14.19

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.19#Ex1	${\displaystyle{\displaystyle x=\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta}}}$ x = \frac{c\sinh@@{\eta}\cos@@{\phi}}{\cosh@@{\eta}-\cos@@{\theta}}		x = (c*sinh(eta)*cos(phi))/(cosh(eta)- cos(theta))	x == Divide[c*Sinh[\[Eta]]*Cos[\[Phi]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Failed [300 / 300] Result: 2.362573279-1.052377925*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: 1.362573279-1.052377925*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[2.3625732791062704, -1.0523779253990262] Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.6505283543319873, -0.046280887188208775] Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19#Ex2	${\displaystyle{\displaystyle y=\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta}}}$ y = \frac{c\sinh@@{\eta}\sin@@{\phi}}{\cosh@@{\eta}-\cos@@{\theta}}		y = (c*sinh(eta)*sin(phi))/(cosh(eta)- cos(theta))	y == Divide[c*Sinh[\[Eta]]*Sin[\[Phi]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Failed [300 / 300] Result: .10381346e-1-.1810305231e-1*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = -3/2} Result: 3.010381346-.1810305231e-1*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = 3/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.010381344893815037, -0.01810305210999985] Test Values: {Rule[c, -1.5], Rule[y, -1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.9871098783639947, 1.7153567749591236] Test Values: {Rule[c, -1.5], Rule[y, -1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19#Ex3	${\displaystyle{\displaystyle z=\frac{c\sin\theta}{\cosh\eta-\cos\theta}}}$ z = \frac{c\sin@@{\theta}}{\cosh@@{\eta}-\cos@@{\theta}}		z = (c*sin(theta))/(cosh(eta)- cos(theta))	z == Divide[c*Sin[\[Theta]],Cosh[\[Eta]]- Cos[\[Theta]]]	Failure	Failure	Failed [300 / 300] Result: 1.948230727-.3664573554*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .5822053230-.4319514e-3*I Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.948230726846754, -0.366457355462031] Test Values: {Rule[c, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[7.733911995808641*^15, 6.041410995179728*^15] Test Values: {Rule[c, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19.E2	${\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-\mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}\right)}}$ \assLegendreP[\mu]{\nu-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{\frac{1}{2}-\mu}}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\hyperOlverF@{\tfrac{1}{2}-\mu}{\tfrac{1}{2}+\nu-\mu}{1-2\mu}{1-e^{-2\xi}}	${\displaystyle{\displaystyle\mu\neq\frac{1}{2},\Re(\frac{1}{2}-\mu)>0}}$	LegendreP(nu -(1)/(2), mu, cosh(xi)) = (GAMMA((1)/(2)- mu))/((Pi)^(1/2)*(1 - exp(- 2*xi))^(mu)* exp((nu +(1/2))*xi))* hypergeom([(1)/(2)- mu, (1)/(2)+ nu - mu], [1 - 2*mu], 1 - exp(- 2*xi))/GAMMA(1 - 2*mu)	LegendreP[\[Nu]-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]] == Divide[Gamma[Divide[1,2]- \[Mu]],(Pi)^(1/2)*(1 - Exp[- 2*\[Xi]])^\[Mu]* Exp[(\[Nu]+(1/2))*\[Xi]]]* Hypergeometric2F1Regularized[Divide[1,2]- \[Mu], Divide[1,2]+ \[Nu]- \[Mu], 1 - 2*\[Mu], 1 - Exp[- 2*\[Xi]]]	Aborted	Failure	Successful [Tested: 200]	Successful [Tested: 200]
