]]] == ArcCos[Divide[x,(1 - \[Alpha]^(2))^(1/2)]]-Divide[\[Alpha],2]*ArcCos[Divide[(1 + \[Alpha]^(2))*(x)^(2)- 1 + \[Alpha]^(2),(1 - \[Alpha]^(2))*(1 - (x)^(2))]]	Error	Failure	-	Failed [3 / 3] Result: 0.2030660835403072 Test Values: {Rule[x, 0.5], Rule[y, 1.5], Rule[α, 0.5]} Result: -0.23253599115284607 Test Values: {Rule[x, 0.5], Rule[y, 0.5], Rule[α, 0.5]} ... skip entries to safe data
14.15.E22	${\displaystyle{\displaystyle{\left(\alpha^{2}-y\right)^{1/2}+\tfrac{1}{2}\alpha\ln|y|-\alpha\ln\left(\left(\alpha^{2}-y\right)^{1/2}+\alpha\right)}={\ln\left(\frac{x+\left(x^{2}-1+\alpha^{2}\right)^{1/2}}{\left(1-\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(\frac{\left(1-\alpha^{2}\right)\left|1-x^{2}\right|}{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}+2\alpha x\left(x^{2}-1+\alpha^{2}\right)^{1/2}}\right)}}}$ {\left(\alpha^{2}-y\right)^{1/2}+\tfrac{1}{2}\alpha\ln@@{|y|}-\alpha\ln@{\left(\alpha^{2}-y\right)^{1/2}+\alpha}} = {\ln@{\frac{x+\left(x^{2}-1+\alpha^{2}\right)^{1/2}}{\left(1-\alpha^{2}\right)^{1/2}}}+\frac{\alpha}{2}\ln@{\frac{\left(1-\alpha^{2}\right)\left|1-x^{2}\right|}{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}+2\alpha x\left(x^{2}-1+\alpha^{2}\right)^{1/2}}}}	${\displaystyle{\displaystyle x\geq\left(1-\alpha^{2}\right)^{1/2},y\leq\alpha^{2}}}$	((alpha)^(2)- y)^(1/2)+(1)/(2)*alpha*ln(abs(y))- alpha*ln(((alpha)^(2)- y)^(1/2)+ alpha) = ln((x +((x)^(2)- 1 + (alpha)^(2))^(1/2))/((1 - (alpha)^(2))^(1/2)))+(alpha)/(2)*ln(((1 - (alpha)^(2))*abs(1 - (x)^(2)))/((1 + (alpha)^(2))*(x)^(2)- 1 + (alpha)^(2)+ 2*alpha*x*((x)^(2)- 1 + (alpha)^(2))^(1/2)))	(\[Alpha]^(2)- y)^(1/2)+Divide[1,2]*\[Alpha]*Log[Abs[y]]- \[Alpha]*Log[(\[Alpha]^(2)- y)^(1/2)+ \[Alpha]] == Log[Divide[x +((x)^(2)- 1 + \[Alpha]^(2))^(1/2),(1 - \[Alpha]^(2))^(1/2)]]+Divide[\[Alpha],2]*Log[Divide[(1 - \[Alpha]^(2))*Abs[1 - (x)^(2)],(1 + \[Alpha]^(2))*(x)^(2)- 1 + \[Alpha]^(2)+ 2*\[Alpha]*x*((x)^(2)- 1 + \[Alpha]^(2))^(1/2)]]	Failure	Aborted	Failed [6 / 6] Result: .3341726928 Test Values: {alpha = 1/2, x = 3/2, y = -3/2} Result: -.2530756688 Test Values: {alpha = 1/2, x = 3/2, y = -1/2} ... skip entries to safe data	Failed [6 / 6] Result: 0.3341726912133833 Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 0.5]} Result: -0.25307566945970117 Test Values: {Rule[x, 1.5], Rule[y, -0.5], Rule[α, 0.5]} ... skip entries to safe data
14.15#Ex3	${\displaystyle{\displaystyle a=\frac{\left(\left(\nu+\mu+\frac{1}{2}\right)\left|\nu-\mu+\frac{1}{2}\right|\right)^{1/2}}{\nu+\frac{1}{2}}}}$ a = \frac{\left(\left(\nu+\mu+\frac{1}{2}\right)\left|\nu-\mu+\frac{1}{2}\right|\right)^{1/2}}{\nu+\frac{1}{2}}		a = (((nu + mu +(1)/(2))*abs(nu - mu +(1)/(2)))^(1/2))/(nu +(1)/(2))	a == Divide[((\[Nu]+ \[Mu]+Divide[1,2])*Abs[\[Nu]- \[Mu]+Divide[1,2]])^(1/2),\[Nu]+Divide[1,2]]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.15#Ex4	${\displaystyle{\displaystyle\alpha=\left(\frac{2\left|\nu-\mu+\frac{1}{2}\right|}{\nu+\frac{1}{2}}\right)^{1/2}}}$ \alpha = \left(\frac{2\left|\nu-\mu+\frac{1}{2}\right|}{\nu+\frac{1}{2}}\right)^{1/2}		alpha = ((2*abs(nu - mu +(1)/(2)))/(nu +(1)/(2)))^(1/2)	\[Alpha] == (Divide[2*Abs[\[Nu]- \[Mu]+Divide[1,2]],\[Nu]+Divide[1,2]])^(1/2)	Skipped - no semantic math	Skipped - no semantic math	-	-
14.15.E27	${\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}-\alpha^{2}\right)^{1/2}-\frac{1}{2}\alpha^{2}\operatorname{arccosh}\left(\frac{\zeta}{\alpha}\right)=\left(1-a^{2}\right)^{1/2}\operatorname{arctanh}\left(\frac{1}{x}\left(\frac{x^{2}-a^{2}}{1-a^{2}}\right)^{1/2}\right)-\operatorname{arccosh}\left(\frac{x}{a}\right)}}$ \frac{1}{2}\zeta\left(\zeta^{2}-\alpha^{2}\right)^{1/2}-\frac{1}{2}\alpha^{2}\acosh@{\frac{\zeta}{\alpha}} = \left(1-a^{2}\right)^{1/2}\atanh@{\frac{1}{x}\left(\frac{x^{2}-a^{2}}{1-a^{2}}\right)^{1/2}}-\acosh@{\frac{x}{a}}	${\displaystyle{\displaystyle a\leq x,x<1,\alpha\leq\zeta,\zeta<\infty}}$	(1)/(2)*zeta*((zeta)^(2)- (alpha)^(2))^(1/2)-(1)/(2)*(alpha)^(2)* arccosh((zeta)/(alpha)) = (1 - (a)^(2))^(1/2)* arctanh((1)/(x)*(((x)^(2)- (a)^(2))/(1 - (a)^(2)))^(1/2))- arccosh((x)/(a))	Divide[1,2]*\[Zeta]*(\[Zeta]^(2)- \[Alpha]^(2))^(1/2)-Divide[1,2]*\[Alpha]^(2)* ArcCosh[Divide[\[Zeta],\[Alpha]]] == (1 - (a)^(2))^(1/2)* ArcTanh[Divide[1,x]*(Divide[(x)^(2)- (a)^(2),1 - (a)^(2)])^(1/2)]- ArcCosh[Divide[x,a]]	Failure	Failure	Failed [21 / 24] Result: -1.756203683+1.443241358*I Test Values: {a = -3/2, alpha = 3/2, x = 1/2, zeta = 3/2} Result: -1.328114170+1.443241358*I Test Values: {a = -3/2, alpha = 3/2, x = 1/2, zeta = 2} ... skip entries to safe data	Failed [21 / 24] Result: Complex[-1.7562036827601817, 1.4432413585571147] Test Values: {Rule[a, -1.5], Rule[x, 0.5], Rule[α, 1.5], Rule[ζ, 1.5]} Result: Complex[-1.32811417110478, 1.4432413585571147] Test Values: {Rule[a, -1.5], Rule[x, 0.5], Rule[α, 1.5], Rule[ζ, 2]} ... skip entries to safe data
14.15.E29	${\displaystyle{\displaystyle\zeta^{2}=-\ln\left(1-x^{2}\right)}}$ \zeta^{2} = -\ln@{1-x^{2}}	${\displaystyle{\displaystyle-1<x,x<1}}$	(zeta)^(2) = - ln(1 - (x)^(2))	\[Zeta]^(2) == - Log[1 - (x)^(2)]	Failure	Failure	Failed [10 / 10] Result: .2123179279+.8660254040*I Test Values: {x = 1/2, zeta = 1/2*3^(1/2)+1/2*I} Result: -.7876820729-.8660254040*I Test Values: {x = 1/2, zeta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [10 / 10] Result: Complex[0.2123179275482192, 0.8660254037844386] Test Values: {Rule[x, 0.5], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.7876820724517807, -0.8660254037844387] Test Values: {Rule[x, 0.5], Rule[ζ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.15.E31	${\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^{1/2}+\frac{1}{2}\alpha^{2}\operatorname{arcsinh}\left(\frac{\zeta}{\alpha}\right)=\left(1+a^{2}\right)^{1/2}\operatorname{arctanh}\left(x\left(\frac{1+a^{2}}{x^{2}+a^{2}}\right)^{1/2}\right)-\operatorname{arcsinh}\left(\frac{x}{a}\right)}}$ \frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^{1/2}+\frac{1}{2}\alpha^{2}\asinh@{\frac{\zeta}{\alpha}} = \left(1+a^{2}\right)^{1/2}\atanh@{x\left(\frac{1+a^{2}}{x^{2}+a^{2}}\right)^{1/2}}-\asinh@{\frac{x}{a}}	${\displaystyle{\displaystyle-1<x,x<1,-\infty<\zeta,\zeta<\infty}}$	(1)/(2)*zeta*((zeta)^(2)+ (alpha)^(2))^(1/2)+(1)/(2)*(alpha)^(2)* arcsinh((zeta)/(alpha)) = (1 + (a)^(2))^(1/2)* arctanh((x((1 + (a)^(2))/((x(+))^(2)*(a)^(2))))^(1/2))- arcsinh((x(a))/($1))	Divide[1,2]*\[Zeta]*(\[Zeta]^(2)+ \[Alpha]^(2))^(1/2)+Divide[1,2]*\[Alpha]^(2)* ArcSinh[Divide[\[Zeta],\[Alpha]]] == (1 + (a)^(2))^(1/2)* ArcTanh[(x[Divide[1 + (a)^(2),(x[+])^(2)*(a)^(2)]])^(1/2)]- ArcSinh[Divide[x[a],$1]]	Failure	Failure	Failed [108 / 108] Result: -4.077558345 Test Values: {a = -3/2, alpha = 3/2, x = 1/2, zeta = -3/2} Result: 1.087512739 Test Values: {a = -3/2, alpha = 3/2, x = 1/2, zeta = 3/2} ... skip entries to safe data	Failed [108 / 108] Result: -4.077558346293386 Test Values: {Rule[a, -1.5], Rule[x, 0.5], Rule[α, 1.5], Rule[ζ, -1.5]} Result: 1.08751273984005 Test Values: {Rule[a, -1.5], Rule[x, 0.5], Rule[α, 1.5], Rule[ζ, 1.5]} ... skip entries to safe data
14.16#Ex1	${\displaystyle{\displaystyle\mu=m+\delta_{\mu}}}$ \mu = m+\delta_{\mu}		mu = m + delta[mu]	\[Mu] == m + Subscript[\[Delta], \[Mu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.16#Ex2	${\displaystyle{\displaystyle\nu=n+\delta_{\nu}}}$ \nu = n+\delta_{\nu}		nu = n + delta[nu]	\[Nu] == n + Subscript[\[Delta], \[Nu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
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]
14.18.E1	${\displaystyle{\displaystyle\mathsf{P}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos\left(m\phi\right)}}$ \FerrersP[]{\nu}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \FerrersP[]{\nu}@{\cos@@{\theta_{1}}}\FerrersP[]{\nu}@{\cos@@{\theta_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\FerrersP[-m]{\nu}@{\cos@@{\theta_{1}}}\FerrersP[m]{\nu}@{\cos@@{\theta_{2}}}\cos@{m\phi}		LegendreP(nu, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)) = LegendreP(nu, cos(theta[1]))*LegendreP(nu, cos(theta[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cos(theta[1]))*LegendreP(nu, m, cos(theta[2]))*cos(m*phi), m = 1..infinity)	LegendreP[\[Nu], Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], Cos[Subscript[\[Theta], 1]]]*LegendreP[\[Nu], Cos[Subscript[\[Theta], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, Cos[Subscript[\[Theta], 1]]]*LegendreP[\[Nu], m, Cos[Subscript[\[Theta], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.18.E2	${\displaystyle{\displaystyle\mathsf{P}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\sum_{m=-n}^{n}(-1)^{m}\mathsf{P}^{-m}_{n}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{n}\left(\cos\theta_{2}\right)\cos\left(m\phi\right)}}$ \FerrersP[]{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \sum_{m=-n}^{n}(-1)^{m}\FerrersP[-m]{n}@{\cos@@{\theta_{1}}}\FerrersP[m]{n}@{\cos@@{\theta_{2}}}\cos@{m\phi}		LegendreP(n, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)) = sum((- 1)^(m)* LegendreP(n, - m, cos(theta[1]))*LegendreP(n, m, cos(theta[2]))*cos(m*phi), m = - n..n)	LegendreP[n, Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]] == Sum[(- 1)^(m)* LegendreP[n, - m, Cos[Subscript[\[Theta], 1]]]*LegendreP[n, m, Cos[Subscript[\[Theta], 2]]]*Cos[m*\[Phi]], {m, - n, n}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.18.E3	${\displaystyle{\displaystyle\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos\left(m\phi\right)}}$ \FerrersQ[]{\nu}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \FerrersP[]{\nu}@{\cos@@{\theta_{1}}}\FerrersQ[]{\nu}@{\cos@@{\theta_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\FerrersP[-m]{\nu}@{\cos@@{\theta_{1}}}\FerrersQ[m]{\nu}@{\cos@@{\theta_{2}}}\cos@{m\phi}		