# 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