Results of Legendre and Related Functions

DLMF	Formula	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.2.E1	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}+\nu(\nu+1)w=0}}$	(1 - (x)^(2))* diff(w, [x$(2)])- 2*x*diff(w, x)+ nu*(nu + 1)* w = 0	(1 - (x)^(2))* D[w, {x, 2}]- 2*x*D[w, x]+ \[Nu]*(\[Nu]+ 1)* w = 0	Failure	Failure	Fail -5.656854245+9.656854243*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)} 9.656854243+5.656854245*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)} 5.656854245-9.656854243*I <- {nu = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)} -9.656854243-5.656854245*I <- {nu = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-5.656854249492381, 9.65685424949238] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[9.65685424949238, -5.656854249492381] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-5.656854249492381, 1.6568542494923806] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.6568542494923806, -5.656854249492381] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.2.E2	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-x^{2}}\right)w=0}}$	(1 - (x)^(2))* diff(w, [x$(2)])- 2*x*diff(w, x)+(nu*(nu + 1)-((mu)^(2))/(1 - (x)^(2)))* w = 0	(1 - (x)^(2))* D[w, {x, 2}]- 2*x*D[w, x]+(\[Nu]*(\[Nu]+ 1)-Divide[(\[Mu])^(2),1 - (x)^(2)])* w = 0	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1} -7.542472327+11.54247233*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2} -6.363961026+10.36396102*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-7.5424723326565095, 11.542472332656509] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-6.36396103067893, 10.36396103067893] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.2.E5	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)=\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)}}}$	LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x)- LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x)=(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 2))	LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu], \[Mu], x]- LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x]=Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 2]]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} .118833e-2-.67509e-3*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Successful
14.3.E1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\left(\frac{1+x}{1-x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$	LegendreP(nu, mu, x)=((1 + x)/(1 - x))^(mu/ 2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)	LegendreP[\[Nu], \[Mu], x]=(Divide[1 + x,1 - x])^(\[Mu]/ 2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} 9.841425439+29.20009169*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 2} 22.82321651+33.19943936*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[9.841425469606474, 29.20009174654549] <- {Rule[x, 2], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[22.823216526761424, 33.199439403579085] <- {Rule[x, 3], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.3.E2	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi}{2\sin\left(\mu\pi\right)}\left(\cos\left(\mu\pi\right)\left(\frac{1+x}{1-x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)}\left(\frac{1-x}{1+x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)\right)}}$	LegendreQ(nu, mu, x)=(Pi)/(2*sin(mu*Pi))*(cos(mu*Pi)*((1 + x)/(1 - x))^(mu/ 2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)-(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1))*((1 - x)/(1 + x))^(mu/ 2)* hypergeom([nu + 1, - nu], [1 + mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 + mu))	LegendreQ[\[Nu], \[Mu], x]=Divide[Pi,2*Sin[\[Mu]*Pi]]*(Cos[\[Mu]*Pi]*(Divide[1 + x,1 - x])^(\[Mu]/ 2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]-Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]]*(Divide[1 - x,1 + x])^(\[Mu]/ 2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 + \[Mu], Divide[1,2]-Divide[1,2]*x])	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} 45.85870096-15.44869178*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 2} 52.14226531-35.83470770*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Skip
14.3.E3	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;x\right)=\frac{1}{\Gamma\left(c\right)}F\left(a,b;c;x\right)}}$	hypergeom([a, b], [c], x)/GAMMA(c)=(1)/(GAMMA(c))*hypergeom([a, b], [c], x)	Hypergeometric2F1Regularized[a, b, c, x]=Divide[1,Gamma[c]]*Hypergeometric2F1[a, b, c, x]	Successful	Successful	-	-
14.3.E4	${\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(1-x^{2}\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$	LegendreP(nu, m, x)=(- 1)^(m)*(GAMMA(nu + m + 1))/((2)^(m)* GAMMA(nu - m + 1))*(1 - (x)^(2))^(m/ 2)* hypergeom([nu + m + 1, m - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)	LegendreP[\[Nu], m, x]=(- 1)^(m)*Divide[Gamma[\[Nu]+ m + 1],(2)^(m)* Gamma[\[Nu]- m + 1]]*(1 - (x)^(2))^(m/ 2)* Hypergeometric2F1Regularized[\[Nu]+ m + 1, m - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Successful	Successful
14.3.E5	${\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{\Gamma\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}\right)^{m/2}\mathbf{F}\left(\nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$	LegendreP(nu, m, x)=(- 1)^(m)*(GAMMA(nu + m + 1))/(GAMMA(nu - m + 1))*((1 - x)/(1 + x))^(m/ 2)* hypergeom([nu + 1, - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)	LegendreP[\[Nu], m, x]=(- 1)^(m)*Divide[Gamma[\[Nu]+ m + 1],Gamma[\[Nu]- m + 1]]*(Divide[1 - x,1 + x])^(m/ 2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Successful	Successful
14.3.E6	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$	LegendreP(nu, mu, x)=((x + 1)/(x - 1))^(mu/ 2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)	LegendreP[\[Nu], \[Mu], 3, x]=(Divide[x + 1,x - 1])^(\[Mu]/ 2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.3.E8	${\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$	LegendreP(nu, m, x)=(GAMMA(nu + m + 1))/((2)^(m)* GAMMA(nu - m + 1))*((x)^(2)- 1)^(m/ 2)* hypergeom([nu + m + 1, m - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)	LegendreP[\[Nu], m, 3, x]=Divide[Gamma[\[Nu]+ m + 1],(2)^(m)* Gamma[\[Nu]- m + 1]]*((x)^(2)- 1)^(m/ 2)* Hypergeometric2F1Regularized[\[Nu]+ m + 1, m - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Successful	Successful
14.3.E9	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$	LegendreP(nu, - mu, x)=((x - 1)/(x + 1))^(mu/ 2)* hypergeom([nu + 1, - nu], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)	LegendreP[\[Nu], - \[Mu], 3, x]=(Divide[x - 1,x + 1])^(\[Mu]/ 2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]	Failure	Successful	Fail Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	-
14.3.E11	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x)}}$	LegendreP(nu, mu, x)= cos((1)/(2)*(nu + mu)* Pi)*w[1]*(nu , mu , x)+ sin((1)/(2)*(nu + mu)* Pi)*w[2]*(nu , mu , x)	LegendreP[\[Nu], \[Mu], x]= Cos[Divide[1,2]*(\[Nu]+ \[Mu])* Pi]*Subscript[w, 1]*(\[Nu], \[Mu], x)+ Sin[Divide[1,2]*(\[Nu]+ \[Mu])* Pi]*Subscript[w, 2]*(\[Nu], \[Mu], x)	Failure	Failure	Error	Error
14.3.E12	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}\pi\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\tfrac{1}{2}\pi\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x)}}$	LegendreQ(nu, mu, x)= -(1)/(2)*Pi*sin((1)/(2)*(nu + mu)* Pi)*w[1]*(nu , mu , x)+(1)/(2)*Pi*cos((1)/(2)*(nu + mu)* Pi)*w[2]*(nu , mu , x)	LegendreQ[\[Nu], \[Mu], x]= -Divide[1,2]*Pi*Sin[Divide[1,2]*(\[Nu]+ \[Mu])* Pi]*Subscript[w, 1]*(\[Nu], \[Mu], x)+Divide[1,2]*Pi*Cos[Divide[1,2]*(\[Nu]+ \[Mu])* Pi]*Subscript[w, 2]*(\[Nu], \[Mu], x)	Failure	Failure	Error	Error
14.3.E13	${\displaystyle{\displaystyle w_{1}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}\left(1-x^{2}\right)^{-\mu/2}\mathbf{F}\left(-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2};\tfrac{1}{2};x^{2}\right)}}$	w[1]*(nu , mu , x)=((2)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu + 1))*(1 - (x)^(2))^(- mu/ 2)* hypergeom([-(1)/(2)*nu -(1)/(2)*mu, (1)/(2)*nu -(1)/(2)*mu +(1)/(2)], [(1)/(2)], (x)^(2))/GAMMA((1)/(2))	Subscript[w, 1]*(\[Nu], \[Mu], x)=Divide[(2)^(\[Mu])* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1]]*(1 - (x)^(2))^(- \[Mu]/ 2)* Hypergeometric2F1Regularized[-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu], Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2], Divide[1,2], (x)^(2)]	Failure	Failure	Error	Error
14.3.E14	${\displaystyle{\displaystyle w_{2}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1-x^{2}\right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right)}}$	w[2]*(nu , mu , x)=((2)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))*x*(1 - (x)^(2))^(- mu/ 2)* hypergeom([(1)/(2)-(1)/(2)*nu -(1)/(2)*mu, (1)/(2)*nu -(1)/(2)*mu + 1], [(3)/(2)], (x)^(2))/GAMMA((3)/(2))	Subscript[w, 2]*(\[Nu], \[Mu], x)=Divide[(2)^(\[Mu])* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]*x*(1 - (x)^(2))^(- \[Mu]/ 2)* Hypergeometric2F1Regularized[Divide[1,2]-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu], Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1, Divide[3,2], (x)^(2)]	Failure	Failure	Error	Error
14.3.E15	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}$	LegendreP(nu, - mu, x)= (2)^(- mu)*((x)^(2)- 1)^(mu/ 2)* hypergeom([mu - nu, nu + mu + 1], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)	LegendreP[\[Nu], - \[Mu], 3, x]= (2)^(- \[Mu])*((x)^(2)- 1)^(\[Mu]/ 2)* Hypergeometric2F1Regularized[\[Mu]- \[Nu], \[Nu]+ \[Mu]+ 1, \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.3.E16	${\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\pi^{1/2}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{\nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right)}}$	cos(nu*Pi)*LegendreP(nu, - mu, x)=((2)^(nu)* (Pi)^(1/ 2)* (x)^(nu - mu)*((x)^(2)- 1)^(mu/ 2))/(GAMMA(nu + mu + 1))*hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*mu -(1)/(2)*nu +(1)/(2)], [(1)/(2)- nu], (1)/((x)^(2)))/GAMMA((1)/(2)- nu)-((Pi)^(1/ 2)*((x)^(2)- 1)^(mu/ 2))/((2)^(nu + 1)* GAMMA(mu - nu)*(x)^(nu + mu + 1))*hypergeom([(1)/(2)*nu +(1)/(2)*mu + 1, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [nu +(3)/(2)], (1)/((x)^(2)))/GAMMA(nu +(3)/(2))	Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], 3, x]=Divide[(2)^(\[Nu])* (Pi)^(1/ 2)* (x)^(\[Nu]- \[Mu])*((x)^(2)- 1)^(\[Mu]/ 2),Gamma[\[Nu]+ \[Mu]+ 1]]*Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], Divide[1,2]- \[Nu], Divide[1,(x)^(2)]]-Divide[(Pi)^(1/ 2)*((x)^(2)- 1)^(\[Mu]/ 2),(2)^(\[Nu]+ 1)* Gamma[\[Mu]- \[Nu]]*(x)^(\[Nu]+ \[Mu]+ 1)]*Hypergeometric2F1Regularized[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Divide[1,(x)^(2)]]	Failure	Failure	Fail Float(undefined)+Float(undefined)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(undefined)+Float(undefined)*I <- {mu = 2^(1/2)-I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Skip