14.19#Ex4	${\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(1-2\mu\right)2^{2\mu}}{\Gamma\left(1-\mu\right)\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;e^{-2\xi}\right)}}$ \assLegendreP[\mu]{\nu-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{1-2\mu}2^{2\mu}}{\EulerGamma@{1-\mu}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\hyperOlverF@{\tfrac{1}{2}-\mu}{\tfrac{1}{2}+\nu-\mu}{1-2\mu}{e^{-2\xi}}		LegendreP(nu -(1)/(2), mu, cosh(xi)) = (GAMMA(1 - 2*mu)*(2)^(2*mu))/(GAMMA(1 - mu)*(1 - exp(- 2*xi))^(mu)* exp((nu +(1/2))*xi))* hypergeom([(1)/(2)- mu, (1)/(2)+ nu - mu], [1 - 2*mu], exp(- 2*xi))/GAMMA(1 - 2*mu)	LegendreP[\[Nu]-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]] == Divide[Gamma[1 - 2*\[Mu]]*(2)^(2*\[Mu]),Gamma[1 - \[Mu]]*(1 - Exp[- 2*\[Xi]])^\[Mu]* Exp[(\[Nu]+(1/2))*\[Xi]]]* Hypergeometric2F1Regularized[Divide[1,2]- \[Mu], Divide[1,2]+ \[Nu]- \[Mu], 1 - 2*\[Mu], Exp[- 2*\[Xi]]]	Failure	Failure	Failed [300 / 300] Result: .2738102545-.736850267e-1*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I} Result: 3.389539010-1.213206227*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.2738102549490508, -0.07368502759104012] Test Values: {Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.38953901122763, -1.2132062234978649] Test Values: {Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19.E3	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\mu+\tfrac{1}{2},\nu+\mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right)}}$ \assLegendreOlverQ[\mu]{\nu-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\hyperOlverF@{\mu+\tfrac{1}{2}}{\nu+\mu+\tfrac{1}{2}}{\nu+1}{e^{-2\xi}}		exp(-(mu)*Pi*I)*LegendreQ(nu -(1)/(2),mu,cosh(xi))/GAMMA(nu -(1)/(2)+mu+1) = ((Pi)^(1/2)*(1 - exp(- 2*xi))^(mu))/(exp((nu +(1/2))*xi))* hypergeom([mu +(1)/(2), nu + mu +(1)/(2)], [nu + 1], exp(- 2*xi))/GAMMA(nu + 1)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu]-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]]/Gamma[\[Nu]-Divide[1,2] + \[Mu] + 1] == Divide[(Pi)^(1/2)*(1 - Exp[- 2*\[Xi]])^\[Mu],Exp[(\[Nu]+(1/2))*\[Xi]]]* Hypergeometric2F1Regularized[\[Mu]+Divide[1,2], \[Nu]+ \[Mu]+Divide[1,2], \[Nu]+ 1, Exp[- 2*\[Xi]]]	Failure	Failure	Failed [20 / 300] Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -2, xi = 1/2*3^(1/2)+1/2*I} Result: Float(undefined)+Float(undefined)*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -2, xi = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Failed [10 / 300] Result: Indeterminate Test Values: {Rule[μ, -1.5], Rule[ν, -2], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[μ, -1.5], Rule[ν, -2], Rule[ξ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