LegendreQ(nu, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)) = LegendreP(nu, cos(theta[1]))*LegendreQ(nu, cos(theta[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cos(theta[1]))*LegendreQ(nu, m, cos(theta[2]))*cos(m*phi), m = 1..infinity)	LegendreQ[\[Nu], Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], Cos[Subscript[\[Theta], 1]]]*LegendreQ[\[Nu], Cos[Subscript[\[Theta], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, Cos[Subscript[\[Theta], 1]]]*LegendreQ[\[Nu], m, Cos[Subscript[\[Theta], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.18.E4	${\displaystyle{\displaystyle P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)P^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right)}}$ \assLegendreP[]{\nu}@{\cosh@@{\xi_{1}}\cosh@@{\xi_{2}}-\sinh@@{\xi_{1}}\sinh@@{\xi_{2}}\cos@@{\phi}} = \assLegendreP[]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreP[]{\nu}@{\cosh@@{\xi_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\assLegendreP[-m]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreP[m]{\nu}@{\cosh@@{\xi_{2}}}\cos@{m\phi}		LegendreP(nu, cosh(xi[1])*cosh(xi[2])- sinh(xi[1])*sinh(xi[2])*cos(phi)) = LegendreP(nu, cosh(xi[1]))*LegendreP(nu, cosh(xi[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cosh(xi[1]))*LegendreP(nu, m, cosh(xi[2]))*cos(m*phi), m = 1..infinity)	LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]*Cosh[Subscript[\[Xi], 2]]- Sinh[Subscript[\[Xi], 1]]*Sinh[Subscript[\[Xi], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreP[\[Nu], m, 3, Cosh[Subscript[\[Xi], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.18.E5	${\displaystyle{\displaystyle Q_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)Q_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)Q^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right)}}$ \assLegendreQ[]{\nu}@{\cosh@@{\xi_{1}}\cosh@@{\xi_{2}}-\sinh@@{\xi_{1}}\sinh@@{\xi_{2}}\cos@@{\phi}} = \assLegendreP[]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreQ[]{\nu}@{\cosh@@{\xi_{2}}}+2\sum_{m=1}^{\infty}(-1)^{m}\assLegendreP[-m]{\nu}@{\cosh@@{\xi_{1}}}\assLegendreQ[m]{\nu}@{\cosh@@{\xi_{2}}}\cos@{m\phi}		LegendreQ(nu, cosh(xi[1])*cosh(xi[2])- sinh(xi[1])*sinh(xi[2])*cos(phi)) = LegendreP(nu, cosh(xi[1]))*LegendreQ(nu, cosh(xi[2]))+ 2*sum((- 1)^(m)* LegendreP(nu, - m, cosh(xi[1]))*LegendreQ(nu, m, cosh(xi[2]))*cos(m*phi), m = 1..infinity)	LegendreQ[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]*Cosh[Subscript[\[Xi], 2]]- Sinh[Subscript[\[Xi], 1]]*Sinh[Subscript[\[Xi], 2]]*Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreQ[\[Nu], 0, 3, Cosh[Subscript[\[Xi], 2]]]+ 2*Sum[(- 1)^(m)* LegendreP[\[Nu], - m, 3, Cosh[Subscript[\[Xi], 1]]]*LegendreQ[\[Nu], m, 3, Cosh[Subscript[\[Xi], 2]]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
14.18.E6	${\displaystyle{\displaystyle(x-y)\sum_{k=0}^{n}(2k+1)P_{k}\left(x\right)P_{k}\left(y\right)=(n+1)\left(P_{n+1}\left(x\right)P_{n}\left(y\right)-P_{n}\left(x\right)P_{n+1}\left(y\right)\right)}}$ (x-y)\sum_{k=0}^{n}(2k+1)\assLegendreP[]{k}@{x}\assLegendreP[]{k}@{y} = (n+1)\left(\assLegendreP[]{n+1}@{x}\assLegendreP[]{n}@{y}-\assLegendreP[]{n}@{x}\assLegendreP[]{n+1}@{y}\right)		(x - y)*sum((2*k + 1)*LegendreP(k, x)*LegendreP(k, y), k = 0..n) = (n + 1)*(LegendreP(n + 1, x)*LegendreP(n, y)- LegendreP(n, x)*LegendreP(n + 1, y))	(x - y)*Sum[(2*k + 1)*LegendreP[k, 0, 3, x]*LegendreP[k, 0, 3, y], {k, 0, n}, GenerateConditions->None] == (n + 1)*(LegendreP[n + 1, 0, 3, x]*LegendreP[n, 0, 3, y]- LegendreP[n, 0, 3, x]*LegendreP[n + 1, 0, 3, y])	Aborted	Aborted	Manual Skip!	Failed [42 / 54] Result: Plus[17.25, Times[0.75, Plus[-28.0625, Times[8.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[, 2], Power[-1.5,<syntaxhighlight lang=mathematica>Result: Plus[-106.73437499999997, Times[0.75, Plus[-28.0625, Times[8.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[, 2], Power[-1.5, 2]], Times[68, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[2, ]]], Times[, Plus[1, ], Plus[-165, Times[-271, ], Times[-162, Power[, 2]], Times[-42, Power[, 3]], Times[-4, Power[, 4]], Times[315, Power[1.5, 2]], Times[531, , Power[1.5, 2]], Times[322, Power[, 2], Power[1.5, 2]], Times[84, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[294, 1.5, -1.5], Times[511, , 1.5, -1.5], Times[318, Power[, 2], 1.5, -1.5], Times[84, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[315, Power[-1.5, 2]], Times[531, , Power[-1.5, 2]], Times[322, Power[, 2], Power[-1.5, 2]], Times[84, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[3, Times[2, ]], Plus[12, Times[7, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[, Plus[1, ], Plus[2, ], Plus[3, ], Plus[4, ], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[1], 0], Equal[[2], Times[1.5, -1.5]], Equal[[3], Plus[Times[1.5, -1.5], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]], Equal[[4], Plus[Times[1.5, -1.5], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]], Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[Times[-2, -1.5], Times[Rational[5, 2], -1.5, Plus[-1, Times[3, Power[-1.5, 2]]]]]]]], Equal[[5], Plus[Times[1.5, -1.5], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]], Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[Times[-2, -1.5], Times[Rational[5, 2], -1.5, Plus[-1, Times[3, Power[-1.5, 2]]]]]], Times[Rational[1, 24], Plus[1, Times[-3, Power[1.5, 2]], Times[-8, Plus[-1, Times[3, Power[1.5, 2]]]], Times[7, 1.5, Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]]]], Plus[1, Times[-3, Power[-1.5, 2]], Times[Rational[1, 2], Plus[1, Times[-3, Power[-1.5, 2]]]], Times[Rational[7, 3], -1.5, Plus[Times[-2, -1.5], Times[Rational[5, 2], -1.5, Plus[-1, Times[3, Power[-1.5, 2]]]]]]]]]]}]][3.0]], Times[4.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Power[Plus[1, ], 2], Plus[7, Times[2, ]], []], Times[Plus[7, Times[2, ]], Plus[1, Times[2, ], Power[, 2], Times[9, 1.5, -1.5], Times[12, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[1, ]]], Times[Plus[55, Times[72, ], Times[30, Power[, 2]], Times[4, Power[, 3]], Times[-105, Power[1.5, 2]], Times[-142, , Power[1.5, 2]], Times[-60, Power[, 2], Power[1.5, 2]], Times[-8, Power[, 3], Power[1.5, 2]], Times[-63, 1.5, -1.5], Times[-102, , 1.5, -1.5], Times[-52, Power[, 2], 1.5, -1.5], Times[-8, Power[, 3], 1.5, -1.5], Times[-105, Power[-1.5, 2]], Times[-142, , Power[-1.5, 2]], Times[-60, Power[, 2], Power[-1.5, 2]], Times[-8, Power[, 3], Power[-1.5, 2]]], [Plus[2, ]]], Times[Plus[-55, Times[-72, ], Times[-30, Power[, 2]], Times[-4, Power[, 3]], Times[105, Power[1.5, 2]], Times[142, , Power[1.5, 2]], Times[60, Power[, 2], Power[1.5, 2]], Times[8, Power[, 3], Power[1.5, 2]], Times[147, 1.5, -1.5], Times[182, , 1.5, -1.5], Times[68, Power[, 2], 1.5, -1.5], Times[8, Power[, 3], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[142, , Power[-1.5, 2]], Times[60, Power[, 2], Power[-1.5, 2]], Times[8, Power[, 3], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, Plus[3, Times[2, ]], Plus[16, Times[8, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[Power[Plus[4, ], 2], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[-3], 0], Equal[[-2], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]], Equal[[-1], Plus[Times[1.5, -1.5], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]], Equal[[0], Plus[1, Times[1.5, -1.5], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]], Equal[[1], Plus[2, Times[1.5, -1.5], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[-1, Times[3, Power[-1.5, 2]]]]]]}]][3.0]]]]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]} ... skip entries to safe data
14.18.E7	${\displaystyle{\displaystyle(x-y)\sum_{k=0}^{n}(2k+1)P_{k}\left(x\right)Q_{k}\left(y\right)=(n+1)\left(P_{n+1}\left(x\right)Q_{n}\left(y\right)-P_{n}\left(x\right)Q_{n+1}\left(y\right)\right)-1}}$ (x-y)\sum_{k=0}^{n}(2k+1)\assLegendreP[]{k}@{x}\assLegendreQ[]{k}@{y} = (n+1)\left(\assLegendreP[]{n+1}@{x}\assLegendreQ[]{n}@{y}-\assLegendreP[]{n}@{x}\assLegendreQ[]{n+1}@{y}\right)-1		(x - y)*sum((2*k + 1)*LegendreP(k, x)*LegendreQ(k, y), k = 0..n) = (n + 1)*(LegendreP(n + 1, x)*LegendreQ(n, y)- LegendreP(n, x)*LegendreQ(n + 1, y))- 1	(x - y)*Sum[(2*k + 1)*LegendreP[k, 0, 3, x]*LegendreQ[k, 0, 3, y], {k, 0, n}, GenerateConditions->None] == (n + 1)*(LegendreP[n + 1, 0, 3, x]*LegendreQ[n, 0, 3, y]- LegendreP[n, 0, 3, x]*LegendreQ[n + 1, 0, 3, y])- 1	Aborted	Aborted	Manual Skip!	