14.3.E17	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\pi\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}}\left(\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}-\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}\right)}}$	LegendreP(nu, - mu, x)=(Pi*((x)^(2)- 1)^(mu/ 2))/((2)^(mu))*((hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [(1)/(2)], (x)^(2))/GAMMA((1)/(2)))/(GAMMA((1)/(2)*mu -(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))-(x*hypergeom([(1)/(2)*mu -(1)/(2)*nu +(1)/(2), (1)/(2)*nu +(1)/(2)*mu + 1], [(3)/(2)], (x)^(2))/GAMMA((3)/(2)))/(GAMMA((1)/(2)*mu -(1)/(2)*nu)*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2))))	LegendreP[\[Nu], - \[Mu], 3, x]=Divide[Pi*((x)^(2)- 1)^(\[Mu]/ 2),(2)^(\[Mu])]*(Divide[Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], Divide[1,2], (x)^(2)],Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]]-Divide[x*Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[3,2], (x)^(2)],Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]]])	Failure	Failure	Fail Float(undefined)+Float(undefined)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(undefined)+Float(undefined)*I <- {mu = 2^(1/2)-I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.3.E18	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\mu+1;1-\frac{1}{x^{2}}\right)}}$	LegendreP(nu, - mu, x)= (2)^(- mu)* (x)^(nu - mu)*((x)^(2)- 1)^(mu/ 2)* hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*mu -(1)/(2)*nu +(1)/(2)], [mu + 1], 1 -(1)/((x)^(2)))/GAMMA(mu + 1)	LegendreP[\[Nu], - \[Mu], 3, x]= (2)^(- \[Mu])* (x)^(\[Nu]- \[Mu])*((x)^(2)- 1)^(\[Mu]/ 2)* Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], \[Mu]+ 1, 1 -Divide[1,(x)^(2)]]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = -2^(1/2)-I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.3.E21	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(1-x^{2}\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right)}}$	LegendreP(nu, mu, x)=((2)^(mu)* GAMMA(1 - 2*mu)*GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*GAMMA(1 - mu)*(1 - (x)^(2))^(mu/ 2))*GegenbauerC(nu + mu, (1)/(2)- mu, x)	LegendreP[\[Nu], \[Mu], x]=Divide[(2)^(\[Mu])* Gamma[1 - 2*\[Mu]]*Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*Gamma[1 - \[Mu]]*(1 - (x)^(2))^(\[Mu]/ 2)]*GegenbauerC[\[Nu]+ \[Mu], Divide[1,2]- \[Mu], x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(-infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.3.E22	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(x^{2}-1\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right)}}$	LegendreP(nu, mu, x)=((2)^(mu)* GAMMA(1 - 2*mu)*GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*GAMMA(1 - mu)*((x)^(2)- 1)^(mu/ 2))*GegenbauerC(nu + mu, (1)/(2)- mu, x)	LegendreP[\[Nu], \[Mu], 3, x]=Divide[(2)^(\[Mu])* Gamma[1 - 2*\[Mu]]*Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*Gamma[1 - \[Mu]]*((x)^(2)- 1)^(\[Mu]/ 2)]*GegenbauerC[\[Nu]+ \[Mu], Divide[1,2]- \[Mu], x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(-infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.3.E23	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{\mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i}(2\nu+1)}\left(\operatorname{arcsinh}\left((\tfrac{1}{2}x-\tfrac{1}{2})^{\ifrac{1}{2}}\right)\right)}}$	LegendreP(nu, mu, x)=(1)/(GAMMA(1 - mu))*((x + 1)/(x - 1))^(mu/ 2)* hypergeom([((- mu)+(mu)+1-I*(- I*(2*nu + 1)))/2, ((- mu)+(mu)+1+I*(- I*(2*nu + 1)))], [(- mu)+1], -sinh(arcsinh(((1)/(2)*x -(1)/(2))^((1)/(2))))^2)	Error	Failure	Error	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} 2.046636964-.4107385956*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 2} 2.134810006+6.018716078*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	-
14.5.E1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(0\right)=\frac{2^{\mu}\pi^{1/2}}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\nu-\frac{1}{2}\mu\right)}}}$	LegendreP(nu, mu, 0)=((2)^(mu)* (Pi)^(1/ 2))/(GAMMA((1)/(2)*nu -(1)/(2)*mu + 1)*GAMMA((1)/(2)-(1)/(2)*nu -(1)/(2)*mu))	LegendreP[\[Nu], \[Mu], 0]=Divide[(2)^(\[Mu])* (Pi)^(1/ 2),Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]]]	Successful	Failure	-	Successful
14.5.E3	${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(0\right)=-\frac{2^{\mu-1}\pi^{1/2}\sin\left(\frac{1}{2}(\nu+\mu)\pi\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}}}$	LegendreQ(nu, mu, 0)= -((2)^(mu - 1)* (Pi)^(1/ 2)* sin((1)/(2)*(nu + mu)* Pi)*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu + 1))	LegendreQ[\[Nu], \[Mu], 0]= -Divide[(2)^(\[Mu]- 1)* (Pi)^(1/ 2)* Sin[Divide[1,2]*(\[Nu]+ \[Mu])* Pi]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1]]	Successful	Failure	-	Successful
14.5.E5	${\displaystyle{\displaystyle\mathsf{P}_{0}\left(x\right)=P_{0}\left(x\right)}}$	LegendreP(0, x)= LegendreP(0, x)	LegendreP[0, x]= LegendreP[0, 0, 3, x]	Successful	Successful	-	-
14.5.E5	${\displaystyle{\displaystyle P_{0}\left(x\right)=1}}$	LegendreP(0, x)= 1	LegendreP[0, 0, 3, x]= 1	Successful	Successful	-	-
14.5.E6	${\displaystyle{\displaystyle\mathsf{P}_{1}\left(x\right)=P_{1}\left(x\right)}}$	LegendreP(1, x)= LegendreP(1, x)	LegendreP[1, x]= LegendreP[1, 0, 3, x]	Successful	Successful	-	-
14.5.E6	${\displaystyle{\displaystyle P_{1}\left(x\right)=x}}$	LegendreP(1, x)= x	LegendreP[1, 0, 3, x]= x	Successful	Successful	-	-
14.5.E7	${\displaystyle{\displaystyle\mathsf{Q}_{0}\left(x\right)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)}}$	LegendreQ(0, x)=(1)/(2)*ln((1 + x)/(1 - x))	LegendreQ[0, x]=Divide[1,2]*Log[Divide[1 + x,1 - x]]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {x = 1} -.2e-9-3.141592654*I <- {x = 2} 0.-3.141592654*I <- {x = 3}	Fail Complex[0.0, -3.141592653589793] <- {Rule[x, 2]} Complex[0.0, -3.141592653589793] <- {Rule[x, 3]}
14.5.E8	${\displaystyle{\displaystyle\mathsf{Q}_{1}\left(x\right)=\frac{x}{2}\ln\left(\frac{1+x}{1-x}\right)-1}}$	LegendreQ(1, x)=(x)/(2)*ln((1 + x)/(1 - x))- 1	LegendreQ[1, x]=Divide[x,2]*Log[Divide[1 + x,1 - x]]- 1	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {x = 1} 0.-6.283185308*I <- {x = 2} 0.-9.424777961*I <- {x = 3}	Fail Complex[0.0, -6.283185307179586] <- {Rule[x, 2]} Complex[0.0, -9.42477796076938] <- {Rule[x, 3]}
14.5.E9	${\displaystyle{\displaystyle\boldsymbol{Q}_{0}\left(x\right)=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)}}$	LegendreQ(0,x)/GAMMA(0+1)=(1)/(2)*ln((x + 1)/(x - 1))	Exp[-0 Pi I] LegendreQ[0, 2, 3, x]/Gamma[0 + 3]=Divide[1,2]*Log[Divide[x + 1,x - 1]]	Successful	Failure	-	Fail Complex[0.11736052233261163, -1.6328623988631373*^-16] <- {Rule[x, 2]} Complex[0.028426409720027357, -9.184850993605148*^-17] <- {Rule[x, 3]}
14.5.E10	${\displaystyle{\displaystyle\boldsymbol{Q}_{1}\left(x\right)=\frac{x}{2}\ln\left(\frac{x+1}{x-1}\right)-1}}$	LegendreQ(1,x)/GAMMA(1+1)=(x)/(2)*ln((x + 1)/(x - 1))- 1	Exp[-1 Pi I] LegendreQ[1, 2, 3, x]/Gamma[1 + 3]=Divide[x,2]*Log[Divide[x + 1,x - 1]]- 1	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {x = 1}	Fail Complex[-0.20972339977922083, 2.7214373314385625*^-17] <- {Rule[x, 2]} Complex[-0.0813874375065845, 1.020538999289461*^-17] <- {Rule[x, 3]}
14.5.E11	${\displaystyle{\displaystyle\mathsf{P}^{1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\cos\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)}}$	LegendreP(nu, 1/ 2, cos(theta))=((2)/(Pi*sin(theta)))^(1/ 2)* cos((nu +(1)/(2))* theta)	LegendreP[\[Nu], 1/ 2, Cos[\[Theta]]]=(Divide[2,Pi*Sin[\[Theta]]])^(1/ 2)* Cos[(\[Nu]+Divide[1,2])* \[Theta]]	Failure	Failure	Fail .36871967-42.38335731*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .4400371092-.3893821086*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} .4400371092+.3893821086*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .36871967+42.38335731*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.3687196755643214, -42.38335740304453] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.4400371109210073, 0.3893821072191709] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[10.19189212608922, -1.5333343011916822] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.6928135017632475, 0.6617977898574373] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E12	${\displaystyle{\displaystyle\mathsf{P}^{-1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\frac{\sin\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}}}}$	LegendreP(nu, - 1/ 2, cos(theta))=((2)/(Pi*sin(theta)))^(1/ 2)*(sin((nu +(1)/(2))* theta))/(nu +(1)/(2))	LegendreP[\[Nu], - 1/ 2, Cos[\[Theta]]]=(Divide[2,Pi*Sin[\[Theta]]])^(1/ 2)*Divide[Sin[(\[Nu]+Divide[1,2])* \[Theta]],\[Nu]+Divide[1,2]]	Failure	Failure	Fail 10.45952059+14.41340860*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -.867303992e-1+.3960930964*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} -.867303992e-1-.3960930964*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 10.45952059-14.41340860*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[10.459520610822572, 14.413408633208103] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.08673039966401005, -0.3960930962039164] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.5351323601542646, 5.567755012428927] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.3292536871449429, -0.1342721773397682] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E13	${\displaystyle{\displaystyle\mathsf{Q}^{1/2}_{\nu}\left(\cos\theta\right)=-\left(\frac{\pi}{2\sin\theta}\right)^{1/2}\sin\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)}}$	LegendreQ(nu, 1/ 2, cos(theta))= -((Pi)/(2*sin(theta)))^(1/ 2)* sin((nu +(1)/(2))* theta)	LegendreQ[\[Nu], 1/ 2, Cos[\[Theta]]]= -(Divide[Pi,2*Sin[\[Theta]]])^(1/ 2)* Sin[(\[Nu]+Divide[1,2])* \[Theta]]	Failure	Failure	Fail .56844255-66.57402099*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 1.140682031-.9983219148*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} 