14.20

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.20.E1	${\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{\mathrm{d}w}{\mathrm{d}x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}\right)w=0}}$ \left(1-x^{2}\right)\deriv[2]{w}{x}-2x\deriv{w}{x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}\right)w = 0		(1 - (x)^(2))*diff(w, [x$(2)])- 2*x*diff(w, x)-((tau)^(2)+(1)/(4)+((mu)^(2))/(1 - (x)^(2)))*w = 0	(1 - (x)^(2))*D[w, {x, 2}]- 2*x*D[w, x]-(\[Tau]^(2)+Divide[1,4]+Divide[\[Mu]^(2),1 - (x)^(2)])*w == 0	Failure	Failure	Failed [300 / 300] Result: -.2165063511-.3250000001*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.2165063516-2.458333334*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.2165063509461097, -0.32499999999999996] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.2165063509461096, 1.675] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E4	${\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right),\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(-x\right)\right\}=\frac{2}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}(1-x^{2})}}}$ \Wronskian@{\FerrersP[-\mu]{-\frac{1}{2}+\iunit\tau}@{x},\FerrersP[-\mu]{-\frac{1}{2}+\iunit\tau}@{-x}} = \frac{2}{|\EulerGamma@{\mu+\frac{1}{2}+\iunit\tau}|^{2}(1-x^{2})}	${\displaystyle{\displaystyle\Re(\mu+\frac{1}{2}+\mathrm{i}\tau)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1}}$	(LegendreP(-(1)/(2)+ I*tau, - mu, x))*diff(LegendreP(-(1)/(2)+ I*tau, - mu, - x), x)-diff(LegendreP(-(1)/(2)+ I*tau, - mu, x), x)*(LegendreP(-(1)/(2)+ I*tau, - mu, - x)) = (2)/((abs(GAMMA(mu +(1)/(2)+ I*tau)))^(2)*(1 - (x)^(2)))	Wronskian[{LegendreP[-Divide[1,2]+ I*\[Tau], - \[Mu], x], LegendreP[-Divide[1,2]+ I*\[Tau], - \[Mu], - x]}, x] == Divide[2,(Abs[Gamma[\[Mu]+Divide[1,2]+ I*\[Tau]]])^(2)*(1 - (x)^(2))]	Failure	Failure	Failed [38 / 56] Result: -17.04997320+4.383607823*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, x = 1/2} Result: .5897199763-1.005797385*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = -1/2+1/2*I*3^(1/2), x = 1/2} ... skip entries to safe data	Failed [38 / 56] Result: Complex[-17.049973187296022, 4.383607825965987] Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.5897199767717201, -1.0057973854572255] Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E6	${\displaystyle{\displaystyle P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{ie^{-\mu\pi i}}{\sinh\left(\tau\pi\right)\left|\Gamma\left(\mu+\frac{1}{2}+i\tau\right)\right|^{2}}\*\left(Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)-Q^{\mu}_{-\frac{1}{2}-i\tau}\left(x\right)\right)}}$ \assLegendreP[-\mu]{-\frac{1}{2}+i\tau}@{x} = \frac{ie^{-\mu\pi i}}{\sinh@{\tau\pi}\left|\EulerGamma@{\mu+\frac{1}{2}+i\tau}\right|^{2}}\*\left(\assLegendreQ[\mu]{-\frac{1}{2}+i\tau}@{x}-\assLegendreQ[\mu]{-\frac{1}{2}-i\tau}@{x}\right)	${\displaystyle{\displaystyle\tau\neq 0,\Re(\mu+\frac{1}{2}+\mathrm{i}\tau)>0}}$	LegendreP(-(1)/(2)+ I*tau, - mu, x) = (I*exp(- mu*Pi*I))/(sinh(tau*Pi)*(abs(GAMMA(mu +(1)/(2)+ I*tau)))^(2))*(LegendreQ(-(1)/(2)+ I*tau, mu, x)- LegendreQ(-(1)/(2)- I*tau, mu, x))	LegendreP[-Divide[1,2]+ I*\[Tau], - \[Mu], 3, x] == Divide[I*Exp[- \[Mu]*Pi*I],Sinh[\[Tau]*Pi]*(Abs[Gamma[\[Mu]+Divide[1,2]+ I*\[Tau]]])^(2)]*(LegendreQ[-Divide[1,2]+ I*\[Tau], \[Mu], 3, x]- LegendreQ[-Divide[1,2]- I*\[Tau], \[Mu], 3, x])	Failure	Failure	Failed [114 / 168] Result: -.1488817069+.9881458426*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.7084727976-.1684769573*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [114 / 168] Result: Complex[-0.14888170656920197, 0.9881458430062731] Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.24375508302595367, -0.3184001443616234] Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E9	${\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{2}{\pi}\int_{0}^{\theta}\frac{\cosh\left(\tau\phi\right)}{\sqrt{2(\cos\phi-\cos\theta)}}\mathrm{d}\phi}}$ \FerrersP[]{-\frac{1}{2}+i\tau}@{\cos@@{\theta}} = \frac{2}{\pi}\int_{0}^{\theta}\frac{\cosh@{\tau\phi}}{\sqrt{2(\cos@@{\phi}-\cos@@{\theta})}}\diff{\phi}		LegendreP(-(1)/(2)+ I*tau, cos(theta)) = (2)/(Pi)*int((cosh(tau*phi))/(sqrt(2*(cos(phi)- cos(theta)))), phi = 0..theta)	LegendreP[-Divide[1,2]+ I*\[Tau], Cos[\[Theta]]] == Divide[2,Pi]*Integrate[Divide[Cosh[\[Tau]*\[Phi]],Sqrt[2*(Cos[\[Phi]]- Cos[\[Theta]])]], {\[Phi], 0, \[Theta]}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.20.E13	${\displaystyle{\displaystyle P_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\cosh\left(\tau\pi\right)}{\pi}\int_{1}^{\infty}\frac{P_{-\frac{1}{2}+i\tau}\left(t\right)}{x+t}\mathrm{d}t}}$ \assLegendreP[]{-\frac{1}{2}+i\tau}@{x} = \frac{\cosh@{\tau\pi}}{\pi}\int_{1}^{\infty}\frac{\assLegendreP[]{-\frac{1}{2}+i\tau}@{t}}{x+t}\diff{t}		LegendreP(-(1)/(2)+ I*tau, x) = (cosh(tau*Pi))/(Pi)*int((LegendreP(-(1)/(2)+ I*tau, t))/(x + t), t = 1..infinity)	LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, x] == Divide[Cosh[\[Tau]*Pi],Pi]*Integrate[Divide[LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, t],x + t], {t, 1, Infinity}, GenerateConditions->None]	Failure	Aborted	Manual Skip!	Skipped - Because timed out
14.20.E14	${\displaystyle{\displaystyle\pi\int_{0}^{\infty}\frac{\tau\tanh\left(\tau\pi\right)}{\cosh\left(\tau\pi\right)}P_{-\frac{1}{2}+i\tau}\left(x\right)P_{-\frac{1}{2}+i\tau}\left(y\right)\mathrm{d}\tau=\frac{1}{y+x}}}$ \pi\int_{0}^{\infty}\frac{\tau\tanh@{\tau\pi}}{\cosh@{\tau\pi}}\assLegendreP[]{-\frac{1}{2}+i\tau}@{x}\assLegendreP[]{-\frac{1}{2}+i\tau}@{y}\diff{\tau} = \frac{1}{y+x}		Pi*int((tau*tanh(tau*Pi))/(cosh(tau*Pi))*LegendreP(-(1)/(2)+ I*tau, x)*LegendreP(-(1)/(2)+ I*tau, y), tau = 0..infinity) = (1)/(y + x)	Pi*Integrate[Divide[\[Tau]*Tanh[\[Tau]*Pi],Cosh[\[Tau]*Pi]]*LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, x]*LegendreP[-Divide[1,2]+ I*\[Tau], 0, 3, y], {\[Tau], 0, Infinity}, GenerateConditions->None] == Divide[1,y + x]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.20.E19	${\displaystyle{\displaystyle\alpha=\mu/\tau}}$ \alpha = \mu/\tau		alpha = mu/tau	\[Alpha] == \[Mu]/\[Tau]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.20.E20	${\displaystyle{\displaystyle\sigma(\mu,\tau)=\frac{\exp\left(\mu-\tau\operatorname{arctan}\alpha\right)}{\left(\mu^{2}+\tau^{2}\right)^{\mu/2}}}}$ \sigma(\mu,\tau) = \frac{\exp@{\mu-\tau\atan@@{\alpha}}}{\left(\mu^{2}+\tau^{2}\right)^{\mu/2}}		sigma(mu , tau) = (exp(mu - tau*arctan(alpha)))/(((mu)^(2)+ (tau)^(2))^(mu/2))	\[Sigma][\[Mu], \[Tau]] == Divide[Exp[\[Mu]- \[Tau]*ArcTan[\[Alpha]]],(\[Mu]^(2)+ \[Tau]^(2))^(\[Mu]/2)]	Failure	Failure	Failed [300 / 300] Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I)-.7960801334+.5660885692*I Test Values: {alpha = 3/2, mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I} Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, -.5000000000+.8660254040*I)+Float(-infinity)+Float(infinity)*I Test Values: {alpha = 3/2, mu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, tau = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Error
