Legendre and Related Functions - 14.19 Toroidal (or Ring) Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.19#Ex1 x = c ⁢ sinh ⁡ η ⁢ cos ⁡ ϕ cosh ⁡ η - cos ⁡ θ 𝑥 𝑐 𝜂 italic-ϕ 𝜂 𝜃 {\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 y = c ⁢ sinh ⁡ η ⁢ sin ⁡ ϕ cosh ⁡ η - cos ⁡ θ 𝑦 𝑐 𝜂 italic-ϕ 𝜂 𝜃 {\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 z = c ⁢ sin ⁡ θ cosh ⁡ η - cos ⁡ θ 𝑧 𝑐 𝜃 𝜂 𝜃 {\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 P ν - 1 2 μ ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( 1 2 - μ ) π 1 / 2 ⁢ ( 1 - e - 2 ⁢ ξ ) μ ⁢ e ( ν + ( 1 / 2 ) ) ⁢ ξ ⁢ 𝐅 ⁡ ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 ⁢ μ ; 1 - e - 2 ⁢ ξ ) Legendre-P-first-kind 𝜇 𝜈 1 2 𝜉 Euler-Gamma 1 2 𝜇 superscript 𝜋 1 2 superscript 1 superscript 𝑒 2 𝜉 𝜇 superscript 𝑒 𝜈 1 2 𝜉 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 𝜇 1 2 𝜇 1 superscript 𝑒 2 𝜉 {\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}}		 μ ≠ 1 2 , ℜ ⁡ ( 1 2 - μ ) > 0 formulae-sequence 𝜇 1 2 1 2 𝜇 0 {\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 P ν - 1 2 μ ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( 1 - 2 ⁢ μ ) ⁢ 2 2 ⁢ μ Γ ⁡ ( 1 - μ ) ⁢ ( 1 - e - 2 ⁢ ξ ) μ ⁢ e ( ν + ( 1 / 2 ) ) ⁢ ξ ⁢ 𝐅 ⁡ ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 ⁢ μ ; e - 2 ⁢ ξ ) Legendre-P-first-kind 𝜇 𝜈 1 2 𝜉 Euler-Gamma 1 2 𝜇 superscript 2 2 𝜇 Euler-Gamma 1 𝜇 superscript 1 superscript 𝑒 2 𝜉 𝜇 superscript 𝑒 𝜈 1 2 𝜉 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 𝜇 1 2 𝜇 superscript 𝑒 2 𝜉 {\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 𝑸 ν - 1 2 μ ⁡ ( cosh ⁡ ξ ) = π 1 / 2 ⁢ ( 1 - e - 2 ⁢ ξ ) μ e ( ν + ( 1 / 2 ) ) ⁢ ξ ⁢ 𝐅 ⁡ ( μ + 1 2 , ν + μ + 1 2 ; ν + 1 ; e - 2 ⁢ ξ ) associated-Legendre-black-Q 𝜇 𝜈 1 2 𝜉 superscript 𝜋 1 2 superscript 1 superscript 𝑒 2 𝜉 𝜇 superscript 𝑒 𝜈 1 2 𝜉 scaled-hypergeometric-bold-F 𝜇 1 2 𝜈 𝜇 1 2 𝜈 1 superscript 𝑒 2 𝜉 {\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 P n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + m + 1 2 ) ⁢ ( sinh ⁡ ξ ) m 2 m ⁢ π 1 / 2 ⁢ Γ ⁡ ( n - m + 1 2 ) ⁢ Γ ⁡ ( m + 1 2 ) ⁢ ∫ 0 π ( sin ⁡ ϕ ) 2 ⁢ m ( cosh ⁡ ξ + cos ⁡ ϕ ⁢ sinh ⁡ ξ ) n + m + ( 1 / 2 ) ⁢ d ϕ Legendre-P-first-kind 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 𝑚 1 2 superscript 𝜉 𝑚 superscript 2 𝑚 superscript 𝜋 1 2 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑚 1 2 superscript subscript 0 𝜋 superscript italic-ϕ 2 𝑚 superscript 𝜉 italic-ϕ 𝜉 𝑛 𝑚 1 2 italic-ϕ {\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}		 ℜ ⁡ ( n + m + 1 2 ) > 0 , ℜ ⁡ ( n - m + 1 2 ) > 0 , ℜ ⁡ ( m + 1 