1.140682031+.9983219148*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .56844255+66.57402099*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.5684425392649075, -66.57402114898068] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.1406820325736367, 0.9983219133728708] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-16.00900172909977, 2.3638891588232163] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.7711003361816383, 0.5385971330629152] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E14	${\displaystyle{\displaystyle\mathsf{Q}^{-1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{\pi}{2\sin\theta}\right)^{1/2}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}}}}$	LegendreQ(nu, - 1/ 2, cos(theta))=((Pi)/(2*sin(theta)))^(1/ 2)*(cos((nu +(1)/(2))* theta))/(nu +(1)/(2))	LegendreQ[\[Nu], - 1/ 2, Cos[\[Theta]]]=(Divide[Pi,2*Sin[\[Theta]]])^(1/ 2)*Divide[Cos[(\[Nu]+Divide[1,2])* \[Theta]],\[Nu]+Divide[1,2]]	Failure	Failure	Fail -16.42654635-22.64375209*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .808817414e-1-.3792805859*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} .808817414e-1+.3792805859*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -16.42654635+22.64375209*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-16.426546382051445, -22.64375214004543] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.08088174281192567, 0.37928058584668234] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-3.960025306655171, 8.760400975986027] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.8692668859401657, 0.20758769049185705] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E15	${\displaystyle{\displaystyle P^{1/2}_{\nu}\left(\cosh\xi\right)=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\cosh\left(\left(\nu+\tfrac{1}{2}\right)\xi\right)}}$	LegendreP(nu, 1/ 2, cosh(xi))=((2)/(Pi*sinh(xi)))^(1/ 2)* cosh((nu +(1)/(2))* xi)	LegendreP[\[Nu], 1/ 2, 3, Cosh[\[Xi]]]=(Divide[2,Pi*Sinh[\[Xi]]])^(1/ 2)* Cosh[(\[Nu]+Divide[1,2])* \[Xi]]	Failure	Failure	Fail -.5864879536-.358184945e-1*I <- {nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} 29.70883518+30.23028354*I <- {nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} 29.70883518-30.23028354*I <- {nu = 2^(1/2)-I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} -.5864879536+.358184945e-1*I <- {nu = 2^(1/2)-I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-0.5864879535933634, -0.03581849661859793] <- {Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[29.708835246197378, 30.230283612094272] <- {Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[29.70883524619738, -30.230283612094276] <- {Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.5864879535933634, 0.03581849661859793] <- {Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E16	${\displaystyle{\displaystyle P^{-1/2}_{\nu}\left(\cosh\xi\right)=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\frac{\sinh\left(\left(\nu+\frac{1}{2}\right)\xi\right)}{\nu+\frac{1}{2}}}}$	LegendreP(nu, - 1/ 2, cosh(xi))=((2)/(Pi*sinh(xi)))^(1/ 2)*(sinh((nu +(1)/(2))* xi))/(nu +(1)/(2))	LegendreP[\[Nu], - 1/ 2, 3, Cosh[\[Xi]]]=(Divide[2,Pi*Sinh[\[Xi]]])^(1/ 2)*Divide[Sinh[(\[Nu]+Divide[1,2])* \[Xi]],\[Nu]+Divide[1,2]]	Failure	Failure	Fail -.2187524610-.3414077677*I <- {nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} 2.795821027-17.58781691*I <- {nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} 2.795821027+17.58781691*I <- {nu = 2^(1/2)-I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} -.2187524610+.3414077677*I <- {nu = 2^(1/2)-I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-0.21875246056952388, -0.34140776804440603] <- {Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[2.7958210326810695, -17.58781693642728] <- {Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[2.7958210326810704, 17.58781693642728] <- {Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.21875246056952388, 0.34140776804440603] <- {Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E18	${\displaystyle{\displaystyle\mathsf{P}^{-\nu}_{\nu}\left(\cos\theta\right)=\frac{(\sin\theta)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}}}$	LegendreP(nu, - nu, cos(theta))=((sin(theta))^(nu))/((2)^(nu)* GAMMA(nu + 1))	LegendreP[\[Nu], - \[Nu], Cos[\[Theta]]]=Divide[(Sin[\[Theta]])^(\[Nu]),(2)^(\[Nu])* Gamma[\[Nu]+ 1]]	Failure	Failure	Fail -43.52472475-91.18820633*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .7853993815-1.577945162*I <- {nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} .7853993815+1.577945162*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -43.52472475+91.18820633*I <- {nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-43.52472498317512, -91.18820649263243] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.7853993822180476, 1.5779451625825405] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-5.58992974788194, -1.258674442064952] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-254.2316906545912, -207.99030953967707] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E19	${\displaystyle{\displaystyle P^{-\nu}_{\nu}\left(\cosh\xi\right)=\frac{(\sinh\xi)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}}}$	LegendreP(nu, - nu, cosh(xi))=((sinh(xi))^(nu))/((2)^(nu)* GAMMA(nu + 1))	LegendreP[\[Nu], - \[Nu], 3, Cosh[\[Xi]]]=Divide[(Sinh[\[Xi]])^(\[Nu]),(2)^(\[Nu])* Gamma[\[Nu]+ 1]]	Failure	Failure	Fail 15.96343012-3.050402005*I <- {nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} -10.72821959-2.234163594*I <- {nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} -10.72821959+2.234163594*I <- {nu = 2^(1/2)-I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} 15.96343012+3.050402005*I <- {nu = 2^(1/2)-I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[15.96343013583397, -3.050402002455038] <- {Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-10.728219617058079, -2.2341635916670013] <- {Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[-10.728219617058079, 2.2341635916670013] <- {Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[15.96343013583397, 3.050402002455038] <- {Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.5.E20	${\displaystyle{\displaystyle\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}\left(2\!E\left(\sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(\sin\left(\tfrac{1}{2}\theta\right)\right)\right)}}$	LegendreP((1)/(2), cos(theta))=(2)/(Pi)*(2*EllipticE(sin((1)/(2)*theta))- EllipticK(sin((1)/(2)*theta)))	LegendreP[Divide[1,2], Cos[\[Theta]]]=Divide[2,Pi]*(2*EllipticE[(Sin[Divide[1,2]*\[Theta]])^2]- EllipticK[(Sin[Divide[1,2]*\[Theta]])^2])	Failure	Failure	Successful	Successful
14.5.E21	${\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}K\left(\sin\left(\tfrac{1}{2}\theta\right)\right)}}$	LegendreP(-(1)/(2), cos(theta))=(2)/(Pi)*EllipticK(sin((1)/(2)*theta))	LegendreP[-Divide[1,2], Cos[\[Theta]]]=Divide[2,Pi]*EllipticK[(Sin[Divide[1,2]*\[Theta]])^2]	Failure	Successful	Successful	-
14.5.E22	${\displaystyle{\displaystyle\mathsf{Q}_{\frac{1}{2}}\left(\cos\theta\right)=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right)-2\!E\left(\cos\left(\tfrac{1}{2}\theta\right)\right)}}$	LegendreQ((1)/(2), cos(theta))= EllipticK(cos((1)/(2)*theta))- 2*EllipticE(cos((1)/(2)*theta))	LegendreQ[Divide[1,2], Cos[\[Theta]]]= EllipticK[(Cos[Divide[1,2]*\[Theta]])^2]- 2*EllipticE[(Cos[Divide[1,2]*\[Theta]])^2]	Failure	Failure	Successful	Successful
14.5.E23	${\displaystyle{\displaystyle\mathsf{Q}_{-\frac{1}{2}}\left(\cos\theta\right)=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right)}}$	LegendreQ(-(1)/(2), cos(theta))= EllipticK(cos((1)/(2)*theta))	LegendreQ[-Divide[1,2], Cos[\[Theta]]]= EllipticK[(Cos[Divide[1,2]*\[Theta]])^2]	Failure	Failure	Successful	Successful
14.5.E24	${\displaystyle{\displaystyle P_{\frac{1}{2}}\left(\cosh\xi\right)=\frac{2}{\pi}e^{\xi/2}E\left(\left(1-e^{-2\xi}\right)^{1/2}\right)}}$	LegendreP((1)/(2), cosh(xi))=(2)/(Pi)*exp(xi/ 2)*EllipticE((1 - exp(- 2*xi))^(1/ 2))	LegendreP[Divide[1,2], 0, 3, Cosh[\[Xi]]]=Divide[2,Pi]*Exp[\[Xi]/ 2]*EllipticE[((1 - Exp[- 2*\[Xi]])^(1/ 2))^2]	Failure	Failure	Successful	Successful
14.5.E25	${\displaystyle{\displaystyle P_{-\frac{1}{2}}\left(\cosh\xi\right)=\frac{2}{\pi\cosh\left(\frac{1}{2}\xi\right)}K\left(\tanh\left(\tfrac{1}{2}\xi\right)\right)}}$	LegendreP(-(1)/(2), cosh(xi))=(2)/(Pi*cosh((1)/(2)*xi))*EllipticK(tanh((1)/(2)*xi))	LegendreP[-Divide[1,2], 0, 3, Cosh[\[Xi]]]=Divide[2,Pi*Cosh[Divide[1,2]*\[Xi]]]*EllipticK[(Tanh[Divide[1,2]*\[Xi]])^2]	Failure	Failure	Successful	Successful
14.5.E26	${\displaystyle{\displaystyle\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}\cosh\xi\operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(\tfrac{1}{2}\xi\right)E\left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right)}}$	LegendreQ((1)/(2),cosh(xi))/GAMMA((1)/(2)+1)= 2*(Pi)^(- 1/ 2)* cosh(xi)*sech((1)/(2)*xi)*EllipticK(sech((1)/(2)*xi))- 4*(Pi)^(- 1/ 2)* cosh((1)/(2)*xi)*EllipticE(sech((1)/(2)*xi))	Exp[-Divide[1,2] Pi I] LegendreQ[Divide[1,2], 2, 3, Cosh[\[Xi]]]/Gamma[Divide[1,2] + 3]= 2*(Pi)^(- 1/ 2)* Cosh[\[Xi]]*Sech[Divide[1,2]*\[Xi]]*EllipticK[(Sech[Divide[1,2]*\[Xi]])^2]- 4*(Pi)^(- 1/ 2)* Cosh[Divide[1,2]*\[Xi]]*EllipticE[(Sech[Divide[1,2]*\[Xi]])^2]	Failure	Failure	Successful	Fail Complex[-0.044625103511119146, 0.2806690096307465] <- {Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.26468966664049587, -0.07266499814523009] <- {Rule[ξ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.044625103511119146, 0.2806690096307465] <- {Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.26468966664049587, -0.07266499814523009] <- {Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
14.5.E27	${\displaystyle{\displaystyle\boldsymbol{Q}_{-\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}e^{-\xi/2}K\left(e^{-\xi}\right)}}$	LegendreQ(-(1)/(2),cosh(xi))/GAMMA(-(1)/(2)+1)= 2*(Pi)^(- 1/ 2)* exp(- xi/ 2)*EllipticK(exp(- xi))	Exp[--Divide[1,2] Pi I] LegendreQ[-Divide[1,2], 2, 3, Cosh[\[Xi]]]/Gamma[-Divide[1,2] + 3]= 2*(Pi)^(- 1/ 2)* Exp[- \[Xi]/ 2]*EllipticK[(Exp[- \[Xi]])^2]	Failure	Failure	Fail -.4187536393-1.678029842*I <- {xi = -2^(1/2)-I*2^(1/2)} -.4187536393+1.678029842*I <- {xi = -2^(1/2)+I*2^(1/2)}	Fail Complex[-1.1386459372175475, 0.07969681822998748] <- {Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.16714638755886108, -1.0459557923897413] <- {Rule[ξ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.5573995767093396, -1.598333024494445] <- {Rule[ξ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.5859000270506534, 0.6320740503346911] <- {Rule[ξ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