Failed [42 / 54] Result: Plus[Complex[-0.38140199474411474, 0.0], Times[3.0, Plus[Times[2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[<syntaxhighlight lang=mathematica>Result: Plus[Complex[2.3599248424792147, 0.0], Times[3.0, Plus[Times[2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Power[Plus[1, ], 3], Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], []], Times[Plus[2, ], Plus[3, ], Plus[7, Times[2, ]], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[9, , 1.5, -1.5], Times[12, Power[, 2], 1.5, -1.5], Times[4, Power[, 3], 1.5, -1.5]], [Plus[1, ]]], Times[-1, , Plus[3, ], Plus[-55, Times[-127, ], Times[-102, Power[, 2]], Times[-34, Power[, 3]], Times[-4, Power[, 4]], Times[105, Power[1.5, 2]], Times[247, , Power[1.5, 2]], Times[202, Power[, 2], Power[1.5, 2]], Times[68, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[126, 1.5, -1.5], Times[267, , 1.5, -1.5], Times[206, Power[, 2], 1.5, -1.5], Times[68, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[247, , Power[-1.5, 2]], Times[202, Power[, 2], Power[-1.5, 2]], Times[68, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[2, ]]], Times[, Plus[1, ], Plus[-165, Times[-271, ], Times[-162, Power[, 2]], Times[-42, Power[, 3]], Times[-4, Power[, 4]], Times[315, Power[1.5, 2]], Times[531, , Power[1.5, 2]], Times[322, Power[, 2], Power[1.5, 2]], Times[84, Power[, 3], Power[1.5, 2]], Times[8, Power[, 4], Power[1.5, 2]], Times[294, 1.5, -1.5], Times[511, , 1.5, -1.5], Times[318, Power[, 2], 1.5, -1.5], Times[84, Power[, 3], 1.5, -1.5], Times[8, Power[, 4], 1.5, -1.5], Times[315, Power[-1.5, 2]], Times[531, , Power[-1.5, 2]], Times[322, Power[, 2], Power[-1.5, 2]], Times[84, Power[, 3], Power[-1.5, 2]], Times[8, Power[, 4], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[3, Times[2, ]], Plus[12, Times[7, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[, Plus[1, ], Plus[2, ], Plus[3, ], Plus[4, ], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[1], 0], Equal[[2], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]]], Equal[[3], Plus[Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]]]], Equal[[4], Plus[Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]]]], Equal[[5], Plus[Times[Rational[1, 6], Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 24], Plus[1, Times[-3, Power[1.5, 2]], Times[-8, Plus[-1, Times[3, Power[1.5, 2]]]], Times[7, 1.5, Plus[Times[-4, 1.5], Times[5, 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]]]], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[7, 3], -1.5, Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[-3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[-3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]]}]][3.0]], DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Power[Plus[1, ], 2], Plus[7, Times[2, ]], []], Times[Plus[7, Times[2, ]], Plus[1, Times[2, ], Power[, 2], Times[9, 1.5, -1.5], Times[12, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[1, ]]], Times[Plus[55, Times[72, ], Times[30, Power[, 2]], Times[4, Power[, 3]], Times[-105, Power[1.5, 2]], Times[-142, , Power[1.5, 2]], Times[-60, Power[, 2], Power[1.5, 2]], Times[-8, Power[, 3], Power[1.5, 2]], Times[-63, 1.5, -1.5], Times[-102, , 1.5, -1.5], Times[-52, Power[, 2], 1.5, -1.5], Times[-8, Power[, 3], 1.5, -1.5], Times[-105, Power[-1.5, 2]], Times[-142, , Power[-1.5, 2]], Times[-60, Power[, 2], Power[-1.5, 2]], Times[-8, Power[, 3], Power[-1.5, 2]]], [Plus[2, ]]], Times[Plus[-55, Times[-72, ], Times[-30, Power[, 2]], Times[-4, Power[, 3]], Times[105, Power[1.5, 2]], Times[142, , Power[1.5, 2]], Times[60, Power[, 2], Power[1.5, 2]], Times[8, Power[, 3], Power[1.5, 2]], Times[147, 1.5, -1.5], Times[182, , 1.5, -1.5], Times[68, Power[, 2], 1.5, -1.5], Times[8, Power[, 3], 1.5, -1.5], Times[105, Power[-1.5, 2]], Times[142, , Power[-1.5, 2]], Times[60, Power[, 2], Power[-1.5, 2]], Times[8, Power[, 3], Power[-1.5, 2]]], [Plus[3, ]]], Times[-1, Plus[3, Times[2, ]], Plus[16, Times[8, ], Power[, 2], Times[49, 1.5, -1.5], Times[28, , 1.5, -1.5], Times[4, Power[, 2], 1.5, -1.5]], [Plus[4, ]]], Times[Power[Plus[4, ], 2], Plus[3, Times[2, ]], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Equal[[2], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Equal[[3], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]], Equal[[4], Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 9], Plus[Times[-2, 1.5], Times[Rational[5, 2], 1.5, Plus[-1, Times[3, Power[1.5, 2]]]]], Plus[2, Times[Rational[5, 2], -1.5, Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[-2, -1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[1, 4], Plus[-1, Times[3, Power[1.5, 2]]], Plus[Times[Rational[1, 2], Log[Plus[-1, -1.5]]], Times[3, -1.5, Plus[-1, Times[-1.5, Plus[Times[Rational[-1, 2], Log[Plus[-1, -1.5]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]]], Times[Rational[-1, 2], Log[Plus[1, -1.5]]]]], Times[Rational[1, 2], Log[Plus[1, -1.5]]]]]}]][3.0]]]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]} ... skip entries to safe data
14.18.E8	${\displaystyle{\displaystyle\mathsf{P}_{\nu}\left(-x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_{n=0}^{\infty}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}_{n}\left(x\right)}}$ \FerrersP[]{\nu}@{-x} = \frac{\sin@{\nu\pi}}{\pi}\sum_{n=0}^{\infty}\frac{2n+1}{(\nu-n)(\nu+n+1)}\FerrersP[]{n}@{x}		LegendreP(nu, - x) = (sin(nu*Pi))/(Pi)*sum((2*n + 1)/((nu - n)*(nu + n + 1))*LegendreP(n, x), n = 0..infinity)	LegendreP[\[Nu], - x] == Divide[Sin[\[Nu]*Pi],Pi]*Sum[Divide[2*n + 1,(\[Nu]- n)*(\[Nu]+ n + 1)]*LegendreP[n, x], {n, 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Failed [3 / 3] Result: Plus[Complex[0.07218102573226806, -2.034342748581157], Times[0.3183098861837907, NSum[Times[Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, 1.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[ν, Rational[3, 2]]} Result: Plus[-0.5703494499205765, Times[0.3183098861837907, NSum[Times[Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, 0.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5], Rule[ν, Rational[3, 2]]} ... skip entries to safe data
14.18.E9	${\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_{n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}^{-\mu}_{n}\left(x\right)}}$ \FerrersP[-\mu]{\nu}@{x} = \frac{\sin@{\nu\pi}}{\pi}\sum_{n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+1)}\FerrersP[-\mu]{n}@{x}	${\displaystyle{\displaystyle-1<x,x\leq 1,\mu\geq 0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, - mu, x) = (sin(nu*Pi))/(Pi)*sum((- 1)^(n)*(2*n + 1)/((nu - n)*(nu + n + 1))*LegendreP(n, - mu, x), n = 0..infinity)	LegendreP[\[Nu], - \[Mu], x] == Divide[Sin[\[Nu]*Pi],Pi]*Sum[(- 1)^(n)*Divide[2*n + 1,(\[Nu]- n)*(\[Nu]+ n + 1)]*LegendreP[n, - \[Mu], x], {n, 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Failed [3 / 3] Result: Plus[0.21434568952624797, Times[0.3183098861837907, NSum[Times[Power[-1, n], Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, -1.5, 0.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5], Rule[μ, 1.5], Rule[ν, Rational[3, 2]]} Result: Plus[0.37125762464284556, Times[0.3183098861837907, NSum[Times[Power[-1, n], Power[Plus[Rational[3, 2], Times[-1, n]], -1], Power[Plus[Rational[5, 2], n], -1], Plus[1, Times[2, n]], LegendreP[n, -0.5, 0.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5], Rule[μ, 0.5], Rule[ν, Rational[3, 2]]} ... skip entries to safe data
14.19#Ex1	${\displaystyle{\displaystyle x=\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta}}}$ x = \frac{c\sinh@@{\eta}\cos@@{\phi}}{\cosh@@{\eta}-\cos@@{\theta}}		x = (c*sinh(eta)*cos(phi))/(cosh(eta)- cos(theta))	x == Divide[c*Sinh[\[Eta]]*Cos[\[Phi]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Failed [300 / 300] Result: 2.362573279-1.052377925*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: 1.362573279-1.052377925*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[2.3625732791062704, -1.0523779253990262] Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.6505283543319873, -0.046280887188208775] Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19#Ex2	${\displaystyle{\displaystyle y=\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta}}}$ y = \frac{c\sinh@@{\eta}\sin@@{\phi}}{\cosh@@{\eta}-\cos@@{\theta}}		y = (c*sinh(eta)*sin(phi))/(cosh(eta)- cos(theta))	y == Divide[c*Sinh[\[Eta]]*Sin[\[Phi]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Failed [300 / 300] Result: .10381346e-1-.1810305231e-1*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = -3/2} Result: 3.010381346-.1810305231e-1*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = 3/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.010381344893815037, -0.01810305210999985] Test Values: {Rule[c, -1.5], Rule[y, -1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.9871098783639947, 1.7153567749591236] Test Values: {Rule[c, -1.5], Rule[y, -1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19#Ex3	${\displaystyle{\displaystyle z=\frac{c\sin\theta}{\cosh\eta-\cos\theta}}}$ z = \frac{c\sin@@{\theta}}{\cosh@@{\eta}-\cos@@{\theta}}		z = (c*sin(theta))/(cosh(eta)- cos(theta))	z == Divide[c*Sin[\[Theta]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Failed [300 / 300] Result: 1.948230727-.3664573554*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .5822053230-.4319514e-3*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.948230726846754, -0.366457355462031] Test Values: {Rule[c, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[7.733911995808641*^15, 6.041410995179728*^15] Test Values: {Rule[c, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19.E2	${\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-\mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}\right)}}$ \assLegendreP[\mu]{\nu-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{\frac{1}{2}-\mu}}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\hyperOlverF@{\tfrac{1}{2}-\mu}{\tfrac{1}{2}+\nu-\mu}{1-2\mu}{1-e^{-2\xi}}	${\displaystyle{\displaystyle\mu\neq\frac{1}{2},\Re(\frac{1}{2}-\mu)>0}}$	LegendreP(nu -(1)/(2), mu, cosh(xi)) = (GAMMA((1)/(2)- mu))/((Pi)^(1/2)*(1 - exp(- 2*xi))^(mu)* exp((nu +(1/2))*xi))* hypergeom([(1)/(2)- mu, (1)/(2)+ nu - mu], [1 - 2*mu], 1 - exp(- 2*xi))/GAMMA(1 - 2*mu)	LegendreP[\[Nu]-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]] == Divide[Gamma[Divide[1,2]- \[Mu]],(Pi)^(1/2)*(1 - Exp[- 2*\[Xi]])^\[Mu]* Exp[(\[Nu]+(1/2))*\[Xi]]]* Hypergeometric2F1Regularized[Divide[1,2]- \[Mu], Divide[1,2]+ \[Nu]- \[Mu], 1 - 2*\[Mu], 1 - Exp[- 2*\[Xi]]]	Aborted	Failure	Successful [Tested: 200]	Successful [Tested: 200]
