Legendre and Related Functions - 14.23 Values on the Cut

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.23.E1 P ν μ ⁡ ( x + i ⁢ 0 ) = e - μ ⁢ π ⁢ i / 2 ⁢ 𝖯 ν μ ⁡ ( x ) Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑖 0 superscript 𝑒 𝜇 𝜋 𝑖 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x+i0\right)=e^{-\mu\pi i/2}% \mathsf{P}^{\mu}_{\nu}\left(x\right)}}
\assLegendreP[\mu]{\nu}@{x+ i0} = e^{-\mu\pi i/2}\FerrersP[\mu]{\nu}@{x}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x + I*0) = exp(- mu*Pi*I/2)*LegendreP(nu, mu, x)

LegendreP[\[Nu], \[Mu], 3, x + I*0] == Exp[- \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], x]
Failure Failure
Failed [295 / 300]
Result: 5.350830664-.896185152*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: 3.575579140-1.800672871*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [159 / 300]
Result: Complex[6.260055630157556, 1.404281972043869]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.1662318532347467, -6.202414130662353]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.23.E1 P ν μ ⁡ ( x - i ⁢ 0 ) = e + μ ⁢ π ⁢ i / 2 ⁢ 𝖯 ν μ ⁡ ( x ) Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑖 0 superscript 𝑒 𝜇 𝜋 𝑖 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x-i0\right)=e^{+\mu\pi i/2}% \mathsf{P}^{\mu}_{\nu}\left(x\right)}}
\assLegendreP[\mu]{\nu}@{x- i0} = e^{+\mu\pi i/2}\FerrersP[\mu]{\nu}@{x}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x - I*0) = exp(+ mu*Pi*I/2)*LegendreP(nu, mu, x)

LegendreP[\[Nu], \[Mu], 3, x - I*0] == Exp[+ \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], x]
Failure Failure
Failed [295 / 300]
Result: -.9092249665-2.300467118*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -1.143434975-1.422772544*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [79 / 300]
Result: Complex[-4.719014112853729, 0.3779003216614092]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.667629477217065, -3.026452547389477]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.23.E2 𝑸 ν μ ⁡ ( x + i ⁢ 0 ) = e + μ ⁢ π ⁢ i / 2 Γ ⁡ ( ν + μ + 1 ) ⁢ ( 𝖰 ν μ ⁡ ( x ) - 1 2 ⁢ π ⁢ i ⁢ 𝖯 ν μ ⁡ ( x ) ) associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 superscript 𝑒 𝜇 𝜋 𝑖 2 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 𝜋 𝑖 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)=\frac{% e^{+\mu\pi i/2}}{\Gamma\left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}% \left(x\right)-\tfrac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right)}}
\assLegendreOlverQ[\mu]{\nu}@{x+ i0} = \frac{e^{+\mu\pi i/2}}{\EulerGamma@{\nu+\mu+1}}\left(\FerrersQ[\mu]{\nu}@{x}-\tfrac{1}{2}\pi i\FerrersP[\mu]{\nu}@{x}\right)		 ℜ ⁡ ( ν + μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 1 2 1 2 𝑥 1 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,% \Re(\nu-\mu+1)>0}} 
exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1) = (exp(+ mu*Pi*I/2))/(GAMMA(nu + mu + 1))*(LegendreQ(nu, mu, x)-(1)/(2)*Pi*I*LegendreP(nu, mu, x))

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[+ \[Mu]*Pi*I/2],Gamma[\[Nu]+ \[Mu]+ 1]]*(LegendreQ[\[Nu], \[Mu], x]-Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], x])
Failure Failure
Failed [120 / 120]
Result: 15.62228457-3.860103415*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: 11.64166640-5.161800279*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [90 / 135]
Result: Complex[2.4984461168598187, 1.2999649891093954]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[5.332631908276789, 3.703974803728466]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.23.E2 𝑸 ν μ ⁡ ( x - i ⁢ 0 ) = e - μ ⁢ π ⁢ i / 2 Γ ⁡ ( ν + μ + 1 ) ⁢ ( 𝖰 ν μ ⁡ ( x ) + 1 2 ⁢ π ⁢ i ⁢ 𝖯 ν μ ⁡ ( x ) ) associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 superscript 𝑒 𝜇 𝜋 𝑖 2 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 𝜋 𝑖 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)=\frac{% e^{-\mu\pi i/2}}{\Gamma\left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}% \left(x\right)+\tfrac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right)}}
\assLegendreOlverQ[\mu]{\nu}@{x- i0} = \frac{e^{-\mu\pi i/2}}{\EulerGamma@{\nu+\mu+1}}\left(\FerrersQ[\mu]{\nu}@{x}+\tfrac{1}{2}\pi i\FerrersP[\mu]{\nu}@{x}\right)		 ℜ ⁡ ( ν + μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 1 2 1 2 𝑥 1 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,% \Re(\nu-\mu+1)>0}} 
exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1) = (exp(- mu*Pi*I/2))/(GAMMA(nu + mu + 1))*(LegendreQ(nu, mu, x)+(1)/(2)*Pi*I*LegendreP(nu, mu, x))