14.5.E28	${\displaystyle{\displaystyle\mathsf{P}_{2}\left(x\right)=P_{2}\left(x\right)}}$	LegendreP(2, x)= LegendreP(2, x)	LegendreP[2, x]= LegendreP[2, 0, 3, x]	Successful	Successful	-	-
14.5.E28	${\displaystyle{\displaystyle P_{2}\left(x\right)=\frac{3x^{2}-1}{2}}}$	LegendreP(2, x)=(3*(x)^(2)- 1)/(2)	LegendreP[2, 0, 3, x]=Divide[3*(x)^(2)- 1,2]	Successful	Successful	-	-
14.5.E29	${\displaystyle{\displaystyle\mathsf{Q}_{2}\left(x\right)=\frac{3x^{2}-1}{4}\ln\left(\frac{1+x}{1-x}\right)-\frac{3}{2}x}}$	LegendreQ(2, x)=(3*(x)^(2)- 1)/(4)*ln((1 + x)/(1 - x))-(3)/(2)*x	LegendreQ[2, x]=Divide[3*(x)^(2)- 1,4]*Log[Divide[1 + x,1 - x]]-Divide[3,2]*x	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {x = 1} -.1e-8-17.27875960*I <- {x = 2} -.1e-8-40.84070450*I <- {x = 3}	Fail Complex[0.0, -17.27875959474386] <- {Rule[x, 2]} Complex[0.0, -40.840704496667314] <- {Rule[x, 3]}
14.5.E30	${\displaystyle{\displaystyle\boldsymbol{Q}_{2}\left(x\right)=\frac{3x^{2}-1}{8}\ln\left(\frac{x+1}{x-1}\right)-\frac{3}{4}x}}$	LegendreQ(2,x)/GAMMA(2+1)=(3*(x)^(2)- 1)/(8)*ln((x + 1)/(x - 1))-(3)/(4)*x	Exp[-2 Pi I] LegendreQ[2, 2, 3, x]/Gamma[2 + 3]=Divide[3*(x)^(2)- 1,8]*Log[Divide[x + 1,x - 1]]-Divide[3,4]*x	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {x = 1}	Successful
14.6.E1	${\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{P}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$	LegendreP(nu, m, x)=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* diff(LegendreP(nu, x), [x$(m)])	LegendreP[\[Nu], m, x]=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* D[LegendreP[\[Nu], x], {x, m}]	Failure	Failure	Successful	Successful
14.6.E2	${\displaystyle{\displaystyle\mathsf{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$	LegendreQ(nu, m, x)=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* diff(LegendreQ(nu, x), [x$(m)])	LegendreQ[\[Nu], m, x]=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* D[LegendreQ[\[Nu], x], {x, m}]	Failure	Failure	Skip	Successful
14.6.E3	${\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}P_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}$	LegendreP(nu, m, x)=((x)^(2)- 1)^(m/ 2)* diff(LegendreP(nu, x), [x$(m)])	LegendreP[\[Nu], m, 3, x]=((x)^(2)- 1)^(m/ 2)* D[LegendreP[\[Nu], 0, 3, x], {x, m}]	Failure	Failure	Successful	Successful
14.7.E1	${\displaystyle{\displaystyle\mathsf{P}^{0}_{n}\left(x\right)=\mathsf{P}_{n}\left(x\right)}}$	LegendreP(n, 0, x)= LegendreP(n, x)	LegendreP[n, 0, x]= LegendreP[n, x]	Successful	Failure	-	Successful
14.7.E1	${\displaystyle{\displaystyle\mathsf{P}_{n}\left(x\right)=P^{0}_{n}\left(x\right)}}$	LegendreP(n, x)= LegendreP(n, 0, x)	LegendreP[n, x]= LegendreP[n, 0, 3, x]	Successful	Failure	-	Successful
14.7.E1	${\displaystyle{\displaystyle P^{0}_{n}\left(x\right)=P_{n}\left(x\right)}}$	LegendreP(n, 0, x)= LegendreP(n, x)	LegendreP[n, 0, 3, x]= LegendreP[n, x]	Successful	Failure	-	Successful
14.7.E2	${\displaystyle{\displaystyle\mathsf{Q}^{0}_{n}\left(x\right)=\mathsf{Q}_{n}\left(x\right)}}$	LegendreQ(n, 0, x)= LegendreQ(n, x)	LegendreQ[n, 0, x]= LegendreQ[n, x]	Successful	Successful	-	-
14.7.E2	${\displaystyle{\displaystyle\mathsf{Q}_{n}\left(x\right)=\frac{1}{2}P_{n}\left(x\right)\ln\left(\frac{1+x}{1-x}\right)-W_{n-1}(x)}}$	LegendreQ(n, x)=(1)/(2)*LegendreP(n, x)*ln((1 + x)/(1 - x))- W[n - 1]*(x)	LegendreQ[n, x]=Divide[1,2]*LegendreP[n, x]*Log[Divide[1 + x,1 - x]]- Subscript[W, n - 1]*(x)	Failure	Failure	Error	Successful
14.7.E3	${\displaystyle{\displaystyle W_{n-1}(x)=\sum_{s=0}^{n-1}\frac{(n+s)!(\psi\left(n+1\right)-\psi\left(s+1\right))}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}}}}$	W[n - 1]*(x)= sum((factorial(n + s)*(Psi(n + 1)- Psi(s + 1)))/((2)^(s)*factorial(n - s)*(factorial(s))^(2))*(x - 1)^(s), s = 0..n - 1)	Subscript[W, n - 1]*(x)= Sum[Divide[(n + s)!*(PolyGamma[n + 1]- PolyGamma[s + 1]),(2)^(s)*(n - s)!*((s)!)^(2)]*(x - 1)^(s), {s, 0, n - 1}]	Failure	Failure	Skip	Successful
14.7.E4	${\displaystyle{\displaystyle W_{n-1}(x)=\sum_{k=1}^{n}\frac{1}{k}P_{k-1}\left(x\right)P_{n-k}\left(x\right)}}$	W[n - 1]*(x)= sum((1)/(k)*LegendreP(k - 1, x)*LegendreP(n - k, x), k = 1..n)	Subscript[W, n - 1]*(x)= Sum[Divide[1,k]*LegendreP[k - 1, x]*LegendreP[n - k, x], {k, 1, n}]	Failure	Failure	Skip	Successful
14.7.E7	${\displaystyle{\displaystyle Q_{n}\left(x\right)=\frac{1}{2}P_{n}\left(x\right)\ln\left(\frac{x+1}{x-1}\right)-W_{n-1}(x)}}$	LegendreQ(n, x)=(1)/(2)*LegendreP(n, x)*ln((x + 1)/(x - 1))- W[n - 1]*(x)	LegendreQ[n, 0, 3, x]=Divide[1,2]*LegendreP[n, x]*Log[Divide[x + 1,x - 1]]- Subscript[W, n - 1]*(x)	Failure	Failure	Error	Successful
14.7.E8	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{P}_{n}\left(x\right)}}$	LegendreP(n, m, x)=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* diff(LegendreP(n, x), [x$(m)])	LegendreP[n, m, x]=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* D[LegendreP[n, x], {x, m}]	Failure	Failure	Successful	Successful
14.7.E9	${\displaystyle{\displaystyle\mathsf{Q}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{Q}_{n}\left(x\right)}}$	LegendreQ(n, m, x)=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* diff(LegendreQ(n, x), [x$(m)])	LegendreQ[n, m, x]=(- 1)^(m)*(1 - (x)^(2))^(m/ 2)* D[LegendreQ[n, x], {x, m}]	Failure	Failure	Skip	Successful
14.7.E10	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(1-x^{2}\right)^{n}}}$	LegendreP(n, m, x)=(- 1)^(m + n)*((1 - (x)^(2))^(m/ 2))/((2)^(n)* factorial(n))*diff((1 - (x)^(2))^(n), [x$(m + n)])	LegendreP[n, m, x]=(- 1)^(m + n)*Divide[(1 - (x)^(2))^(m/ 2),(2)^(n)* (n)!]*D[(1 - (x)^(2))^(n), {x, m + n}]	Failure	Failure	Fail Float(undefined)+Float(undefined)*I <- {m = 1, n = 1, x = 1} Float(undefined)+Float(undefined)*I <- {m = 1, n = 1, x = 2} Float(undefined)+Float(undefined)*I <- {m = 1, n = 1, x = 3} Float(undefined)+Float(undefined)*I <- {m = 1, n = 2, x = 1} ... skip entries to safe data	Successful
14.7.E11	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}P_{n}\left(x\right)}}$	LegendreP(n, m, x)=((x)^(2)- 1)^(m/ 2)* diff(LegendreP(n, x), [x$(m)])	LegendreP[n, m, 3, x]=((x)^(2)- 1)^(m/ 2)* D[LegendreP[n, x], {x, m}]	Failure	Failure	Successful	Successful
14.7.E13	${\displaystyle{\displaystyle P_{n}\left(x\right)=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{2}-1\right)^{n}}}$	LegendreP(n, x)=(1)/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(n)])	LegendreP[n, x]=Divide[1,(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, n}]	Failure	Failure	Error	Successful
14.7.E14	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(x^{2}-1\right)^{n}}}$	LegendreP(n, m, x)=(((x)^(2)- 1)^(m/ 2))/((2)^(n)* factorial(n))*diff(((x)^(2)- 1)^(n), [x$(m + n)])	LegendreP[n, m, 3, x]=Divide[((x)^(2)- 1)^(m/ 2),(2)^(n)* (n)!]*D[((x)^(2)- 1)^(n), {x, m + n}]	Failure	Failure	Fail Float(undefined)+Float(undefined)*I <- {m = 1, n = 1, x = 1} Float(undefined)+Float(undefined)*I <- {m = 1, n = 1, x = 2} Float(undefined)+Float(undefined)*I <- {m = 1, n = 1, x = 3} Float(undefined)+Float(undefined)*I <- {m = 1, n = 2, x = 1} ... skip entries to safe data	Successful
14.7.E15	${\displaystyle{\displaystyle P^{m}_{m}\left(x\right)=\frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}}}$	LegendreP(m, m, x)=(factorial(2*m))/((2)^(m)* factorial(m))*((x)^(2)- 1)^(m/ 2)	LegendreP[m, m, 3, x]=Divide[(2*m)!,(2)^(m)* (m)!]*((x)^(2)- 1)^(m/ 2)	Failure	Failure	Successful	Successful
14.7.E16	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)=P^{m}_{n}\left(x\right)}}$	LegendreP(n, m, x)= LegendreP(n, m, x)	LegendreP[n, m, x]= LegendreP[n, m, 3, x]	Successful	Failure	-	Successful
14.7.E16	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=0}}$	LegendreP(n, m, x)= 0	LegendreP[n, m, 3, x]= 0	Failure	Failure	Skip	Successful
14.7.E17	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(-x\right)=(-1)^{n-m}\mathsf{P}^{m}_{n}\left(x\right)}}$	LegendreP(n, m, - x)=(- 1)^(n - m)* LegendreP(n, m, x)	LegendreP[n, m, - x]=(- 1)^(n - m)* LegendreP[n, m, x]	Failure	Failure	Successful	Successful
14.7.E18	${\displaystyle{\displaystyle\mathsf{Q}^{+m}_{n}\left(-x\right)=(-1)^{n-m-1}\mathsf{Q}^{+m}_{n}\left(x\right)}}$	LegendreQ(n, + m, - x)=(- 1)^(n - m - 1)* LegendreQ(n, + m, x)	LegendreQ[n, + m, - x]=(- 1)^(n - m - 1)* LegendreQ[n, + m, x]	Failure	Failure	Error	Successful
14.7.E18	${\displaystyle{\displaystyle\mathsf{Q}^{-m}_{n}\left(-x\right)=(-1)^{n-m-1}\mathsf{Q}^{-m}_{n}\left(x\right)}}$	LegendreQ(n, - m, - x)=(- 1)^(n - m - 1)* LegendreQ(n, - m, x)	LegendreQ[n, - m, - x]=(- 1)^(n - m - 1)* LegendreQ[n, - m, x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {m = 1, n = 1, x = 1} Float(infinity)+Float(infinity)*I <- {m = 1, n = 2, x = 1} Float(infinity)+Float(infinity)*I <- {m = 1, n = 3, x = 1} Float(infinity)+Float(infinity)*I <- {m = 2, n = 1, x = 1} ... skip entries to safe data	Successful
14.7.E19	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{n}=\left(1-2xh+h^{2}\right)^{-1/2}}}$	sum(LegendreP(n, x)*(h)^(n), n = 0..infinity)=(1 - 2*x*h + (h)^(2))^(- 1/ 2)	Sum[LegendreP[n, x]*(h)^(n), {n, 0, Infinity}]=(1 - 2*x*h + (h)^(2))^(- 1/ 2)	Failure	Successful	Skip	-
14.7.E20	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{Q}_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}\right)}}$	sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity)=(1)/((1 - 2*x*h + (h)^(2))^(1/ 2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/ 2))/((1 - (x)^(2))^(1/ 2)))	Sum[LegendreQ[n, x]*(h)^(n), {n, 0, Infinity}]=Divide[1,(1 - 2*x*h + (h)^(2))^(1/ 2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/ 2),(1 - (x)^(2))^(1/ 2)]]	Failure	Failure	Skip	Skip
14.7.E21	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{-n-1}=\left(1-2xh+h^{2}\right)^{-1/2}}}$	sum(LegendreP(n, x)*(h)^(- n - 1), n = 0..infinity)=(1 - 2*x*h + (h)^(2))^(- 1/ 2)	Sum[LegendreP[n, x]*(h)^(- n - 1), {n, 0, Infinity}]=(1 - 2*x*h + (h)^(2))^(- 1/ 2)	Failure	Failure	Skip	Fail Complex[-0.15300174890586637, -0.8864791595823325] <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]} Complex[-0.18053688047160102, -0.6525291152630062] <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]} Complex[-0.15300174890586637, 0.8864791595823325] <- {Rule[h, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 2]} Complex[-0.18053688047160102, 0.6525291152630062] <- {Rule[h, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 3]} ... skip entries to safe data