14.19.E4	${\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\frac{1}{2}\right)(\sinh\xi)^{m}}{2^{m}\pi^{1/2}\Gamma\left(n-m+\frac{1}{2}\right)\Gamma\left(m+\frac{1}{2}\right)}\*\int_{0}^{\pi}\frac{(\sin\phi)^{2m}}{(\cosh\xi+\cos\phi\sinh\xi)^{n+m+(1/2)}}\mathrm{d}\phi}}$ \assLegendreP[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{n+m+\frac{1}{2}}(\sinh@@{\xi})^{m}}{2^{m}\pi^{1/2}\EulerGamma@{n-m+\frac{1}{2}}\EulerGamma@{m+\frac{1}{2}}}\*\int_{0}^{\pi}\frac{(\sin@@{\phi})^{2m}}{(\cosh@@{\xi}+\cos@@{\phi}\sinh@@{\xi})^{n+m+(1/2)}}\diff{\phi}	${\displaystyle{\displaystyle\Re(n+m+\frac{1}{2})>0,\Re(n-m+\frac{1}{2})>0,\Re(m+\frac{1}{2})>0}}$	LegendreP(n -(1)/(2), m, cosh(xi)) = (GAMMA(n + m +(1)/(2))*(sinh(xi))^(m))/((2)^(m)* (Pi)^(1/2)* GAMMA(n - m +(1)/(2))*GAMMA(m +(1)/(2)))* int(((sin(phi))^(2*m))/((cosh(xi)+ cos(phi)*sinh(xi))^(n + m +(1/2))), phi = 0..Pi)	LegendreP[n -Divide[1,2], m, 3, Cosh[\[Xi]]] == Divide[Gamma[n + m +Divide[1,2]]*(Sinh[\[Xi]])^(m),(2)^(m)* (Pi)^(1/2)* Gamma[n - m +Divide[1,2]]*Gamma[m +Divide[1,2]]]* Integrate[Divide[(Sin[\[Phi]])^(2*m),(Cosh[\[Xi]]+ Cos[\[Phi]]*Sinh[\[Xi]])^(n + m +(1/2))], {\[Phi], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
14.19.E5	${\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}\right)\Gamma\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(mt\right)}{(\cosh\xi+\cosh t\sinh\xi)^{n+(1/2)}}\mathrm{d}t}}$ \assLegendreOlverQ[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{n+\frac{1}{2}}}{\EulerGamma@{n+m+\tfrac{1}{2}}\EulerGamma@{n-m+\frac{1}{2}}}\*\int_{0}^{\infty}\frac{\cosh@{mt}}{(\cosh@@{\xi}+\cosh@@{t}\sinh@@{\xi})^{n+(1/2)}}\diff{t}	${\displaystyle{\displaystyle m<n+\tfrac{1}{2},\Re(n+\frac{1}{2})>0,\Re(n+m+\tfrac{1}{2})>0,\Re(n-m+\frac{1}{2})>0}}$	exp(-(m)*Pi*I)*LegendreQ(n -(1)/(2),m,cosh(xi))/GAMMA(n -(1)/(2)+m+1) = (GAMMA(n +(1)/(2)))/(GAMMA(n + m +(1)/(2))*GAMMA(n - m +(1)/(2)))* int((cosh(m*t))/((cosh(xi)+ cosh(t)*sinh(xi))^(n +(1/2))), t = 0..infinity)	Exp[-(m) Pi I] LegendreQ[n -Divide[1,2], m, 3, Cosh[\[Xi]]]/Gamma[n -Divide[1,2] + m + 1] == Divide[Gamma[n +Divide[1,2]],Gamma[n + m +Divide[1,2]]*Gamma[n - m +Divide[1,2]]]* Integrate[Divide[Cosh[m*t],(Cosh[\[Xi]]+ Cosh[t]*Sinh[\[Xi]])^(n +(1/2))], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
14.19.E6	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)+2\sum_{n=1}^{\infty}\frac{\Gamma\left(\mu+n+\tfrac{1}{2}\right)}{\Gamma\left(\mu+\tfrac{1}{2}\right)}\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right)\cos\left(n\phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu}}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}}}}$ \assLegendreOlverQ[\mu]{-\frac{1}{2}}@{\cosh@@{\xi}}+2\sum_{n=1}^{\infty}\frac{\EulerGamma@{\mu+n+\tfrac{1}{2}}}{\EulerGamma@{\mu+\tfrac{1}{2}}}\assLegendreOlverQ[\mu]{n-\frac{1}{2}}@{\cosh@@{\xi}}\cos@{n\phi} = \dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh@@{\xi}\right)^{\mu}}{\left(\cosh@@{\xi}-\cos@@{\phi}\right)^{\mu+(1/2)}}	${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re(\mu+n+\tfrac{1}{2})>0,\Re(\mu+\tfrac{1}{2})>0}}$	exp(-(mu)*Pi*I)*LegendreQ(-(1)/(2),mu,cosh(xi))/GAMMA(-(1)/(2)+mu+1)+ 2*sum((GAMMA(mu + n +(1)/(2)))/(GAMMA(mu +(1)/(2)))*exp(-(mu)*Pi*I)*LegendreQ(n -(1)/(2),mu,cosh(xi))/GAMMA(n -(1)/(2)+mu+1)*cos(n*phi), n = 1..infinity) = (((1)/(2)*Pi)^(1/2)*(sinh(xi))^(mu))/((cosh(xi)- cos(phi))^(mu +(1/2)))	Exp[-(\[Mu]) Pi I] LegendreQ[-Divide[1,2], \[Mu], 3, Cosh[\[Xi]]]/Gamma[-Divide[1,2] + \[Mu] + 1]+ 2*Sum[Divide[Gamma[\[Mu]+ n +Divide[1,2]],Gamma[\[Mu]+Divide[1,2]]]*Exp[-(\[Mu]) Pi I] LegendreQ[n -Divide[1,2], \[Mu], 3, Cosh[\[Xi]]]/Gamma[n -Divide[1,2] + \[Mu] + 1]*Cos[n*\[Phi]], {n, 1, Infinity}, GenerateConditions->None] == Divide[(Divide[1,2]*Pi)^(1/2)*(Sinh[\[Xi]])^\[Mu],(Cosh[\[Xi]]- Cos[\[Phi]])^(\[Mu]+(1/2))]	Failure	Failure	Skipped - Because timed out	Skipped - Because timed out