2 ) > 0 formulae-sequence 𝑛 𝑚 1 2 0 formulae-sequence 𝑛 𝑚 1 2 0 𝑚 1 2 0 {\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 𝑸 n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + 1 2 ) Γ ⁡ ( n + m + 1 2 ) ⁢ Γ ⁡ ( n - m + 1 2 ) ⁢ ∫ 0 ∞ cosh ⁡ ( m ⁢ t ) ( cosh ⁡ ξ + cosh ⁡ t ⁢ sinh ⁡ ξ ) n + ( 1 / 2 ) ⁢ d t associated-Legendre-black-Q 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 1 2 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑛 𝑚 1 2 superscript subscript 0 𝑚 𝑡 superscript 𝜉 𝑡 𝜉 𝑛 1 2 𝑡 {\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}		 m < n + 1 2 , ℜ ⁡ ( n + 1 2 ) > 0 , ℜ ⁡ ( n + m + 1 2 ) > 0 , ℜ ⁡ ( n - m + 1 2 ) > 0 formulae-sequence 𝑚 𝑛 1 2 formulae-sequence 𝑛 1 2 0 formulae-sequence 𝑛 𝑚 1 2 0 𝑛 𝑚 1 2 0 {\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 𝑸 - 1 2 μ ⁡ ( cosh ⁡ ξ ) + 2 ⁢ ∑ n = 1 ∞ Γ ⁡ ( μ + n + 1 2 ) Γ ⁡ ( μ + 1 2 ) ⁢ 𝑸 n - 1 2 μ ⁡ ( cosh ⁡ ξ ) ⁢ cos ⁡ ( n ⁢ ϕ ) = ( 1 2 ⁢ π ) 1 / 2 ⁢ ( sinh ⁡ ξ ) μ ( cosh ⁡ ξ - cos ⁡ ϕ ) μ + ( 1 / 2 ) associated-Legendre-black-Q 𝜇 1 2 𝜉 2 superscript subscript 𝑛 1 Euler-Gamma 𝜇 𝑛 1 2 Euler-Gamma 𝜇 1 2 associated-Legendre-black-Q 𝜇 𝑛 1 2 𝜉 𝑛 italic-ϕ superscript 1 2 𝜋 1 2 superscript 𝜉 𝜇 superscript 𝜉 italic-ϕ 𝜇 1 2 {\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)}}		 ℜ ⁡ μ > - 1 2 , ℜ ⁡ ( μ + n + 1 2 ) > 0 , ℜ ⁡ ( μ + 1 2 ) > 0 formulae-sequence 𝜇 1 2 formulae-sequence 𝜇 𝑛 1 2 0 𝜇 1 2 0 {\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 P n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + m + 1 2 ) Γ ⁡ ( n - m + 1 2 ) ⁢ ( 2 π ⁢ sinh ⁡ ξ ) 1 / 2 ⁢ 𝑸 m - 1 2 n ⁡ ( coth ⁡ ξ ) Legendre-P-first-kind 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑛 𝑚 1 2 superscript 2 𝜋 𝜉 1 2 associated-Legendre-black-Q 𝑛 𝑚 1 2 hyperbolic-cotangent 𝜉 {\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}}		 ℜ ⁡ ( n + m + 1 2 ) > 0 , ℜ ⁡ ( n - m + 1 2 ) > 0 formulae-sequence 𝑛 𝑚 1 2 0 𝑛 𝑚 1 2 0 {\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 𝑸 n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( m - n + 1 2 ) Γ ⁡ ( m + n + 1 2 ) ⁢ ( π 2 ⁢ sinh ⁡ ξ ) 1 / 2 ⁢ P m - 1 2 n ⁡ ( coth ⁡ ξ ) associated-Legendre-black-Q 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑚 𝑛 1 2 Euler-Gamma 𝑚 𝑛 1 2 superscript 𝜋 2 𝜉 1 2 Legendre-P-first-kind 𝑛 𝑚 1 2 hyperbolic-cotangent 𝜉 {\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}}		 ℜ ⁡ ( m - n + 1 2 ) > 0 , ℜ ⁡ ( m + n + 1 2 ) > 0 formulae-sequence 𝑚 𝑛 1 2 0 𝑚 𝑛 1 2 0 {\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
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