14.7.E22	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}Q_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right)^{1/2}}\right)}}$	sum(LegendreQ(n, x)*(h)^(n), n = 0..infinity)=(1)/((1 - 2*x*h + (h)^(2))^(1/ 2))* ln((x - h +(1 - 2*x*h + (h)^(2))^(1/ 2))/(((x)^(2)- 1)^(1/ 2)))	Sum[LegendreQ[n, 0, 3, x]*(h)^(n), {n, 0, Infinity}]=Divide[1,(1 - 2*x*h + (h)^(2))^(1/ 2)]* Log[Divide[x - h +(1 - 2*x*h + (h)^(2))^(1/ 2),((x)^(2)- 1)^(1/ 2)]]	Failure	Failure	Skip	Skip
14.9.E1	${\displaystyle{\displaystyle\frac{\pi\sin\left(\mu\pi\right)}{2\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)+\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right)}}$	(Pi*sin(mu*Pi))/(2*GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x)= -(1)/(GAMMA(nu + mu + 1))*LegendreQ(nu, mu, x)+(cos(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreQ(nu, - mu, x)	Divide[Pi*Sin[\[Mu]*Pi],2*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x]= -Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreQ[\[Nu], \[Mu], x]+Divide[Cos[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreQ[\[Nu], - \[Mu], x]	Successful	Successful	-	-
14.9.E2	${\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)}}$	(2*sin(mu*Pi))/(Pi*GAMMA(nu - mu + 1))*LegendreQ(nu, - mu, x)=(1)/(GAMMA(nu + mu + 1))*LegendreP(nu, mu, x)-(cos(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x)	Divide[2*Sin[\[Mu]*Pi],Pi*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreQ[\[Nu], - \[Mu], x]=Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreP[\[Nu], \[Mu], x]-Divide[Cos[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x]	Successful	Successful	-	-
14.9.E3	${\displaystyle{\displaystyle\mathsf{P}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\mathsf{P}^{m}_{\nu}\left(x\right)}}$	LegendreP(nu, - m, x)=(- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreP(nu, m, x)	LegendreP[\[Nu], - m, x]=(- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreP[\[Nu], m, x]	Failure	Failure	Successful	Successful
14.9.E4	${\displaystyle{\displaystyle\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right)}}$	LegendreQ(nu, - m, x)=(- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreQ(nu, m, x)	LegendreQ[\[Nu], - m, x]=(- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreQ[\[Nu], m, x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {nu = 2^(1/2)+I*2^(1/2), m = 1, x = 1} Float(infinity)+Float(infinity)*I <- {nu = 2^(1/2)+I*2^(1/2), m = 2, x = 1} Float(infinity)+Float(infinity)*I <- {nu = 2^(1/2)+I*2^(1/2), m = 3, x = 1} Float(infinity)+Float(infinity)*I <- {nu = 2^(1/2)-I*2^(1/2), m = 1, x = 1} ... skip entries to safe data	Successful
14.9#Ex1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$	LegendreP(- nu - 1, mu, x)= LegendreP(nu, mu, x)	LegendreP[- \[Nu]- 1, \[Mu], x]= LegendreP[\[Nu], \[Mu], x]	Successful	Failure	-	Successful
14.9#Ex2	${\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{-\mu}_{\nu}\left(x\right)}}$	LegendreP(- nu - 1, - mu, x)= LegendreP(nu, - mu, x)	LegendreP[- \[Nu]- 1, - \[Mu], x]= LegendreP[\[Nu], - \[Mu], x]	Successful	Failure	-	Successful
14.9.E6	${\displaystyle{\displaystyle\pi\cos\left(\nu\pi\right)\cos\left(\mu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)=\sin\left((\nu+\mu)\pi\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)}}$	Pi*cos(nu*Pi)*cos(mu*Pi)*LegendreP(nu, mu, x)= sin((nu + mu)* Pi)*LegendreQ(nu, mu, x)- sin((nu - mu)* Pi)*LegendreQ(- nu - 1, mu, x)	Pi*Cos[\[Nu]*Pi]*Cos[\[Mu]*Pi]*LegendreP[\[Nu], \[Mu], x]= Sin[(\[Nu]+ \[Mu])* Pi]*LegendreQ[\[Nu], \[Mu], x]- Sin[(\[Nu]- \[Mu])* Pi]*LegendreQ[- \[Nu]- 1, \[Mu], x]	Successful	Failure	-	Successful
14.9.E7	${\displaystyle{\displaystyle\frac{\sin\left((\nu-\mu)\pi\right)}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\frac{\sin\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(-x\right)}}$	(sin((nu - mu)* Pi))/(GAMMA(nu + mu + 1))*LegendreP(nu, mu, x)=(sin(nu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x)-(sin(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, - x)	Divide[Sin[(\[Nu]- \[Mu])* Pi],Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreP[\[Nu], \[Mu], x]=Divide[Sin[\[Nu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x]-Divide[Sin[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], - x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Successful
14.9.E8	${\displaystyle{\displaystyle\tfrac{1}{2}\pi\sin\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\cos\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)-\mathsf{Q}^{-\mu}_{\nu}\left(-x\right)}}$	(1)/(2)*Pi*sin((nu - mu)* Pi)*LegendreP(nu, - mu, x)= - cos((nu - mu)* Pi)*LegendreQ(nu, - mu, x)- LegendreQ(nu, - mu, - x)	Divide[1,2]*Pi*Sin[(\[Nu]- \[Mu])* Pi]*LegendreP[\[Nu], - \[Mu], x]= - Cos[(\[Nu]- \[Mu])* Pi]*LegendreQ[\[Nu], - \[Mu], x]- LegendreQ[\[Nu], - \[Mu], - x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Successful
14.9.E9	${\displaystyle{\displaystyle\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right)}}$	(2)/(GAMMA(nu + mu + 1)*GAMMA(mu - nu))*LegendreQ(nu, mu, x)= - cos(nu*Pi)*LegendreP(nu, - mu, x)+ cos(mu*Pi)*LegendreP(nu, - mu, - x)	Divide[2,Gamma[\[Nu]+ \[Mu]+ 1]*Gamma[\[Mu]- \[Nu]]]*LegendreQ[\[Nu], \[Mu], x]= - Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], x]+ Cos[\[Mu]*Pi]*LegendreP[\[Nu], - \[Mu], - x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.9.E10	${\displaystyle{\displaystyle(2/\pi)\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\cos\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\mathsf{P}^{-\mu}_{\nu}\left(-x\right)}}$	(2/ Pi)* sin((nu - mu)* Pi)*LegendreQ(nu, - mu, x)= cos((nu - mu)* Pi)*LegendreP(nu, - mu, x)- LegendreP(nu, - mu, - x)	(2/ Pi)* Sin[(\[Nu]- \[Mu])* Pi]*LegendreQ[\[Nu], - \[Mu], x]= Cos[(\[Nu]- \[Mu])* Pi]*LegendreP[\[Nu], - \[Mu], x]- LegendreP[\[Nu], - \[Mu], - x]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)-I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.9#Ex3	${\displaystyle{\displaystyle P^{-\mu}_{-\nu-1}\left(x\right)=P^{-\mu}_{\nu}\left(x\right)}}$	LegendreP(- nu - 1, - mu, x)= LegendreP(nu, - mu, x)	LegendreP[- \[Nu]- 1, - \[Mu], 3, x]= LegendreP[\[Nu], - \[Mu], 3, x]	Successful	Successful	-	-
14.9#Ex4	${\displaystyle{\displaystyle P^{\mu}_{-\nu-1}\left(x\right)=P^{\mu}_{\nu}\left(x\right)}}$	LegendreP(- nu - 1, mu, x)= LegendreP(nu, mu, x)	LegendreP[- \[Nu]- 1, \[Mu], 3, x]= LegendreP[\[Nu], \[Mu], 3, x]	Successful	Successful	-	-
14.9.E13	${\displaystyle{\displaystyle P^{-m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}P^{m}_{\nu}\left(x\right)}}$	LegendreP(nu, - m, x)=(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreP(nu, m, x)	LegendreP[\[Nu], - m, 3, x]=Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreP[\[Nu], m, 3, x]	Failure	Failure	Successful	Successful
14.10.E1	${\displaystyle{\displaystyle{\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}$	LegendreP(nu, mu + 2, x)+ 2*(mu + 1)* x*(1 - (x)^(2))^(- 1/ 2)* LegendreP(nu, mu + 1, x)+(nu - mu)*(nu + mu + 1)* LegendreP(nu, mu, x)= 0	LegendreP[\[Nu], \[Mu]+ 2, x]+ 2*(\[Mu]+ 1)* x*(1 - (x)^(2))^(- 1/ 2)* LegendreP[\[Nu], \[Mu]+ 1, x]+(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)* LegendreP[\[Nu], \[Mu], x]= 0	Failure	Successful	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	-
14.10.E2	${\displaystyle{\displaystyle{\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}$	(1 - (x)^(2))^(1/ 2)* LegendreP(nu, mu + 1, x)-(nu - mu + 1)* LegendreP(nu + 1, mu, x)+(nu + mu + 1)* x*LegendreP(nu, mu, x)= 0	(1 - (x)^(2))^(1/ 2)* LegendreP[\[Nu], \[Mu]+ 1, x]-(\[Nu]- \[Mu]+ 1)* LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ \[Mu]+ 1)* x*LegendreP[\[Nu], \[Mu], x]= 0	Failure	Successful	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(-infinity)-Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	-
14.10.E3	${\displaystyle{\displaystyle{(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}$	(nu - mu + 2)* LegendreP(nu + 2, mu, x)-(2*nu + 3)* x*LegendreP(nu + 1, mu, x)+(nu + mu + 1)* LegendreP(nu, mu, x)= 0	(\[Nu]- \[Mu]+ 2)* LegendreP[\[Nu]+ 2, \[Mu], x]-(2*\[Nu]+ 3)* x*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ \[Mu]+ 1)* LegendreP[\[Nu], \[Mu], x]= 0	Successful	Successful	-	-
14.10.E4	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x\mathsf{P}^{\mu}_{\nu}\left(x\right)}}}$	(1 - (x)^(2))* diff(LegendreP(nu, mu, x), x)=(mu - nu - 1)* LegendreP(nu + 1, mu, x)+(nu + 1)* x*LegendreP(nu, mu, x)	(1 - (x)^(2))* D[LegendreP[\[Nu], \[Mu], x], x]=(\[Mu]- \[Nu]- 1)* LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ 1)* x*LegendreP[\[Nu], \[Mu], x]	Successful	Successful	-	-
14.10.E5	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(x\right)-\nu x\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$	(1 - (x)^(2))* diff(LegendreP(nu, mu, x), x)=(nu + mu)* LegendreP(nu - 1, mu, x)- nu*x*LegendreP(nu, mu, x)	(1 - (x)^(2))* D[LegendreP[\[Nu], \[Mu], x], x]=(\[Nu]+ \[Mu])* LegendreP[\[Nu]- 1, \[Mu], x]- \[Nu]*x*LegendreP[\[Nu], \[Mu], x]	Successful	Successful	-	-
14.10.E6	${\displaystyle{\displaystyle{P^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}P^{\mu+1}_{\nu}\left(x\right)}-(\nu-\mu)(\nu+\mu+1)P^{\mu}_{\nu}\left(x\right)=0}}$	LegendreP(nu, mu + 2, x)+ 2*(mu + 1)* x*((x)^(2)- 1)^(- 1/ 2)* LegendreP(nu, mu + 1, x)-(nu - mu)*(nu + mu + 1)* LegendreP(nu, mu, x)= 0	LegendreP[\[Nu], \[Mu]+ 2, 3, x]+ 2*(\[Mu]+ 1)* x*((x)^(2)- 1)^(- 1/ 2)* LegendreP[\[Nu], \[Mu]+ 1, 3, x]-(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)* LegendreP[\[Nu], \[Mu], 3, x]= 0	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.10.E7	${\displaystyle{\displaystyle{\left(x^{2}-1\right)^{1/2}P^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)P^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)xP^{\mu}_{\nu}\left(x\right)=0}}$	((x)^(2)- 1)^(1/ 2)* LegendreP(nu, mu + 1, x)-(nu - mu + 1)* LegendreP(nu + 1, mu, x)+(nu + mu + 1)* x*LegendreP(nu, mu, x)= 0	((x)^(2)- 1)^(1/ 2)* LegendreP[\[Nu], \[Mu]+ 1, 3, x]-(\[Nu]- \[Mu]+ 1)* LegendreP[\[Nu]+ 1, \[Mu], 3, x]+(\[Nu]+ \[Mu]+ 1)* x*LegendreP[\[Nu], \[Mu], 3, x]= 0	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} Float(-infinity)-Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)-I*2^(1/2), x = 1} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = -2^(1/2)+I*2^(1/2), x = 1} ... skip entries to safe data	Successful