14.19#Ex4	${\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(1-2\mu\right)2^{2\mu}}{\Gamma\left(1-\mu\right)\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;e^{-2\xi}\right)}}$ \assLegendreP[\mu]{\nu-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{1-2\mu}2^{2\mu}}{\EulerGamma@{1-\mu}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\hyperOlverF@{\tfrac{1}{2}-\mu}{\tfrac{1}{2}+\nu-\mu}{1-2\mu}{e^{-2\xi}}		LegendreP(nu -(1)/(2), mu, cosh(xi)) = (GAMMA(1 - 2*mu)*(2)^(2*mu))/(GAMMA(1 - mu)*(1 - exp(- 2*xi))^(mu)* exp((nu +(1/2))*xi))* hypergeom([(1)/(2)- mu, (1)/(2)+ nu - mu], [1 - 2*mu], exp(- 2*xi))/GAMMA(1 - 2*mu)	LegendreP[\[Nu]-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]] == Divide[Gamma[1 - 2*\[Mu]]*(2)^(2*\[Mu]),Gamma[1 - \[Mu]]*(1 - Exp[- 2*\[Xi]])^\[Mu]* Exp[(\[Nu]+(1/2))*\[Xi]]]* Hypergeometric2F1Regularized[Divide[1,2]- \[Mu], Divide[1,2]+ \[Nu]- \[Mu], 1 - 2*\[Mu], Exp[- 2*\[Xi]]]	Failure	Failure	Failed [300 / 300] Result: .2738102545-.736850267e-1*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I} Result: 3.389539010-1.213206227*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.2738102549490508, -0.07368502759104012] Test Values: {Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.38953901122763, -1.2132062234978649] Test Values: {Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19.E3	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\mu+\tfrac{1}{2},\nu+\mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right)}}$ \assLegendreOlverQ[\mu]{\nu-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\hyperOlverF@{\mu+\tfrac{1}{2}}{\nu+\mu+\tfrac{1}{2}}{\nu+1}{e^{-2\xi}}		exp(-(mu)*Pi*I)*LegendreQ(nu -(1)/(2),mu,cosh(xi))/GAMMA(nu -(1)/(2)+mu+1) = ((Pi)^(1/2)*(1 - exp(- 2*xi))^(mu))/(exp((nu +(1/2))*xi))* hypergeom([mu +(1)/(2), nu + mu +(1)/(2)], [nu + 1], exp(- 2*xi))/GAMMA(nu + 1)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu]-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]]/Gamma[\[Nu]-Divide[1,2] + \[Mu] + 1] == Divide[(Pi)^(1/2)*(1 - Exp[- 2*\[Xi]])^\[Mu],Exp[(\[Nu]+(1/2))*\[Xi]]]* Hypergeometric2F1Regularized[\[Mu]+Divide[1,2], \[Nu]+ \[Mu]+Divide[1,2], \[Nu]+ 1, Exp[- 2*\[Xi]]]	Failure	Failure	Failed [20 / 300] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -2, xi = 1/2*3^(1/2)+1/2*I} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -2, xi = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Failed [10 / 300] Result: Indeterminate Test Values: {Rule[μ, -1.5], Rule[ν, -2], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[μ, -1.5], Rule[ν, -2], Rule[ξ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
14.19.E4	${\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\frac{1}{2}\right)(\sinh\xi)^{m}}{2^{m}\pi^{1/2}\Gamma\left(n-m+\frac{1}{2}\right)\Gamma\left(m+\frac{1}{2}\right)}\*\int_{0}^{\pi}\frac{(\sin\phi)^{2m}}{(\cosh\xi+\cos\phi\sinh\xi)^{n+m+(1/2)}}\mathrm{d}\phi}}$ \assLegendreP[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{n+m+\frac{1}{2}}(\sinh@@{\xi})^{m}}{2^{m}\pi^{1/2}\EulerGamma@{n-m+\frac{1}{2}}\EulerGamma@{m+\frac{1}{2}}}\*\int_{0}^{\pi}\frac{(\sin@@{\phi})^{2m}}{(\cosh@@{\xi}+\cos@@{\phi}\sinh@@{\xi})^{n+m+(1/2)}}\diff{\phi}	${\displaystyle{\displaystyle\Re(n+m+\frac{1}{2})>0,\Re(n-m+\frac{1}{2})>0,\Re(m+\frac{1}{2})>0}}$	LegendreP(n -(1)/(2), m, cosh(xi)) = (GAMMA(n + m +(1)/(2))*(sinh(xi))^(m))/((2)^(m)* (Pi)^(1/2)* GAMMA(n - m +(1)/(2))*GAMMA(m +(1)/(2)))* int(((sin(phi))^(2*m))/((cosh(xi)+ cos(phi)*sinh(xi))^(n + m +(1/2))), phi = 0..Pi)	LegendreP[n -Divide[1,2], m, 3, Cosh[\[Xi]]] == Divide[Gamma[n + m +Divide[1,2]]*(Sinh[\[Xi]])^(m),(2)^(m)* (Pi)^(1/2)* Gamma[n - m +Divide[1,2]]*Gamma[m +Divide[1,2]]]* Integrate[Divide[(Sin[\[Phi]])^(2*m),(Cosh[\[Xi]]+ Cos[\[Phi]]*Sinh[\[Xi]])^(n + m +(1/2))], {\[Phi], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.19.E5	${\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}\right)\Gamma\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(mt\right)}{(\cosh\xi+\cosh t\sinh\xi)^{n+(1/2)}}\mathrm{d}t}}$ \assLegendreOlverQ[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{n+\frac{1}{2}}}{\EulerGamma@{n+m+\tfrac{1}{2}}\EulerGamma@{n-m+\frac{1}{2}}}\*\int_{0}^{\infty}\frac{\cosh@{mt}}{(\cosh@@{\xi}+\cosh@@{t}\sinh@@{\xi})^{n+(1/2)}}\diff{t}	${\displaystyle{\displaystyle m<n+\tfrac{1}{2},\Re(n+\frac{1}{2})>0,\Re(n+m+\tfrac{1}{2})>0,\Re(n-m+\frac{1}{2})>0}}$	exp(-(m)*Pi*I)*LegendreQ(n -(1)/(2),m,cosh(xi))/GAMMA(n -(1)/(2)+m+1) = (GAMMA(n +(1)/(2)))/(GAMMA(n + m +(1)/(2))*GAMMA(n - m +(1)/(2)))* int((cosh(m*t))/((cosh(xi)+ cosh(t)*sinh(xi))^(n +(1/2))), t = 0..infinity)	Exp[-(m) Pi I] LegendreQ[n -Divide[1,2], m, 3, Cosh[\[Xi]]]/Gamma[n -Divide[1,2] + m + 1] == Divide[Gamma[n +Divide[1,2]],Gamma[n + m +Divide[1,2]]*Gamma[n - m +Divide[1,2]]]* Integrate[Divide[Cosh[m*t],(Cosh[\[Xi]]+ Cosh[t]*Sinh[\[Xi]])^(n +(1/2))], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
14.19.E6	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)+2\sum_{n=1}^{\infty}\frac{\Gamma\left(\mu+n+\tfrac{1}{2}\right)}{\Gamma\left(\mu+\tfrac{1}{2}\right)}\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right)\cos\left(n\phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu}}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}}}}$ \assLegendreOlverQ[\mu]{-\frac{1}{2}}@{\cosh@@{\xi}}+2\sum_{n=1}^{\infty}\frac{\EulerGamma@{\mu+n+\tfrac{1}{2}}}{\EulerGamma@{\mu+\tfrac{1}{2}}}\assLegendreOlverQ[\mu]{n-\frac{1}{2}}@{\cosh@@{\xi}}\cos@{n\phi} = \dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh@@{\xi}\right)^{\mu}}{\left(\cosh@@{\xi}-\cos@@{\phi}\right)^{\mu+(1/2)}}	${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re(\mu+n+\tfrac{1}{2})>0,\Re(\mu+\tfrac{1}{2})>0}}$	exp(-(mu)*Pi*I)*LegendreQ(-(1)/(2),mu,cosh(xi))/GAMMA(-(1)/(2)+mu+1)+ 2*sum((GAMMA(mu + n +(1)/(2)))/(GAMMA(mu +(1)/(2)))*exp(-(mu)*Pi*I)*LegendreQ(n -(1)/(2),mu,cosh(xi))/GAMMA(n -(1)/(2)+mu+1)*cos(n*phi), n = 1..infinity) = (((1)/(2)*Pi)^(1/2)*(sinh(xi))^(mu))/((cosh(xi)- cos(phi))^(mu +(1/2)))	Exp[-(\[Mu]) Pi I] LegendreQ[-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]]/Gamma[-Divide[1,2] + \[Mu] + 1]+ 2*Sum[Divide[Gamma[\[Mu]+ n +Divide[1,2]],Gamma[\[Mu]+Divide[1,2]]]*Exp[-(\[Mu]) Pi I] LegendreQ[n -Divide[1,2], \[Mu], 3, Cosh[\[Xi]]]/Gamma[n -Divide[1,2] + \[Mu] + 1]*Cos[n*\[Phi]], {n, 1, Infinity}, GenerateConditions->None] == Divide[(Divide[1,2]*Pi)^(1/2)*(Sinh[\[Xi]])^\[Mu],(Cosh[\[Xi]]- Cos[\[Phi]])^(\[Mu]+(1/2))]	Failure	Failure	Skipped - Because timed out	Skipped - Because timed out
14.19.E7	${\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\tfrac{1}{2}\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}\left(\coth\xi\right)}}$ \assLegendreP[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{n+m+\tfrac{1}{2}}}{\EulerGamma@{n-m+\tfrac{1}{2}}}\*\left(\frac{2}{\pi\sinh@@{\xi}}\right)^{1/2}\assLegendreOlverQ[n]{m-\frac{1}{2}}@{\coth@@{\xi}}	${\displaystyle{\displaystyle\Re(n+m+\tfrac{1}{2})>0,\Re(n-m+\tfrac{1}{2})>0}}$	LegendreP(n -(1)/(2), m, cosh(xi)) = (GAMMA(n + m +(1)/(2)))/(GAMMA(n - m +(1)/(2)))*((2)/(Pi*sinh(xi)))^(1/2)* exp(-(n)*Pi*I)*LegendreQ(m -(1)/(2),n,coth(xi))/GAMMA(m -(1)/(2)+n+1)	LegendreP[n -Divide[1,2], m, 3, Cosh[\[Xi]]] == Divide[Gamma[n + m +Divide[1,2]],Gamma[n - m +Divide[1,2]]]*(Divide[2,Pi*Sinh[\[Xi]]])^(1/2)* Exp[-(n) Pi I] LegendreQ[m -Divide[1,2], n, 3, Coth[\[Xi]]]/Gamma[m -Divide[1,2] + n + 1]	Failure	Failure	Failed [20 / 60] Result: .3683324082-.6470690126*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1, n = 1} Result: .5135733695-3.117174531*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1, n = 2} ... skip entries to safe data	Failed [20 / 60] Result: Complex[0.36833240837635506, -0.6470690125104284] Test Values: {Rule[m, 1], Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.5135733718660924, -3.117174532097865] Test Values: {Rule[m, 1], Rule[n, 2], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19.E8	${\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(m-n+\tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2}\right)}\*\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}P^{n}_{m-\frac{1}{2}}\left(\coth\xi\right)}}$ \assLegendreOlverQ[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{m-n+\tfrac{1}{2}}}{\EulerGamma@{m+n+\tfrac{1}{2}}}\*\left(\frac{\pi}{2\sinh@@{\xi}}\right)^{1/2}\assLegendreP[n]{m-\frac{1}{2}}@{\coth@@{\xi}}	${\displaystyle{\displaystyle\Re(m-n+\tfrac{1}{2})>0,\Re(m+n+\tfrac{1}{2})>0}}$	exp(-(m)*Pi*I)*LegendreQ(n -(1)/(2),m,cosh(xi))/GAMMA(n -(1)/(2)+m+1) = (GAMMA(m - n +(1)/(2)))/(GAMMA(m + n +(1)/(2)))*((Pi)/(2*sinh(xi)))^(1/2)* LegendreP(m -(1)/(2), n, coth(xi))	Exp[-(m) Pi I] LegendreQ[n -Divide[1,2], m, 3, Cosh[\[Xi]]]/Gamma[n -Divide[1,2] + m + 1] == Divide[Gamma[m - n +Divide[1,2]],Gamma[m + n +Divide[1,2]]]*(Divide[Pi,2*Sinh[\[Xi]]])^(1/2)* LegendreP[m -Divide[1,2], n, 3, Coth[\[Xi]]]	Failure	Failure	Failed [30 / 60] Result: .7427758821+1.946023521*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1, n = 1} Result: -.1057063209+.477539648e-1*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 2, n = 1} ... skip entries to safe data	Failed [30 / 60] Result: Complex[0.7427758815190426, 1.9460235199869547] Test Values: {Rule[m, 1], Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.10570632113064243, 0.04775396399318543] Test Values: {Rule[m, 2], Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E1	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}\right)w=0}}$ \left(1-x^{2}\right)\deriv[2]{w}{x}-2x\deriv{w}{x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}\right)w = 0		(1 - (x)^(2))*diff(w, [x$(2)])- 2*x*diff(w, x)-((tau)^(2)+(1)/(4)+((mu)^(2))/(1 - (x)^(2)))*w = 0	(1 - (x)^(2))*D[w, {x, 2}]- 2*x*D[w, x]-(\[Tau]^(2)+Divide[1,4]+Divide[\[Mu]^(2),1 - (x)^(2)])*w == 0	Failure	Failure	Failed [300 / 300] Result: -.2165063511-.3250000001*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.2165063516-2.458333334*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 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[-0.2165063509461097, -0.32499999999999996] 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[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.2165063509461096, 1.675] 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[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E4	${\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right),\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(-x\right)\right\}=\frac{2}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}(1-x^{2})}}}$ \Wronskian@{\FerrersP[-\mu]{-\frac{1}{2}+\iunit\tau}@{x},\FerrersP[-\mu]{-\frac{1}{2}+\iunit\tau}@{-x}} = \frac{2}{|\EulerGamma@{\mu+\frac{1}{2}+\iunit\tau}|^{2}(1-x^{2})}	${\displaystyle{\displaystyle\Re(\mu+\frac{1}{2}+\mathrm{i}\tau)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1}}$	(LegendreP(-(1)/(2)+ I*tau, - mu, x))*diff(LegendreP(-(1)/(2)+ I*tau, - mu, - x), x)-diff(LegendreP(-(1)/(2)+ I*tau, - mu, x), x)*(LegendreP(-(1)/(2)+ I*tau, - mu, - x)) = (2)/((abs(GAMMA(mu +(1)/(2)+ I*tau)))^(2)*(1 - (x)^(2)))	Wronskian[{LegendreP[-Divide[1,2]+ I*\[Tau], - \[Mu], x], LegendreP[-Divide[1,2]+ I*\[Tau], - \[Mu], - x]}, x] == Divide[2,(Abs[Gamma[\[Mu]+Divide[1,2]+ I*\[Tau]]])^(2)*(1 - (x)^(2))]	Failure	Failure	Failed [38 / 56] Result: -17.04997320+4.383607823*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, x = 1/2} Result: .5897199763-1.005797385*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = -1/2+1/2*I*3^(1/2), x = 1/2} ... skip entries to safe data	Failed [38 / 56] Result: Complex[-17.049973187296022, 4.383607825965987] Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.5897199767717201, -1.0057973854572255] Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E6	${\displaystyle{\displaystyle P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{ie^{-\mu\pi i}}{\sinh\left(\tau\pi\right)\left|\Gamma\left(\mu+\frac{1}{2}+i\tau\right)\right|^{2}}\*\left(Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)-Q^{\mu}_{-\frac{1}{2}-i\tau}\left(x\right)\right)}}$ \assLegendreP[-\mu]{-\frac{1}{2}+i\tau}@{x} = \frac{ie^{-\mu\pi i}}{\sinh@{\tau\pi}\left|\EulerGamma@{\mu+\frac{1}{2}+i\tau}\right|^{2}}\*\left(\assLegendreQ[\mu]{-\frac{1}{2}+i\tau}@{x}-\assLegendreQ[\mu]{-\frac{1}{2}-i\tau}@{x}\right)	${\displaystyle{\displaystyle\tau\neq 0,\Re(\mu+\frac{1}{2}+\mathrm{i}\tau)>0}}$	LegendreP(-(1)/(2)+ I*tau, - mu, x) = (I*exp(- mu*Pi*I))/(sinh(tau*Pi)*(abs(GAMMA(mu +(1)/(2)+ I*tau)))^(2))*(LegendreQ(-(1)/(2)+ I*tau, mu, x)- LegendreQ(-(1)/(2)- I*tau, mu, x))	LegendreP[-Divide[1,2]+ I*\[Tau], - \[Mu], 3, x] == Divide[I*Exp[- \[Mu]*Pi*I],Sinh[\[Tau]*Pi]*(Abs[Gamma[\[Mu]+Divide[1,2]+ I*\[Tau]]])^(2)]*(LegendreQ[-Divide[1,2]+ I*\[Tau], \[Mu], 3, x]- LegendreQ[-Divide[1,2]- I*\[Tau], \[Mu], 3, x])	Failure	Failure	Failed [114 / 168] Result: -.1488817069+.9881458426*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.7084727976-.1684769573*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [114 / 168] Result: Complex[-0.14888170656920197, 0.9881458430062731] Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.24375508302595367, -0.3184001443616234] Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E9	${\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{2}{\pi}\int_{0}^{\theta}\frac{\cosh\left(\tau\phi\right)}{\sqrt{2(\cos\phi-\cos\theta)}}\mathrm{d}\phi}}$ \FerrersP[]{-\frac{1}{2}+i\tau}@{\cos@@{\theta}} = \frac{2}{\pi}\int_{0}^{\theta}\frac{\cosh@{\tau\phi}}{\sqrt{2(\cos@@{\phi}-\cos@@{\theta})}}\diff{\phi}		LegendreP(-(1)/(2)+ I*tau, cos(theta)) = (2)/(Pi)*int((cosh(tau*phi))/(sqrt(2*(cos(phi)- cos(theta)))), phi = 0..theta)	LegendreP[-Divide[1,2]+ I*\[Tau], Cos[\[Theta]]] == Divide[2,Pi]*Integrate[Divide[Cosh[\[Tau]*\[Phi]],Sqrt[2*(Cos[\[Phi]]- Cos[\[Theta]])]], {\[Phi], 0, \[Theta]}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.20.E13	${\displaystyle{\displaystyle P_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\cosh\left(\tau\pi\right)}{\pi}\int_{1}^{\infty}\frac{P_{-\frac{1}{2}+i\tau}\left(t\right)}{x+t}\mathrm{d}t}}$ \assLegendreP[]{-\frac{1}{2}+i\tau}@{x} = \frac{\cosh@{\tau\pi}}{\pi}\int_{1}^{\infty}\frac{\assLegendreP[]{-\frac{1}{2}+i\tau}@{t}}{x+t}\diff{t}		LegendreP(-(1)/(2)+ I*tau, x) = (cosh(tau*Pi))/(Pi)*int((LegendreP(-(1)/(2)+ I*tau, t))/(x + t), t = 1..infinity)	LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, x] == Divide[Cosh[\[Tau]*Pi],Pi]*Integrate[Divide[LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, t],x + t], {t, 1, Infinity}, GenerateConditions->None]	Failure	Aborted	Manual Skip!	Skipped - Because timed out
14.20.E14	${\displaystyle{\displaystyle\pi\int_{0}^{\infty}\frac{\tau\tanh\left(\tau\pi\right)}{\cosh\left(\tau\pi\right)}P_{-\frac{1}{2}+i\tau}\left(x\right)P_{-\frac{1}{2}+i\tau}\left(y\right)\mathrm{d}\tau=\frac{1}{y+x}}}$ \pi\int_{0}^{\infty}\frac{\tau\tanh@{\tau\pi}}{\cosh@{\tau\pi}}\assLegendreP[]{-\frac{1}{2}+i\tau}@{x}\assLegendreP[]{-\frac{1}{2}+i\tau}@{y}\diff{\tau} = \frac{1}{y+x}		Pi*int((tau*tanh(tau*Pi))/(cosh(tau*Pi))*LegendreP(-(1)/(2)+ I*tau, x)*LegendreP(-(1)/(2)+ I*tau, y), tau = 0..infinity) = (1)/(y + x)	Pi*Integrate[Divide[\[Tau]*Tanh[\[Tau]*Pi],Cosh[\[Tau]*Pi]]*LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, x]*LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, y], {\[Tau], 0, Infinity}, GenerateConditions->None] == Divide[1,y + x]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.20.E19	${\displaystyle{\displaystyle\alpha=\mu/\tau}}$ \alpha = \mu/\tau		alpha = mu/tau	\[Alpha] == \[Mu]/\[Tau]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.20.E20	${\displaystyle{\displaystyle\sigma(\mu,\tau)=\frac{\exp\left(\mu-\tau\operatorname{arctan}\alpha\right)}{\left(\mu^{2}+\tau^{2}\right)^{\mu/2}}}}$ \sigma(\mu,\tau) = \frac{\exp@{\mu-\tau\atan@@{\alpha}}}{\left(\mu^{2}+\tau^{2}\right)^{\mu/2}}		sigma(mu , tau) = (exp(mu - tau*arctan(alpha)))/(((mu)^(2)+ (tau)^(2))^(mu/2))	\[Sigma][\[Mu], \[Tau]] == Divide[Exp[\[Mu]- \[Tau]*ArcTan[\[Alpha]]],(\[Mu]^(2)+ \[Tau]^(2))^(\[Mu]/2)]	Failure	Failure	Failed [300 / 300] Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)-.7960801334+.5660885692*I Test Values: {alpha = 3/2, mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I} Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, -.5000000000+.8660254040*I)+Float(-infinity)+Float(infinity)*I Test Values: {alpha = 3/2, mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Error
14.20.E21	${\displaystyle{\displaystyle{\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln\eta-\alpha\ln\left(\left(\alpha^{2}+\eta\right)^{1/2}+\alpha\right)}={\operatorname{arccos}\left(\frac{x}{\left(1+\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(\frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}\right)}}}$ {\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln@@{\eta}-\alpha\ln@{\left(\alpha^{2}+\eta\right)^{1/2}+\alpha}} = {\acos@{\frac{x}{\left(1+\alpha^{2}\right)^{1/2}}}+\frac{\alpha}{2}\ln@{\frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}}}		((alpha)^(2)+ eta)^(1/2)+(1)/(2)*alpha*ln(eta)- alpha*ln(((alpha)^(2)+ eta)^(1/2)+ alpha) = arccos((x)/((1 + (alpha)^(2))^(1/2)))+(alpha)/(2)*ln((1 + (alpha)^(2)+((alpha)^(2)- 1)*(x)^(2)- 2*alpha*x*(1 + (alpha)^(2)- (x)^(2))^(1/2))/((1 + (alpha)^(2))*(1 - (x)^(2))))	(\[Alpha]^(2)+ \[Eta])^(1/2)+Divide[1,2]*\[Alpha]*Log[\[Eta]]- \[Alpha]*Log[(\[Alpha]^(2)+ \[Eta])^(1/2)+ \[Alpha]] == ArcCos[Divide[x,(1 + \[Alpha]^(2))^(1/2)]]+Divide[\[Alpha],2]*Log[Divide[1 + \[Alpha]^(2)+(\[Alpha]^(2)- 1)*(x)^(2)- 2*\[Alpha]*x*(1 + \[Alpha]^(2)- (x)^(2))^(1/2),(1 + \[Alpha]^(2))*(1 - (x)^(2))]]	Failure	Failure	Failed [90 / 90] Result: .1205172872-1.887022822*I Test Values: {alpha = 3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.6024770750+.4691716681*I Test Values: {alpha = 3/2, eta = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[0.12051728613742685, -1.887022822024303] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.09653321282632854, -0.6333444267807768] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E23	${\displaystyle{\displaystyle\beta=\tau/\mu}}$ \beta = \tau/\mu		beta = tau/mu	\[Beta] == \[Tau]/\[Mu]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.20.E24	${\displaystyle{\displaystyle\rho=\frac{1}{2}\ln\left(\frac{\left(1-\beta^{2}\right)x^{2}+1+\beta^{2}+2x\left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^{2}}\right)+\beta\operatorname{arctan}\left(\frac{\beta x}{\sqrt{1+\beta^{2}-\beta^{2}x^{2}}}\right)-\frac{1}{2}\ln\left(1+\beta^{2}\right)}}$ \rho = \frac{1}{2}\ln@{\frac{\left(1-\beta^{2}\right)x^{2}+1+\beta^{2}+2x\left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^{2}}}+\beta\atan@{\frac{\beta x}{\sqrt{1+\beta^{2}-\beta^{2}x^{2}}}}-\frac{1}{2}\ln@{1+\beta^{2}}		rho = (1)/(2)*ln(((1 - (beta)^(2))*(x)^(2)+ 1 + (beta)^(2)+ 2*x*(1 + (beta)^(2)- (beta)^(2)* (x)^(2))^(1/2))/(1 - (x)^(2)))+ beta*arctan((beta*x)/(sqrt(1 + (beta)^(2)- (beta)^(2)* (x)^(2))))-(1)/(2)*ln(1 + (beta)^(2))	\[Rho] == Divide[1,2]*Log[Divide[(1 - \[Beta]^(2))*(x)^(2)+ 1 + \[Beta]^(2)+ 2*x*(1 + \[Beta]^(2)- \[Beta]^(2)* (x)^(2))^(1/2),1 - (x)^(2)]]+ \[Beta]*ArcTan[Divide[\[Beta]*x,Sqrt[1 + \[Beta]^(2)- \[Beta]^(2)* (x)^(2)]]]-Divide[1,2]*Log[1 + \[Beta]^(2)]	Failure	Failure	Failed [90 / 90] Result: 3.222219894+2.375212337*I Test Values: {beta = 3/2, rho = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.925994550e-1+.5000000000*I Test Values: {beta = 3/2, rho = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[3.2222198939767837, 2.37521233732194] Test Values: {Rule[x, 1.5], Rule[β, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.856194490192345, 2.741237741106379] Test Values: {Rule[x, 1.5], Rule[β, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.21.E1	${\displaystyle{\displaystyle\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0}}$ \left(1-z^{2}\right)\deriv[2]{w}{z}-2z\deriv{w}{z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w = 0		(1 - (z)^(2))*diff(w, [z$(2)])- 2*z*diff(w, z)+(nu*(nu + 1)-((mu)^(2))/(1 - (z)^(2)))*w = 0	(1 - (z)^(2))*D[w, {z, 2}]- 2*z*D[w, z]+(\[Nu]*(\[Nu]+ 1)-Divide[\[Mu]^(2),1 - (z)^(2)])*w == 0	Failure	Failure	Failed [300 / 300] Result: 1.366025404+1.366025404*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, z = 1/2*3^(1/2)+1/2*I} Result: .2113248651+1.366025405*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, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.3660254037844388, 1.3660254037844386] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.7755575615628914*^-16, -0.9999999999999997] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 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