14.19.E7	${\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\tfrac{1}{2}\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}\left(\coth\xi\right)}}$ \assLegendreP[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{n+m+\tfrac{1}{2}}}{\EulerGamma@{n-m+\tfrac{1}{2}}}\*\left(\frac{2}{\pi\sinh@@{\xi}}\right)^{1/2}\assLegendreOlverQ[n]{m-\frac{1}{2}}@{\coth@@{\xi}}	${\displaystyle{\displaystyle\Re(n+m+\tfrac{1}{2})>0,\Re(n-m+\tfrac{1}{2})>0}}$	LegendreP(n -(1)/(2), m, cosh(xi)) = (GAMMA(n + m +(1)/(2)))/(GAMMA(n - m +(1)/(2)))*((2)/(Pi*sinh(xi)))^(1/2)* exp(-(n)*Pi*I)*LegendreQ(m -(1)/(2),n,coth(xi))/GAMMA(m -(1)/(2)+n+1)	LegendreP[n -Divide[1,2], m, 3, Cosh[\[Xi]]] == Divide[Gamma[n + m +Divide[1,2]],Gamma[n - m +Divide[1,2]]]*(Divide[2,Pi*Sinh[\[Xi]]])^(1/2)* Exp[-(n) Pi I] LegendreQ[m -Divide[1,2], n, 3, Coth[\[Xi]]]/Gamma[m -Divide[1,2] + n + 1]	Failure	Failure	Failed [20 / 60] Result: .3683324082-.6470690126*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1, n = 1} Result: .5135733695-3.117174531*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1, n = 2} ... skip entries to safe data	Failed [20 / 60] Result: Complex[0.36833240837635506, -0.6470690125104284] Test Values: {Rule[m, 1], Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.5135733718660924, -3.117174532097865] Test Values: {Rule[m, 1], Rule[n, 2], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
14.19.E8	${\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(m-n+\tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2}\right)}\*\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}P^{n}_{m-\frac{1}{2}}\left(\coth\xi\right)}}$ \assLegendreOlverQ[m]{n-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{\EulerGamma@{m-n+\tfrac{1}{2}}}{\EulerGamma@{m+n+\tfrac{1}{2}}}\*\left(\frac{\pi}{2\sinh@@{\xi}}\right)^{1/2}\assLegendreP[n]{m-\frac{1}{2}}@{\coth@@{\xi}}	${\displaystyle{\displaystyle\Re(m-n+\tfrac{1}{2})>0,\Re(m+n+\tfrac{1}{2})>0}}$	exp(-(m)*Pi*I)*LegendreQ(n -(1)/(2),m,cosh(xi))/GAMMA(n -(1)/(2)+m+1) = (GAMMA(m - n +(1)/(2)))/(GAMMA(m + n +(1)/(2)))*((Pi)/(2*sinh(xi)))^(1/2)* LegendreP(m -(1)/(2), n, coth(xi))	Exp[-(m) Pi I] LegendreQ[n -Divide[1,2], m, 3, Cosh[\[Xi]]]/Gamma[n -Divide[1,2] + m + 1] == Divide[Gamma[m - n +Divide[1,2]],Gamma[m + n +Divide[1,2]]]*(Divide[Pi,2*Sinh[\[Xi]]])^(1/2)* LegendreP[m -Divide[1,2], n, 3, Coth[\[Xi]]]	Failure	Failure	Failed [30 / 60] Result: .7427758821+1.946023521*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1, n = 1} Result: -.1057063209+.477539648e-1*I Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 2, n = 1} ... skip entries to safe data	Failed [30 / 60] Result: Complex[0.7427758815190426, 1.9460235199869547] Test Values: {Rule[m, 1], Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.10570632113064243, 0.04775396399318543] Test Values: {Rule[m, 2], Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data