14.11.E1	${\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\pi\cot\left(\nu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{1}{\pi}\mathsf{A}_{\nu}^{\mu}(x)}}$	diff(LegendreP(nu, mu, x), nu)= Pi*cot(nu*Pi)*LegendreP(nu, mu, x)-(1)/(Pi)*(A[nu])^(mu)*(x)	D[LegendreP[\[Nu], \[Mu], x], \[Nu]]= Pi*Cot[\[Nu]*Pi]*LegendreP[\[Nu], \[Mu], x]-Divide[1,Pi]*(Subscript[A, \[Nu]])^(\[Mu])*(x)	Failure	Failure	Skip	Skip
14.11.E2	${\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}\pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{\pi\sin\left(\mu\pi\right)}{\sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}(-x)}}$	diff(LegendreQ(nu, mu, x), nu)= -(1)/(2)*(Pi)^(2)* LegendreP(nu, mu, x)+(Pi*sin(mu*Pi))/(sin(nu*Pi)*sin((nu + mu)* Pi))*LegendreQ(nu, mu, x)-(1)/(2)*cot((nu + mu)* Pi)*(A[nu])^(mu)*(x)+(1)/(2)*csc((nu + mu)* Pi)*(A[nu])^(mu)*(- x)	D[LegendreQ[\[Nu], \[Mu], x], \[Nu]]= -Divide[1,2]*(Pi)^(2)* LegendreP[\[Nu], \[Mu], x]+Divide[Pi*Sin[\[Mu]*Pi],Sin[\[Nu]*Pi]*Sin[(\[Nu]+ \[Mu])* Pi]]*LegendreQ[\[Nu], \[Mu], x]-Divide[1,2]*Cot[(\[Nu]+ \[Mu])* Pi]*(Subscript[A, \[Nu]])^(\[Mu])*(x)+Divide[1,2]*Csc[(\[Nu]+ \[Mu])* Pi]*(Subscript[A, \[Nu]])^(\[Mu])*(- x)	Failure	Failure	Skip	Skip
14.11.E3	${\displaystyle{\displaystyle\mathsf{A}_{\nu}^{\mu}(x)=\sin\left(\nu\pi\right)\left(\frac{1+x}{1-x}\right)^{\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}\Gamma\left(k-\nu\right)\Gamma\left(k+\nu+1\right)}{k!\Gamma\left(k-\mu+1\right)}\*\left(\psi\left(k+\nu+1\right)-\psi\left(k-\nu\right)\right)}}$	(A[nu])^(mu)*(x)= sin(nu*Pi)*((1 + x)/(1 - x))^(mu/ 2)* sum((((1)/(2)-(1)/(2)*x)^(k)* GAMMA(k - nu)*GAMMA(k + nu + 1))/(factorial(k)*GAMMA(k - mu + 1))*(Psi(k + nu + 1)- Psi(k - nu)), k = 0..infinity)	(Subscript[A, \[Nu]])^(\[Mu])*(x)= Sin[\[Nu]*Pi]*(Divide[1 + x,1 - x])^(\[Mu]/ 2)* Sum[Divide[(Divide[1,2]-Divide[1,2]*x)^(k)* Gamma[k - \[Nu]]*Gamma[k + \[Nu]+ 1],(k)!*Gamma[k - \[Mu]+ 1]]*(PolyGamma[k + \[Nu]+ 1]- PolyGamma[k - \[Nu]]), {k, 0, Infinity}]	Failure	Failure	Skip	Error
14.12.E1	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{1/2}(\sin\theta)^{\mu}}{\pi^{1/2}\Gamma\left(\frac{1}{2}-\mu\right)}\int_{0}^{\theta}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)t\right)}{(\cos t-\cos\theta)^{\mu+(1/2)}}\mathrm{d}t}}$	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]}]	Failure	Failure	Skip	Error
14.12.E2	${\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\left(1-x^{2}\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{x}^{1}\mathsf{P}_{\nu}\left(t\right)(t-x)^{\mu-1}\mathrm{d}t}}$	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}]	Failure	Failure	Skip	Skip
14.12.E4	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{2^{1/2}\Gamma\left(\mu+\frac{1}{2}\right)\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\*\int_{0}^{\infty}\frac{\cosh\left(\left(\nu+\frac{1}{2}\right)t\right)}{(x+\cosh t)^{\mu+(1/2)}}\mathrm{d}t}}$	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}]	Failure	Failure	Skip	Error
14.12.E5	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\mathrm{d}t}}$	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}]	Failure	Failure	Skip	Skip
14.12.E7	${\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{{\left(\nu+1\right)_{m}}}{\pi}\*\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi\right)^{\nu}\cos\left(m\phi\right)\mathrm{d}\phi}}$	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}]	Failure	Failure	Skip	Skip
14.12.E8	${\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{2^{m}m!(n+m)!\left(x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi\right)^{n-m}(\sin\phi)^{2m}\mathrm{d}\phi}}$	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}]	Failure	Failure	Skip	Error
14.12.E10	${\displaystyle{\displaystyle u=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)}}$	u =(1)/(2)*ln((x + 1)/(x - 1))	u =Divide[1,2]*Log[Divide[x + 1,x - 1]]	Failure	Failure	Fail Float(-infinity)+1.414213562*I <- {u = 2^(1/2)+I*2^(1/2), x = 1} .8649074175+1.414213562*I <- {u = 2^(1/2)+I*2^(1/2), x = 2} 1.067639972+1.414213562*I <- {u = 2^(1/2)+I*2^(1/2), x = 3} Float(-infinity)-1.414213562*I <- {u = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[-1] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]} Complex[0.8649074180390403, 1.4142135623730951] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]} Complex[1.0676399720931224, 1.4142135623730951] <- {Rule[u, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]} DirectedInfinity[-1] <- {Rule[u, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]} ... skip entries to safe data
14.12.E13	${\displaystyle{\displaystyle\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}\int_{-1}^{1}\frac{P_{n}\left(t\right)}{x-t}\mathrm{d}t}}$	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}]	Failure	Failure	Skip	Error
14.12.E14	${\displaystyle{\displaystyle\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{\infty}\frac{\mathrm{d}t}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{n+1}}}}$	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}]	Failure	Failure	Skip	Error
14.13#Ex1	${\displaystyle{\displaystyle+\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)+\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}\Gamma\left(\nu+\mu+1\right)(2\sin\theta)^{\mu}e^{+(\nu+\mu+1)i\theta}\*\mathbf{F}\left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{+2i\theta}\right)}}$	+(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	Fail -16028.04070+41946.17697*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -.1085313814+.1259393077*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} -10838.77596-7620.380119*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -62.65836322-72.59645613*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Skip
14.13#Ex1	${\displaystyle{\displaystyle-\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)+\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}\Gamma\left(\nu+\mu+1\right)(2\sin\theta)^{\mu}e^{-(\nu+\mu+1)i\theta}\*\mathbf{F}\left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{-2i\theta}\right)}}$	-(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	Fail -525.7608359-50.44260442*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 2.999684768+4.050396905*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} 56.23820942-132.0163440*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .4211699594-1.140171137*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Skip
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 =(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	Fail -.781353228+2.096391260*I <- {alpha = 2^(1/2)+I*2^(1/2), p = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2)} -.781353228-.732035864*I <- {alpha = 2^(1/2)+I*2^(1/2), p = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)-I*2^(1/2)} -3.609780352-.732035864*I <- {alpha = 2^(1/2)+I*2^(1/2), p = 2^(1/2)+I*2^(1/2), rho = -2^(1/2)-I*2^(1/2)} -3.609780352+2.096391260*I <- {alpha = 2^(1/2)+I*2^(1/2), p = 2^(1/2)+I*2^(1/2), rho = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-0.7813532286851879, 2.0963912619832947] <- {Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.7813532286851879, -0.7320358627628958] <- {Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-3.6097803534313786, -0.7320358627628958] <- {Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-3.6097803534313786, 2.0963912619832947] <- {Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... 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(((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	Fail Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), x = 1} -.1981647463-3.477041044*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), x = 2} -.8454592291-4.091101615*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), x = 3} Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail Complex[-0.19816474611075496, -3.4770410458684844] <- {Rule[x, 2], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.8454592292078034, -4.091101617492367] <- {Rule[x, 3], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.666056695470399, -0.5020500570697619] <- {Rule[x, 2], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.3133511785674474, -1.116110628693645] <- {Rule[x, 3], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.15.E21	${\displaystyle{\displaystyle\left(y-\alpha^{2}\right)^{1/2}-\alpha\operatorname{arctan}\left(\frac{\left(y-\alpha^{2}\right)^{1/2}}{\alpha}\right)=\operatorname{arccos}\left(\frac{x}{\left(1-\alpha^{2}\right)^{1/2}}\right)-\frac{\alpha}{2}\operatorname{arccos}\left(\frac{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}}{\left(1-\alpha^{2}\right)\left(1-x^{2}\right)}\right)}}$	(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))]]	Failure	Failure	Skip	Successful
14.15.E22	${\displaystyle{\displaystyle{\left(\alpha^{2}-y\right)^{1/2}+\tfrac{1}{2}\alpha\ln|y|-\alpha\ln\left(\left(\alpha^{2}-y\right)^{1/2}+\alpha\right)}={\ln\left(\frac{x+\left(x^{2}-1+\alpha^{2}\right)^{1/2}}{\left(1-\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(\frac{\left(1-\alpha^{2}\right)\left|1-x^{2}\right|}{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}+2\alpha x\left(x^{2}-1+\alpha^{2}\right)^{1/2}}\right)}}}$	((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	Failure	Skip	Successful
14.15#Ex1	${\displaystyle{\displaystyle\begin{cases}\left({U^{2}}\left(-c,x\right)+{\overline{U}^{2}}\left(-c,x\right)\right)^{1/2},&0<=x}}\)\@add@PDF@RDFa@triples\end{document}\end{cases}$			Error	Failure	-	Error
14.15#Ex2	${\displaystyle{\displaystyle\begin{cases}\left({U^{2}}\left(-c,x\right)+{\overline{U}^{2}}\left(-c,x\right)\right)^{1/2},&0<=x}}\)\@add@PDF@RDFa@triples\end{document}\end{cases}$			Error	Error	-	-
14.15.E27	${\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}-\alpha^{2}\right)^{1/2}-\frac{1}{2}\alpha^{2}\operatorname{arccosh}\left(\frac{\zeta}{\alpha}\right)=\left(1-a^{2}\right)^{1/2}\operatorname{arctanh}\left(\frac{1}{x}\left(\frac{x^{2}-a^{2}}{1-a^{2}}\right)^{1/2}\right)-\operatorname{arccosh}\left(\frac{x}{a}\right)}}$	(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	Skip	Error
14.15.E28	${\displaystyle{\displaystyle\frac{1}{2}\alpha^{2}\operatorname{arcsin}\left(\frac{\zeta}{\alpha}\right)+\frac{1}{2}\zeta\left(\alpha^{2}-\zeta^{2}\right)^{1/2}=\operatorname{arcsin}\left(\frac{x}{a}\right)-\left(1-a^{2}\right)^{1/2}\operatorname{arctan}\left(x\left(\frac{1-a^{2}}{a^{2}-x^{2}}\right)^{1/2}\right)}}$	(1)/(2)*(alpha)^(2)* arcsin((zeta)/(alpha))+(1)/(2)*zeta*((alpha)^(2)- (zeta)^(2))^(1/ 2)= arcsin((x)/(a))-(1 - (a)^(2))^(1/ 2)* arctan(x*((1 - (a)^(2))/((a)^(2)- (x)^(2)))^(1/ 2))	Divide[1,2]*(\[Alpha])^(2)* ArcSin[Divide[\[zeta],\[Alpha]]]+Divide[1,2]*\[zeta]*((\[Alpha])^(2)- (\[zeta])^(2))^(1/ 2)= ArcSin[Divide[x,a]]-(1 - (a)^(2))^(1/ 2)* ArcTan[x*(Divide[1 - (a)^(2),(a)^(2)- (x)^(2)])^(1/ 2)]	Failure	Failure	Skip	Error