14.23.E1	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x+i0\right)=e^{-\mu\pi i/2}\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$ \assLegendreP[\mu]{\nu}@{x+ i0} = e^{-\mu\pi i/2}\FerrersP[\mu]{\nu}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x + I*0) = exp(- mu*Pi*I/2)*LegendreP(nu, mu, x)	LegendreP[\[Nu], \[Mu], 3, x + I*0] == Exp[- \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], x]	Failure	Failure	Failed [295 / 300] Result: 5.350830664-.896185152*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: 3.575579140-1.800672871*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 [159 / 300] Result: Complex[6.260055630157556, 1.404281972043869] 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[3.1662318532347467, -6.202414130662353] 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
14.23.E1	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x-i0\right)=e^{+\mu\pi i/2}\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$ \assLegendreP[\mu]{\nu}@{x- i0} = e^{+\mu\pi i/2}\FerrersP[\mu]{\nu}@{x}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x - I*0) = exp(+ mu*Pi*I/2)*LegendreP(nu, mu, x)	LegendreP[\[Nu], \[Mu], 3, x - I*0] == Exp[+ \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], x]	Failure	Failure	Failed [295 / 300] Result: -.9092249665-2.300467118*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: -1.143434975-1.422772544*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 [79 / 300] Result: Complex[-4.719014112853729, 0.3779003216614092] 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
14.23.E2	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)=\frac{e^{+\mu\pi i/2}}{\Gamma\left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\tfrac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right)}}$ \assLegendreOlverQ[\mu]{\nu}@{x+ i0} = \frac{e^{+\mu\pi i/2}}{\EulerGamma@{\nu+\mu+1}}\left(\FerrersQ[\mu]{\nu}@{x}-\tfrac{1}{2}\pi i\FerrersP[\mu]{\nu}@{x}\right)	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu-\mu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1) = (exp(+ mu*Pi*I/2))/(GAMMA(nu + mu + 1))*(LegendreQ(nu, mu, x)-(1)/(2)*Pi*I*LegendreP(nu, mu, x))	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[+ \[Mu]*Pi*I/2],Gamma[\[Nu]+ \[Mu]+ 1]]*(LegendreQ[\[Nu], \[Mu], x]-Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], x])	Failure	Failure	Failed [120 / 120] Result: 15.62228457-3.860103415*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: 11.64166640-5.161800279*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 [90 / 135] Result: Complex[2.4984461168598187, 1.2999649891093954] 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[5.332631908276789, 3.703974803728466] 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
14.23.E2	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)=\frac{e^{-\mu\pi i/2}}{\Gamma\left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}\left(x\right)+\tfrac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right)}}$ \assLegendreOlverQ[\mu]{\nu}@{x- i0} = \frac{e^{-\mu\pi i/2}}{\EulerGamma@{\nu+\mu+1}}\left(\FerrersQ[\mu]{\nu}@{x}+\tfrac{1}{2}\pi i\FerrersP[\mu]{\nu}@{x}\right)	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu-\mu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1) = (exp(- mu*Pi*I/2))/(GAMMA(nu + mu + 1))*(LegendreQ(nu, mu, x)+(1)/(2)*Pi*I*LegendreP(nu, mu, x))	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[- \[Mu]*Pi*I/2],Gamma[\[Nu]+ \[Mu]+ 1]]*(LegendreQ[\[Nu], \[Mu], x]+Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], x])	Failure	Failure	Failed [120 / 120] Result: 13.12383845-5.160068402*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: 9.802483176-6.415524146*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 [45 / 135] Result: Complex[-1.839183222440096, -1.2537238668211261] 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[1.419436191421772, -4.262017463676762] 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, 3]], Pi]]]} ... skip entries to safe data
14.23.E3	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)=\frac{e^{-\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}-i\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}\right)}}$ \assLegendreOlverQ[\mu]{\nu}@{x+ i0} = \frac{e^{-\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\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}\nu-\frac{1}{2}\mu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}- i\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}\nu-\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}\right)	${\displaystyle{\displaystyle\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+1)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,|(x^{2})|<1}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1) = (exp(- nu*Pi*I/2)*(Pi)^(3/2)*(1 - (x)^(2))^(mu/2))/((2)^(nu + 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)*nu -(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))- I*(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)*nu -(1)/(2)*mu + 1)*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1)))	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[- \[Nu]*Pi*I/2]*(Pi)^(3/2)*(1 - (x)^(2))^(\[Mu]/2),(2)^(\[Nu]+ 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]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]]]- I*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]*\[Nu]-Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]])	Failure	Failure	Successful [Tested: 40]	Successful [Tested: 45]
14.23.E3	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)=\frac{e^{+\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}+i\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}\right)}}$ \assLegendreOlverQ[\mu]{\nu}@{x- i0} = \frac{e^{+\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\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}\nu-\frac{1}{2}\mu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}+ i\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}\nu-\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}\right)	${\displaystyle{\displaystyle\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+1)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,|(x^{2})|<1}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1) = (exp(+ nu*Pi*I/2)*(Pi)^(3/2)*(1 - (x)^(2))^(mu/2))/((2)^(nu + 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)*nu -(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))+ I*(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)*nu -(1)/(2)*mu + 1)*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1)))	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[+ \[Nu]*Pi*I/2]*(Pi)^(3/2)*(1 - (x)^(2))^(\[Mu]/2),(2)^(\[Nu]+ 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]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]]]+ I*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]*\[Nu]-Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]])	Failure	Failure	Failed [40 / 40] Result: -1.839183223-1.253723866*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: 1.419436198-4.262017468*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 [45 / 45] Result: Complex[-1.8391832224400957, -1.2537238668211277] 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[1.4194361914217857, -4.2620174636767665] 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, 3]], Pi]]]} ... skip entries to safe data
14.23.E4	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=e^{+\mu\pi i/2}P^{\mu}_{\nu}\left(x+i0\right)}}$ \FerrersP[\mu]{\nu}@{x} = e^{+\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x+ i0}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x) = exp(+ mu*Pi*I/2)*LegendreP(nu, mu, x + I*0)	LegendreP[\[Nu], \[Mu], x] == Exp[+ \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x + I*0]	Failure	Failure	Failed [295 / 300] Result: -.9092249665-2.300467118*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: -1.143434975-1.422772544*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 [159 / 300] Result: Complex[0.02990691582525623, -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, -0.8210135056644176] 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