14.20.E21	${\displaystyle{\displaystyle{\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln\eta-\alpha\ln\left(\left(\alpha^{2}+\eta\right)^{1/2}+\alpha\right)}={\operatorname{arccos}\left(\frac{x}{\left(1+\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(\frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}\right)}}}$ {\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln@@{\eta}-\alpha\ln@{\left(\alpha^{2}+\eta\right)^{1/2}+\alpha}} = {\acos@{\frac{x}{\left(1+\alpha^{2}\right)^{1/2}}}+\frac{\alpha}{2}\ln@{\frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}}}		((alpha)^(2)+ eta)^(1/2)+(1)/(2)*alpha*ln(eta)- alpha*ln(((alpha)^(2)+ eta)^(1/2)+ alpha) = arccos((x)/((1 + (alpha)^(2))^(1/2)))+(alpha)/(2)*ln((1 + (alpha)^(2)+((alpha)^(2)- 1)*(x)^(2)- 2*alpha*x*(1 + (alpha)^(2)- (x)^(2))^(1/2))/((1 + (alpha)^(2))*(1 - (x)^(2))))	(\[Alpha]^(2)+ \[Eta])^(1/2)+Divide[1,2]*\[Alpha]*Log[\[Eta]]- \[Alpha]*Log[(\[Alpha]^(2)+ \[Eta])^(1/2)+ \[Alpha]] == ArcCos[Divide[x,(1 + \[Alpha]^(2))^(1/2)]]+Divide[\[Alpha],2]*Log[Divide[1 + \[Alpha]^(2)+(\[Alpha]^(2)- 1)*(x)^(2)- 2*\[Alpha]*x*(1 + \[Alpha]^(2)- (x)^(2))^(1/2),(1 + \[Alpha]^(2))*(1 - (x)^(2))]]	Failure	Failure	Failed [90 / 90] Result: .1205172872-1.887022822*I Test Values: {alpha = 3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.6024770750+.4691716681*I Test Values: {alpha = 3/2, eta = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[0.12051728613742685, -1.887022822024303] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.09653321282632854, -0.6333444267807768] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.20.E23	${\displaystyle{\displaystyle\beta=\tau/\mu}}$ \beta = \tau/\mu		beta = tau/mu	\[Beta] == \[Tau]/\[Mu]	Skipped - no semantic math	Skipped - no semantic math	-	-
14.20.E24	${\displaystyle{\displaystyle\rho=\frac{1}{2}\ln\left(\frac{\left(1-\beta^{2}\right)x^{2}+1+\beta^{2}+2x\left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^{2}}\right)+\beta\operatorname{arctan}\left(\frac{\beta x}{\sqrt{1+\beta^{2}-\beta^{2}x^{2}}}\right)-\frac{1}{2}\ln\left(1+\beta^{2}\right)}}$ \rho = \frac{1}{2}\ln@{\frac{\left(1-\beta^{2}\right)x^{2}+1+\beta^{2}+2x\left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^{2}}}+\beta\atan@{\frac{\beta x}{\sqrt{1+\beta^{2}-\beta^{2}x^{2}}}}-\frac{1}{2}\ln@{1+\beta^{2}}		rho = (1)/(2)*ln(((1 - (beta)^(2))*(x)^(2)+ 1 + (beta)^(2)+ 2*x*(1 + (beta)^(2)- (beta)^(2)* (x)^(2))^(1/2))/(1 - (x)^(2)))+ beta*arctan((beta*x)/(sqrt(1 + (beta)^(2)- (beta)^(2)* (x)^(2))))-(1)/(2)*ln(1 + (beta)^(2))	\[Rho] == Divide[1,2]*Log[Divide[(1 - \[Beta]^(2))*(x)^(2)+ 1 + \[Beta]^(2)+ 2*x*(1 + \[Beta]^(2)- \[Beta]^(2)* (x)^(2))^(1/2),1 - (x)^(2)]]+ \[Beta]*ArcTan[Divide[\[Beta]*x,Sqrt[1 + \[Beta]^(2)- \[Beta]^(2)* (x)^(2)]]]-Divide[1,2]*Log[1 + \[Beta]^(2)]	Failure	Failure	Failed [90 / 90] Result: 3.222219894+2.375212337*I Test Values: {beta = 3/2, rho = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.925994550e-1+.5000000000*I Test Values: {beta = 3/2, rho = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[3.2222198939767837, 2.37521233732194] Test Values: {Rule[x, 1.5], Rule[β, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.856194490192345, 2.741237741106379] Test Values: {Rule[x, 1.5], Rule[β, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data