14.15.E29	${\displaystyle{\displaystyle\zeta^{2}=-\ln\left(1-x^{2}\right)}}$	(zeta)^(2)= - ln(1 - (x)^(2))	(\[zeta])^(2)= - Log[1 - (x)^(2)]	Failure	Failure	Fail -.2876820725+3.999999998*I <- {zeta = 2^(1/2)+I*2^(1/2), x = 1/2} -.2876820725-3.999999998*I <- {zeta = 2^(1/2)-I*2^(1/2), x = 1/2} -.2876820725+3.999999998*I <- {zeta = -2^(1/2)-I*2^(1/2), x = 1/2} -.2876820725-3.999999998*I <- {zeta = -2^(1/2)+I*2^(1/2), x = 1/2}	Error
14.15.E31	${\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^{1/2}+\frac{1}{2}\alpha^{2}\operatorname{arcsinh}\left(\frac{\zeta}{\alpha}\right)=\left(1+a^{2}\right)^{1/2}\operatorname{arctanh}\left(x\left(\frac{1+a^{2}}{x^{2}+a^{2}}\right)^{1/2}\right)-\operatorname{arcsinh}\left(\frac{x}{a}\right)}}$	(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))	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]]	Failure	Failure	Skip	Error
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((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]=-(1 - (x)^(2))^(-(\[Mu]- 1)/ 2)* LegendreP[\[Nu], \[Mu]- 1, x]	Failure	Failure	Skip	Successful
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((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]=Divide[(1 - (x)^(2))^((\[Mu]+ 1)/ 2),(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)]*LegendreP[\[Nu], \[Mu]+ 1, x]	Error	Failure	-	Successful
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*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]=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	Failure	-	Successful
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((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]=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	Failure	Skip	Error
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((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}]=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	Skip	Successful
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(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}]=Divide[(n + m)!,(n - m)!*(n +Divide[1,2])]*KroneckerDelta[l, n]	Failure	Failure	Skip	Error
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(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}]=Divide[(- 1)^(m),l +Divide[1,2]]*KroneckerDelta[l, n]	Failure	Failure	Skip	Successful
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((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}]=Divide[(n + m)!,(n - m)!*m]*KroneckerDelta[l, m]	Failure	Failure	Skip	Error
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((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}]=Divide[(- 1)^(l),l]*KroneckerDelta[l, m]	Failure	Failure	Skip	Error
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(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}]=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	Failure	-	Successful
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((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}]=Divide[(Pi)^(2)- 2*(Sin[\[Nu]*Pi])^(2)* (D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),(Pi)^(2)*(\[Nu]+Divide[1,2])]	Failure	Failure	Skip	Successful
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(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}]=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)]	Failure	Failure	Skip	Skip
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((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}]=Divide[(Pi)^(2)- 2*(1 + (Cos[\[Nu]*Pi])^(2))* (D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),2*(2*\[Nu]+ 1)]	Failure	Failure	Skip	Error
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(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}]=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)]	Failure	Failure	Skip	Error
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(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}]= -Divide[Sin[2*\[Nu]*Pi]*(D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),Pi*(2*\[Nu]+ 1)]	Failure	Failure	Skip	Error
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(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}]=Divide[(1 -(- 1)^(l + n))*(l + m)!,(l - n)*(l + n + 1)*(l - m)!]	Failure	Failure	Skip	Skip
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(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}]= 0	Failure	Failure	Skip	Error
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(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}]=Divide[1,(\[Lambda]- \[Nu])*(\[Nu]+ \[Lambda]+ 1)]	Failure	Failure	Skip	Successful
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(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}]=Divide[PolyGamma[\[Lambda]+ 1]- PolyGamma[\[Nu]+ 1],(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]	Failure	Failure	Skip	Skip
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((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}]=Divide[D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1,2*\[Nu]+ 1]	Failure	Failure	Skip	Successful
14.18.E1	${\displaystyle{\displaystyle\mathsf{P}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos\left(m\phi\right)}}$	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}]	Error	Failure	-	Skip
14.18.E2	${\displaystyle{\displaystyle\mathsf{P}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\sum_{m=-n}^{n}(-1)^{m}\mathsf{P}^{-m}_{n}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{n}\left(\cos\theta_{2}\right)\cos\left(m\phi\right)}}$	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}]	Failure	Failure	Skip	Skip
14.18.E3	${\displaystyle{\displaystyle\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos\left(m\phi\right)}}$	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}]	Error	Failure	-	Skip
14.18.E4	${\displaystyle{\displaystyle P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)P^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right)}}$	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}]	Error	Failure	-	Skip
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((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}]=(n + 1)*(LegendreP[n + 1, 0, 3, x]*LegendreP[n, 0, 3, y]- LegendreP[n, 0, 3, x]*LegendreP[n + 1, 0, 3, y])	Failure	Failure	Skip	Skip
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((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}]=(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	Failure	Failure	Skip	Successful
14.18.E8	${\displaystyle{\displaystyle\mathsf{P}_{\nu}\left(-x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_{n=0}^{\infty}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}_{n}\left(x\right)}}$	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}]	Failure	Failure	Skip	Skip
14.18.E9	${\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_{n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}^{-\mu}_{n}\left(x\right)}}$	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}]	Failure	Failure	Skip	Error
14.19#Ex1	${\displaystyle{\displaystyle x=\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta}}}$	x =(c*sinh(eta)*cos(phi))/(cosh(eta)- cos(theta))	x =Divide[c*Sinh[\[Eta]]*Cos[\[Phi]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Fail -.616269251+1.502300221*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), x = 1} .383730749+1.502300221*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), x = 2} 1.383730749+1.502300221*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Skip
14.19#Ex2	${\displaystyle{\displaystyle y=\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta}}}$	y =(c*sinh(eta)*sin(phi))/(cosh(eta)- cos(theta))	y =Divide[c*Sinh[\[Eta]]*Sin[\[Phi]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Fail -.746192753-1.746192753*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), y = 1} .253807247-1.746192753*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), y = 2} 1.253807247-1.746192753*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), y = 3} Float(infinity)+Float(infinity)*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), y = 1} ... skip entries to safe data	Skip
14.19#Ex3	${\displaystyle{\displaystyle z=\frac{c\sin\theta}{\cosh\eta-\cos\theta}}}$	z =(c*sin(theta))/(cosh(eta)- cos(theta))	z =Divide[c*Sin[\[Theta]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Fail .5066331465+2.098524802*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} .5066331465-.7299023223*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -2.321793978-.7299023223*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -2.321793978+2.098524802*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.5066331469868752, 2.0985248025073626] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[2.321793977759315, 0.7299023222388277] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.19.E2	${\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-\mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}\right)}}$	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]]]	Failure	Failure	Fail -17.12741418-18.21426284*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2)} 6.524638641-39.40236575*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)-I*2^(1/2)} -2.696896498+.2815203921*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} 325.5260470-172.1893792*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Skip
14.19#Ex4	${\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(1-2\mu\right)2^{2\mu}}{\Gamma\left(1-\mu\right)\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;e^{-2\xi}\right)}}$	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	Fail -14.40303680-12.48223319*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2)} 8.755806503-36.24067863*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)-I*2^(1/2)} -2.762869792-.3736752023e-1*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)-I*2^(1/2)} -23.95584924-45.55470526*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Skip