14.23.E4	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i/2}P^{\mu}_{\nu}\left(x-i0\right)}}$ \FerrersP[\mu]{\nu}@{x} = e^{-\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x- i0}	${\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(nu, mu, x) = exp(- mu*Pi*I/2)*LegendreP(nu, mu, x - I*0)	LegendreP[\[Nu], \[Mu], x] == Exp[- \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x - I*0]	Failure	Failure	Failed [295 / 300] Result: 5.350830664-.896185152*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: 3.575579140-1.800672871*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 [79 / 300] Result: Complex[1.351552463852863, -10.294914164956062] 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[7.255468107198464, -2.190256047354226] 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
14.23.E5	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\tfrac{1}{2}\Gamma\left(\nu+\mu+1\right)\left(e^{-\mu\pi i/2}\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)+e^{\mu\pi i/2}\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)\right)}}$ \FerrersQ[\mu]{\nu}@{x} = \tfrac{1}{2}\EulerGamma@{\nu+\mu+1}\left(e^{-\mu\pi i/2}\assLegendreOlverQ[\mu]{\nu}@{x+i0}+e^{\mu\pi i/2}\assLegendreOlverQ[\mu]{\nu}@{x-i0}\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) = (1)/(2)*GAMMA(nu + mu + 1)*(exp(- mu*Pi*I/2)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1)+ exp(mu*Pi*I/2)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1))	LegendreQ[\[Nu], \[Mu], x] == Divide[1,2]*Gamma[\[Nu]+ \[Mu]+ 1]*(Exp[- \[Mu]*Pi*I/2]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1]+ Exp[\[Mu]*Pi*I/2]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1])	Failure	Failure	Failed [120 / 120] Result: -15.30496809+11.59724304*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: -10.41616244+10.97902682*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 [135 / 135] Result: Complex[-3.9489024974094016, 0.15503510169416979] 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[-4.5992221195498555, 6.976681726631964] 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
14.23.E6	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i/2}\Gamma\left(\nu+\mu+1\right)\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)+\tfrac{1}{2}\pi ie^{+\mu\pi i/2}P^{\mu}_{\nu}\left(x+i0\right)}}$ \FerrersQ[\mu]{\nu}@{x} = e^{-\mu\pi i/2}\EulerGamma@{\nu+\mu+1}\assLegendreOlverQ[\mu]{\nu}@{x+ i0}+\tfrac{1}{2}\pi ie^{+\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x+ i0}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(nu, mu, x) = exp(- mu*Pi*I/2)*GAMMA(nu + mu + 1)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1)+(1)/(2)*Pi*I*exp(+ mu*Pi*I/2)*LegendreP(nu, mu, x + I*0)	LegendreQ[\[Nu], \[Mu], x] == Exp[- \[Mu]*Pi*I/2]*Gamma[\[Nu]+ \[Mu]+ 1]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1]+Divide[1,2]*Pi*I*Exp[+ \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x + I*0]	Failure	Failure	Failed [120 / 120] Result: -29.08177200+29.72441292*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: -18.94845706+26.98747914*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 [90 / 135] Result: Complex[-3.303261395604329, 0.35704787691241624] 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[-5.262064714407579, 5.6951304506187865] 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
14.23.E6	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{+\mu\pi i/2}\Gamma\left(\nu+\mu+1\right)\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)-\tfrac{1}{2}\pi ie^{-\mu\pi i/2}P^{\mu}_{\nu}\left(x-i0\right)}}$ \FerrersQ[\mu]{\nu}@{x} = e^{+\mu\pi i/2}\EulerGamma@{\nu+\mu+1}\assLegendreOlverQ[\mu]{\nu}@{x- i0}-\tfrac{1}{2}\pi ie^{-\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x- i0}	${\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreQ(nu, mu, x) = exp(+ mu*Pi*I/2)*GAMMA(nu + mu + 1)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1)-(1)/(2)*Pi*I*exp(- mu*Pi*I/2)*LegendreP(nu, mu, x - I*0)	LegendreQ[\[Nu], \[Mu], x] == Exp[+ \[Mu]*Pi*I/2]*Gamma[\[Nu]+ \[Mu]+ 1]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1]-Divide[1,2]*Pi*I*Exp[- \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x - I*0]	Failure	Failure	Failed [120 / 120] Result: .677676788-16.36319923*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: -2.477472256-12.44203554*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 [45 / 135] Result: Complex[-17.39472965859494, -1.6880401639683693] 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[-2.8057990956489687, 0.19849176253311906] 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, 3]], Pi]]]} ... skip entries to safe data
14.24.E1	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)=e^{s\nu\pi i}P^{-\mu}_{\nu}\left(z\right)+\frac{2i\sin\left(\left(\nu+\frac{1}{2}\right)s\pi\right)e^{-s\pi i/2}}{\cos\left(\nu\pi\right)\Gamma\left(\mu-\nu\right)}\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)}}$ \assLegendreP[-\mu]{\nu}@{ze^{s\pi i}} = e^{s\nu\pi i}\assLegendreP[-\mu]{\nu}@{z}+\frac{2i\sin@{\left(\nu+\frac{1}{2}\right)s\pi}e^{-s\pi i/2}}{\cos@{\nu\pi}\EulerGamma@{\mu-\nu}}\assLegendreOlverQ[\mu]{\nu}@{z}	${\displaystyle{\displaystyle\Re(\mu-\nu)>0}}$	LegendreP(nu, - mu, z*exp(s*Pi*I)) = exp(s*nu*Pi*I)*LegendreP(nu, - mu, z)+(2*I*sin((nu +(1)/(2))*s*Pi)*exp(- s*Pi*I/2))/(cos(nu*Pi)*GAMMA(mu - nu))*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1)	LegendreP[\[Nu], - \[Mu], 3, z*Exp[s*Pi*I]] == Exp[s*\[Nu]*Pi*I]*LegendreP[\[Nu], - \[Mu], 3, z]+Divide[2*I*Sin[(\[Nu]+Divide[1,2])*s*Pi]*Exp[- s*Pi*I/2],Cos[\[Nu]*Pi]*Gamma[\[Mu]- \[Nu]]]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1]	Failure	Failure	Manual Skip!	Failed [299 / 300] Result: Complex[-21.32728052513349, -8.911336897051166] Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[13.892460412350314, 1.7999110613880858] Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 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
14.24.E2	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)=(-1)^{s}e^{-s\nu\pi i}\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)}}$ \assLegendreOlverQ[\mu]{\nu}@{ze^{s\pi i}} = (-1)^{s}e^{-s\nu\pi i}\assLegendreOlverQ[\mu]{\nu}@{z}		exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z*exp(s*Pi*I))/GAMMA(nu+mu+1) = (- 1)^(s)* exp(- s*nu*Pi*I)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z*Exp[s*Pi*I]]/Gamma[\[Nu] + \[Mu] + 1] == (- 1)^(s)* Exp[- s*\[Nu]*Pi*I]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1]	Failure	Failure	Failed [300 / 300] Result: -.2140796977+.7286338337*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, s = -3/2, z = 1/2*3^(1/2)+1/2*I} Result: -.1549543426-.1299026639*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, s = -3/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.2140796979538467, 0.7286338343398007] Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.2472082058834166, -8.359397493451592] Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 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
14.24.E3	${\displaystyle{\displaystyle P^{-\mu}_{\nu,s}\left(z\right)=e^{s\mu\pi i}P^{-\mu}_{\nu}\left(z\right)}}$ \assLegendreP[-\mu]{\nu,s}@{z} = e^{s\mu\pi i}\assLegendreP[-\mu]{\nu}@{z}		LegendreP(nu , s, - mu, z) = exp(s*mu*Pi*I)*LegendreP(nu, - mu, z)	LegendreP[\[Nu], s, - \[Mu], 3, z] == Exp[s*\[Mu]*Pi*I]*LegendreP[\[Nu], - \[Mu], 3, z]	Error	Failure	-	Successful [Tested: 300]
14.24.E4	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)=e^{-s\mu\pi i}\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)-\frac{\pi i\sin\left(s\mu\pi\right)}{\sin\left(\mu\pi\right)\Gamma\left(\nu-\mu+1\right)}P^{-\mu}_{\nu}\left(z\right)}}$ \assLegendreOlverQ[\mu]{\nu,s}@{z} = e^{-s\mu\pi i}\assLegendreOlverQ[\mu]{\nu}@{z}-\frac{\pi i\sin@{s\mu\pi}}{\sin@{\mu\pi}\EulerGamma@{\nu-\mu+1}}\assLegendreP[-\mu]{\nu}@{z}	${\displaystyle{\displaystyle\Re(\nu-\mu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu , s,mu,z)/GAMMA(nu , s+mu+1) = exp(- s*mu*Pi*I)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1)-(Pi*I*sin(s*mu*Pi))/(sin(mu*Pi)*GAMMA(nu - mu + 1))*LegendreP(nu, - mu, z)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], s, \[Mu], 3, z]/Gamma[\[Nu], s + \[Mu] + 1] == Exp[- s*\[Mu]*Pi*I]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1]-Divide[Pi*I*Sin[s*\[Mu]*Pi],Sin[\[Mu]*Pi]*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], 3, z]	Error	Failure	-	Failed [69 / 300] Result: Indeterminate Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]} Result: Indeterminate Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]} ... skip entries to safe data
14.25.E1	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(z\right)=\frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\Gamma\left(\mu-\nu\right)\Gamma\left(\nu+1\right)}\int_{0}^{\infty}\frac{(\sinh t)^{2\nu+1}}{(z+\cosh t)^{\nu+\mu+1}}\mathrm{d}t}}$ \assLegendreP[-\mu]{\nu}@{z} = \frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\EulerGamma@{\mu-\nu}\EulerGamma@{\nu+1}}\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\nu+1}}{(z+\cosh@@{t})^{\nu+\mu+1}}\diff{t}	${\displaystyle{\displaystyle\Re\mu>\Re\nu,\Re\nu>-1,\Re(\mu-\nu)>0,\Re(\nu+1)>0}}$	LegendreP(nu, - mu, z) = (((z)^(2)- 1)^(mu/2))/((2)^(nu)* GAMMA(mu - nu)*GAMMA(nu + 1))*int(((sinh(t))^(2*nu + 1))/((z + cosh(t))^(nu + mu + 1)), t = 0..infinity)	LegendreP[\[Nu], - \[Mu], 3, z] == Divide[((z)^(2)- 1)^(\[Mu]/2),(2)^\[Nu]* Gamma[\[Mu]- \[Nu]]*Gamma[\[Nu]+ 1]]*Integrate[Divide[(Sinh[t])^(2*\[Nu]+ 1),(z + Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