14.20.E1	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}\right)w=0}}$	(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	Fail Float(-infinity)-Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), tau = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1} 3.417682772-4.124789553*I <- {mu = 2^(1/2)+I*2^(1/2), tau = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2} 4.596194074-5.303300855*I <- {mu = 2^(1/2)+I*2^(1/2), tau = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3} Float(-infinity)-Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), tau = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.417682775734979, -4.1247895569215265] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[4.596194077712559, -5.303300858899106] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[τ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[τ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.20.E9	${\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{2}{\pi}\int_{0}^{\theta}\frac{\cosh\left(\tau\phi\right)}{\sqrt{2(\cos\phi-\cos\theta)}}\mathrm{d}\phi}}$	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]}]	Failure	Failure	Skip	Skip
14.20.E13	${\displaystyle{\displaystyle P_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\cosh\left(\tau\pi\right)}{\pi}\int_{1}^{\infty}\frac{P_{-\frac{1}{2}+i\tau}\left(t\right)}{x+t}\mathrm{d}t}}$	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}]	Failure	Failure	Skip	Error
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((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}]=Divide[1,y + x]	Failure	Failure	Skip	Error
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)=(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	Error	Error
14.20.E21	${\displaystyle{\displaystyle{\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln\eta-\alpha\ln\left(\left(\alpha^{2}+\eta\right)^{1/2}+\alpha\right)}={\operatorname{arccos}\left(\frac{x}{\left(1+\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(\frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}\right)}}}$	((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	Fail Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), x = 1} 2.509204677-2.403472660*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), x = 2} 2.335929278-2.883411364*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), x = 3} Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[Complex[-0.7071067811865475, -0.7071067811865475]] <- {Rule[x, 1], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.5092046786645588, -2.4034726594210074] <- {Rule[x, 2], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.3359292790943744, -2.883411363373449] <- {Rule[x, 3], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[Complex[-0.7071067811865475, -0.7071067811865475]] <- {Rule[x, 1], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
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 =(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	Fail Float(-infinity)-.631329830e-1*I <- {beta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), x = 1} .8588287874-2.880074289*I <- {beta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), x = 2} 1.506123270-3.494134856*I <- {beta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), x = 3} Float(-infinity)-2.891560107*I <- {beta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail DirectedInfinity[-1] <- {Rule[x, 1], Rule[β, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.8588287887643107, -2.8800742905707475] <- {Rule[x, 2], Rule[β, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.5061232718613613, -3.4941348621946324] <- {Rule[x, 3], Rule[β, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[-1] <- {Rule[x, 1], Rule[β, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.21.E1	${\displaystyle{\displaystyle\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0}}$	(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	Fail -3.993073584+10.65512264*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -6.655122641+7.993073582*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -3.993073584+10.65512264*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -6.655122641+7.993073582*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-3.9930735878769763, 10.655122646461624] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[11.320634911107787, -4.658585852523139] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-3.9930735878769754, 2.655122646461624] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[3.320634911107787, -4.658585852523139] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.23.E1	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x+i0\right)=e^{-\mu\pi i/2}\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$	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	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} -145.9645465+265.3087326*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 2} -88.63555579+385.8611656*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail Complex[-159.27099992888859, 235.28464740712568] <- {Rule[x, 2], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-116.1110150711135, 352.86522728793915] <- {Rule[x, 3], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-4037.028253953607, -377.07365549484035] <- {Rule[x, 2], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-5888.751551022363, 3195.540326205004] <- {Rule[x, 3], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.23.E1	${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x-i0\right)=e^{+\mu\pi i/2}\mathsf{P}^{\mu}_{\nu}\left(x\right)}}$	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	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} 13.30645315+30.02408580*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 2} 27.47545907+32.99593875*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Successful
14.23.E4	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=e^{+\mu\pi i/2}P^{\mu}_{\nu}\left(x+i0\right)}}$	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	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} 13.30645315+30.02408580*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 2} 27.47545907+32.99593875*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail Complex[9.841425469606474, 29.20009174654549] <- {Rule[x, 2], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[22.823216526761424, 33.199439403579085] <- {Rule[x, 3], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-297.7310254998052, 323.60566796262134] <- {Rule[x, 2], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-111.07139755868064, 718.0843910470185] <- {Rule[x, 3], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
14.23.E4	${\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i/2}P^{\mu}_{\nu}\left(x-i0\right)}}$	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	Fail Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 1} -145.9645465+265.3087326*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 2} -88.63555579+385.8611656*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), x = 3} Float(infinity)+Float(infinity)*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Successful
14.24.E3	${\displaystyle{\displaystyle P^{-\mu}_{\nu,s}\left(z\right)=e^{s\mu\pi i}P^{-\mu}_{\nu}\left(z\right)}}$	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	-	Skip
14.25.E1	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(z\right)=\frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\Gamma\left(\mu-\nu\right)\Gamma\left(\nu+1\right)}\int_{0}^{\infty}\frac{(\sinh t)^{2\nu+1}}{(z+\cosh t)^{\nu+\mu+1}}\mathrm{d}t}}$	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}]	Failure	Failure	Skip	Error
14.28.E1	${\displaystyle{\displaystyle P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right)}}$	LegendreP(nu, z[1]*z[2]-(z(z[1])^(2)- 1)^(1/ 2)*(z(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]-(z(Subscript[z, 1])^(2)- 1)^(1/ 2)*(z(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}]	Error	Failure	-	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((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}]=Divide[1,Subscript[z, 1]- Subscript[z, 2]]	Failure	Failure	Skip	Successful
14.29.E1	${\displaystyle{\displaystyle\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{\mathrm{d}w}{\mathrm{d}z}+{\left(\nu(\nu+1)-\frac{\mu_{1}^{2}}{2(1-z)}-\frac{\mu_{2}^{2}}{2(1+z)}\right)w}=0}}$	(1 - (z)^(2))* diff(w, [z$(2)])- 2*z*(nu*(nu + 1)-(mu(mu[1])^(2))/(2*(1 - z))-(mu(mu[2])^(2))/(2*(1 + z)))* w*= 0	(1 - (z)^(2))* D[w, {z, 2}]- 2*z*(\[Nu]*(\[Nu]+ 1)-Divide[\[Mu](Subscript[\[Mu], 1])^(2),2*(1 - z)]-Divide[\[Mu](Subscript[\[Mu], 2])^(2),2*(1 + z)])* w*= 0	Failure	Failure	Skip	Successful
14.30.E1	${\displaystyle{\displaystyle Y_{{l},{m}}\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\mathsf{P}^{m}_{l}\left(\cos\theta\right)}}$	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	-	Skip
14.30.E6	${\displaystyle{\displaystyle Y_{{l},{-m}}\left(\theta,\phi\right)=(-1)^{m}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}}}$	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	Fail 2.770814494+.8972490350*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 1, m = 1} 5.966262911-12.72066041*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 2, m = 1} 25.49303911+17.20629452*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 2, m = 2} -42.95507842-34.73059764*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 3, m = 1} ... skip entries to safe data	Skip
14.30.E7	${\displaystyle{\displaystyle Y_{{l},{m}}\left(\pi-\theta,\phi+\pi\right)=(-1)^{l}Y_{{l},{m}}\left(\theta,\phi\right)}}$	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	Fail -.3649216406+.6291037293e-2*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 1, m = 1} .2502825188-1.564455147*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 2, m = 1} 6.418170178+2.106248885*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 3, m = 1} -.1228000333+.6356041761e-2*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), l = 3, m = 3} ... skip entries to safe data	Skip
14.30.E9	${\displaystyle{\displaystyle\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}\overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_{2},\phi_{2}\right)}}$	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}]	Failure	Failure	Skip	Skip
14.30.E10	${\displaystyle{\displaystyle{\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial W}{\partial\rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial W}{\partial\theta}\right)}+\frac{1}{\rho^{2}{\sin^{2}}\theta}\frac{{\partial}^{2}W}{{\partial\phi}^{2}}=0}}$	(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	-	-