14.25.E2	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)=\frac{\pi^{1/2}\left(z^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(z+(z^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\mathrm{d}t}}$ \assLegendreOlverQ[\mu]{\nu}@{z} = \frac{\pi^{1/2}\left(z^{2}-1\right)^{\mu/2}}{2^{\mu}\EulerGamma@{\mu+\frac{1}{2}}\EulerGamma@{\nu-\mu+1}}\*\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\mu}}{\left(z+(z^{2}-1)^{1/2}\cosh@@{t}\right)^{\nu+\mu+1}}\diff{t}	${\displaystyle{\displaystyle\Re\left(\nu+1\right)>\Re\mu,\Re\mu>-\tfrac{1}{2},\Re(\mu+\frac{1}{2})>0,\Re(\nu-\mu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1) = ((Pi)^(1/2)*((z)^(2)- 1)^(mu/2))/((2)^(mu)* GAMMA(mu +(1)/(2))*GAMMA(nu - mu + 1))* int(((sinh(t))^(2*mu))/((z +((z)^(2)- 1)^(1/2)* cosh(t))^(nu + mu + 1)), t = 0..infinity)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(Pi)^(1/2)*((z)^(2)- 1)^(\[Mu]/2),(2)^\[Mu]* Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]- \[Mu]+ 1]]* Integrate[Divide[(Sinh[t])^(2*\[Mu]),(z +((z)^(2)- 1)^(1/2)* Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
14.28.E1	${\displaystyle{\displaystyle P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right)}}$ \assLegendreP[]{\nu}@{z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos@@{\phi}} = \assLegendreP[]{\nu}@{z_{1}}\assLegendreP[]{\nu}@{z_{2}}+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\*\assLegendreP[m]{\nu}@{z_{1}}\assLegendreP[m]{\nu}(z_{2})\cos@{m\phi}	${\displaystyle{\displaystyle\Re(\nu-m+1)>0,\Re(\nu+m+1)>0}}$	LegendreP(nu, z[1]*z[2]-((z[1])^(2)- 1)^(1/2)*((z[2])^(2)- 1)^(1/2)* cos(phi)) = LegendreP(nu, z[1])*LegendreP(nu, z[2])+ 2*sum((- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))* LegendreP(nu, m, z[1])*LegendreP(nu, m, z[2])*cos(m*phi), m = 1..infinity)	LegendreP[\[Nu], 0, 3, Subscript[z, 1]*Subscript[z, 2]-((Subscript[z, 1])^(2)- 1)^(1/2)*((Subscript[z, 2])^(2)- 1)^(1/2)* Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Subscript[z, 1]]*LegendreP[\[Nu], 0, 3, Subscript[z, 2]]+ 2*Sum[(- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]* LegendreP[\[Nu], m, 3, Subscript[z, 1]]*LegendreP[\[Nu], m, 3, Subscript[z, 2]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]	Translation Error	Translation Error	-	-
14.28.E2	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}(2n+1)Q_{n}\left(z_{1}\right)P_{n}\left(z_{2}\right)=\frac{1}{z_{1}-z_{2}}}}$ \sum_{n=0}^{\infty}(2n+1)\assLegendreQ[]{n}@{z_{1}}\assLegendreP[]{n}@{z_{2}} = \frac{1}{z_{1}-z_{2}}		sum((2*n + 1)*LegendreQ(n, z[1])*LegendreP(n, z[2]), n = 0..infinity) = (1)/(z[1]- z[2])	Sum[(2*n + 1)*LegendreQ[n, 0, 3, Subscript[z, 1]]*LegendreP[n, 0, 3, Subscript[z, 2]], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Subscript[z, 1]- Subscript[z, 2]]	Failure	Failure	Skipped - Because timed out	Failed [100 / 100] Result: Plus[DirectedInfinity[], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.6830127018922194, -0.18301270189221946], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.29.E1	${\displaystyle{\displaystyle\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{\mathrm{d}w}{\mathrm{d}z}+{\left(\nu(\nu+1)-\frac{\mu_{1}^{2}}{2(1-z)}-\frac{\mu_{2}^{2}}{2(1+z)}\right)w}=0}}$ \left(1-z^{2}\right)\deriv[2]{w}{z}-2z\deriv{w}{z}+{\left(\nu(\nu+1)-\frac{\mu_{1}^{2}}{2(1-z)}-\frac{\mu_{2}^{2}}{2(1+z)}\right)w} = 0		(1 - (z)^(2))*diff(w, [z$(2)])- 2*z*diff(w, z)+(nu*(nu + 1)-((mu[1])^(2))/(2*(1 - z))-((mu[2])^(2))/(2*(1 + z)))*w = 0	(1 - (z)^(2))*D[w, {z, 2}]- 2*z*D[w, z]+(\[Nu]*(\[Nu]+ 1)-Divide[(Subscript[\[Mu], 1])^(2),2*(1 - z)]-Divide[(Subscript[\[Mu], 2])^(2),2*(1 + z)])*w == 0	Failure	Failure	Failed [300 / 300] Result: -1.000000001-3.732050810*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, z = 1/2*3^(1/2)+1/2*I, mu[1] = 1/2*3^(1/2)+1/2*I, mu[2] = 1/2*3^(1/2)+1/2*I} Result: -1.000000001-3.732050810*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, z = 1/2*3^(1/2)+1/2*I, mu[1] = 1/2*3^(1/2)+1/2*I, mu[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [296 / 300] Result: Complex[-0.7320508075688783, -4.732050807568878] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[μ, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[μ, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.3322676295501878*^-15, -5.464101615137755] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[μ, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[μ, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.30.E1	${\displaystyle{\displaystyle Y_{{l},{m}}\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\mathsf{P}^{m}_{l}\left(\cos\theta\right)}}$ \sphharmonicY{l}{m}@{\theta}{\phi} = \left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\FerrersP[m]{l}@{\cos@@{\theta}}		SphericalY(l, m, theta, phi) = ((factorial(l - m)*(2*l + 1))/(4*Pi*factorial(l + m)))^(1/2)* exp(I*m*phi)*LegendreP(l, m, cos(theta))	SphericalHarmonicY[l, m, \[Theta], \[Phi]] == (Divide[(l - m)!*(2*l + 1),4*Pi*(l + m)!])^(1/2)* Exp[I*m*\[Phi]]*LegendreP[l, m, Cos[\[Theta]]]	Failure	Failure	Failed [234 / 300] Result: .1254512786+.3659009168*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, l = 1, m = 1} Result: Float(undefined)+Float(undefined)*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, l = 1, m = 2} ... skip entries to safe data	Failed [154 / 300] Result: Indeterminate Test Values: {Rule[l, 1], Rule[m, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[l, 1], Rule[m, 3], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.30.E6	${\displaystyle{\displaystyle Y_{{l},{-m}}\left(\theta,\phi\right)=(-1)^{m}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}}}$ \sphharmonicY{l}{-m}@{\theta}{\phi} = (-1)^{m}\conj{\sphharmonicY{l}{m}@{\theta}{\phi}}		SphericalY(l, - m, theta, phi) = (- 1)^(m)* conjugate(SphericalY(l, m, theta, phi))	SphericalHarmonicY[l, - m, \[Theta], \[Phi]] == (- 1)^(m)* Conjugate[SphericalHarmonicY[l, m, \[Theta], \[Phi]]]	Failure	Failure	Failed [199 / 300] Result: .651899905e-1+.4007576287*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, l = 1, m = 1} Result: .5735569852+.2720162074*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, l = 2, m = 1} ... skip entries to safe data	Failed [199 / 300] Result: Complex[0.4007576286123945, -0.06518999054786037] Test Values: {Rule[l, 1], Rule[m, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.2720162074039931, -0.5735569852255453] Test Values: {Rule[l, 2], Rule[m, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
14.30.E7	${\displaystyle{\displaystyle Y_{{l},{m}}\left(\pi-\theta,\phi+\pi\right)=(-1)^{l}Y_{{l},{m}}\left(\theta,\phi\right)}}$ \sphharmonicY{l}{m}@{\pi-\theta}{\phi+\pi} = (-1)^{l}\sphharmonicY{l}{m}@{\theta}{\phi}		SphericalY(l, m, Pi - theta, phi + Pi) = (- 1)^(l)* SphericalY(l, m, theta, phi)	SphericalHarmonicY[l, m, Pi - \[Theta], \[Phi]+ Pi] == (- 1)^(l)* SphericalHarmonicY[l, m, \[Theta], \[Phi]]	Failure	Failure	Failed [114 / 300] Result: -.3659009168+.1254512785*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, l = 1, m = 1} Result: .4863638630-.5297060789*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, l = 2, m = 1} ... skip entries to safe data	Successful [Tested: 300]
14.30.E8	${\displaystyle{\displaystyle\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}}}$ \int_{0}^{2\pi}\!\!\int_{0}^{\pi}\conj{\sphharmonicY{l_{1}}{m_{1}}@{\theta}{\phi}}\sphharmonicY{l_{2}}{m_{2}}@{\theta}{\phi}\sin@@{\theta}\diff{\theta}\diff{\phi} = \Kroneckerdelta{l_{1}}{l_{2}}\Kroneckerdelta{m_{1}}{m_{2}}		int(int(conjugate(SphericalY(l[1], m[1], theta, phi))*SphericalY(l[2], m[2], theta, phi)*sin(theta), theta = 0..Pi), phi = 0..2*Pi) = KroneckerDelta[l[1], l[2]]*KroneckerDelta[m[1], m[2]]	Integrate[Integrate[Conjugate[SphericalHarmonicY[Subscript[l, 1], Subscript[m, 1], \[Theta], \[Phi]]]*SphericalHarmonicY[Subscript[l, 2], Subscript[m, 2], \[Theta], \[Phi]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None], {\[Phi], 0, 2*Pi}, GenerateConditions->None] == KroneckerDelta[Subscript[l, 1], Subscript[l, 2]]*KroneckerDelta[Subscript[m, 1], Subscript[m, 2]]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.30.E9	${\displaystyle{\displaystyle\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}\overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_{2},\phi_{2}\right)}}$ \FerrersP[]{l}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@{\phi_{1}-\phi_{2}}} = \frac{4\pi}{2l+1}\sum_{m=-l}^{l}\conj{\sphharmonicY{l}{m}@{\theta_{1}}{\phi_{1}}}\sphharmonicY{l}{m}@{\theta_{2}}{\phi_{2}}		LegendreP(l, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi[1]- phi[2])) = (4*Pi)/(2*l + 1)*sum(conjugate(SphericalY(l, m, theta[1], phi[1]))*SphericalY(l, m, theta[2], phi[2]), m = - l..l)	LegendreP[l, Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[Subscript[\[Phi], 1]- Subscript[\[Phi], 2]]] == Divide[4*Pi,2*l + 1]*Sum[Conjugate[SphericalHarmonicY[l, m, Subscript[\[Theta], 1], Subscript[\[Phi], 1]]]*SphericalHarmonicY[l, m, Subscript[\[Theta], 2], Subscript[\[Phi], 2]], {m, - l, l}, GenerateConditions->None]	Aborted	Failure	Error	Skipped - Because timed out
14.30.E10	${\displaystyle{\displaystyle{\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial W}{\partial\rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial W}{\partial\theta}\right)}+\frac{1}{\rho^{2}{\sin^{2}}\theta}\frac{{\partial}^{2}W}{{\partial\phi}^{2}}=0}}$ {\frac{1}{\rho^{2}}\pderiv{}{\rho}\left(\rho^{2}\pderiv{W}{\rho}\right)+\frac{1}{\rho^{2}\sin@@{\theta}}\pderiv{}{\theta}\left(\sin@@{\theta}\pderiv{W}{\theta}\right)}+\frac{1}{\rho^{2}\sin^{2}@@{\theta}}\pderiv[2]{W}{\phi} = 0		(1)/((rho)^(2))*diff(((rho)^(2)* diff(W, rho))+(1)/((rho)^(2)* sin(theta))*diff(sin(theta)*diff(W, theta), theta), rho)+(1)/((rho)^(2)* (sin(theta))^(2))*diff(W, [phi$(2)]) = 0	Divide[1,\[Rho]^(2)]*D[(\[Rho]^(2)* D[W, \[Rho]])+Divide[1,\[Rho]^(2)* Sin[\[Theta]]]*D[Sin[\[Theta]]*D[W, \[Theta]], \[Theta]], \[Rho]]+Divide[1,\[Rho]^(2)* (Sin[\[Theta]])^(2)]*D[W, {\[Phi], 2}] == 0	Successful	Successful	-	Successful [Tested: 300]