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[- \[Mu]*Pi*I/2],Gamma[\[Nu]+ \[Mu]+ 1]]*(LegendreQ[\[Nu], \[Mu], x]+Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], x])
Failure Failure
Failed [120 / 120]
Result: 13.12383845-5.160068402*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: 9.802483176-6.415524146*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [45 / 135]
Result: Complex[-1.839183222440096, -1.2537238668211261]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.419436191421772, -4.262017463676762]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.23.E3 𝑸 ν μ ⁡ ( x + i ⁢ 0 ) = e - ν ⁢ π ⁢ i / 2 ⁢ π 3 / 2 ⁢ ( 1 - x 2 ) μ / 2 2 ν + 1 ⁢ ( x ⁢ 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ; 3 2 ; x 2 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) - i ⁢ 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ; 1 2 ; x 2 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) ) associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 superscript 𝑒 𝜈 𝜋 𝑖 2 superscript 𝜋 3 2 superscript 1 superscript 𝑥 2 𝜇 2 superscript 2 𝜈 1 𝑥 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 1 2 𝜈 1 2 𝜇 1 3 2 superscript 𝑥 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝑖 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 𝜈 1 2 𝜇 1 2 1 2 superscript 𝑥 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)=\frac{% e^{-\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\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}\nu-\frac{1}{2}% \mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}% \right)}-i\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}% \nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}% \right)}}
\assLegendreOlverQ[\mu]{\nu}@{x+ i0} = \frac{e^{-\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\hyperOlverF@{\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}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}- i\frac{\hyperOlverF@{\frac{1}{2}\mu-\frac{1}{2}\nu}{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}{\frac{1}{2}}{x^{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}\right)		 ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) > 0 , | ( x 2 ) | < 1 formulae-sequence 1 2 𝜈 1 2 𝜇 1 2 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 2 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 0 superscript 𝑥 2 1 {\displaystyle{\displaystyle\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2})>0,% \Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2% }\mu+1)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,|(x^{2})|<1}} 
exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1) = (exp(- nu*Pi*I/2)*(Pi)^(3/2)*(1 - (x)^(2))^(mu/2))/((2)^(nu + 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)*nu -(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))- I*(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)*nu -(1)/(2)*mu + 1)*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1)))

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[- \[Nu]*Pi*I/2]*(Pi)^(3/2)*(1 - (x)^(2))^(\[Mu]/2),(2)^(\[Nu]+ 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]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]]]- I*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]*\[Nu]-Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]])
Failure Failure Successful [Tested: 40] Successful [Tested: 45]
14.23.E3 𝑸 ν μ ⁡ ( x - i ⁢ 0 ) = e + ν ⁢ π ⁢ i / 2 ⁢ π 3 / 2 ⁢ ( 1 - x 2 ) μ / 2 2 ν + 1 ⁢ ( x ⁢ 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ; 3 2 ; x 2 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) + i ⁢ 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ; 1 2 ; x 2 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) ) associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 superscript 𝑒 𝜈 𝜋 𝑖 2 superscript 𝜋 3 2 superscript 1 superscript 𝑥 2 𝜇 2 superscript 2 𝜈 1 𝑥 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 1 2 𝜈 1 2 𝜇 1 3 2 superscript 𝑥 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝑖 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 𝜈 1 2 𝜇 1 2 1 2 superscript 𝑥 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)=\frac{% e^{+\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\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}\nu-\frac{1}{2}% \mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}% \right)}+i\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}% \nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}% \right)}}
\assLegendreOlverQ[\mu]{\nu}@{x- i0} = \frac{e^{+\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\hyperOlverF@{\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}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}+ i\frac{\hyperOlverF@{\frac{1}{2}\mu-\frac{1}{2}\nu}{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}{\frac{1}{2}}{x^{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}\right)		 ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) > 0 , | ( x 2 ) | < 1 formulae-sequence 1 2 𝜈 1 2 𝜇 1 2 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 2 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 0 formulae-sequence 1 2 𝜈 1 2 𝜇 1 0 superscript 𝑥 2 1 {\displaystyle{\displaystyle\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2})>0,% \Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2% }\mu+1)>0,\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,|(x^{2})|<1}} 
exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1) = (exp(+ nu*Pi*I/2)*(Pi)^(3/2)*(1 - (x)^(2))^(mu/2))/((2)^(nu + 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)*nu -(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))+ I*(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)*nu -(1)/(2)*mu + 1)*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1)))

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1] == Divide[Exp[+ \[Nu]*Pi*I/2]*(Pi)^(3/2)*(1 - (x)^(2))^(\[Mu]/2),(2)^(\[Nu]+ 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]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]]]+ I*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]*\[Nu]-Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]])
Failure Failure
Failed [40 / 40]
Result: -1.839183223-1.253723866*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}


Result: 1.419436198-4.262017468*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2-1/2*I*3^(1/2), x = 1/2}

... skip entries to safe data
Failed [45 / 45]
Result: Complex[-1.8391832224400957, -1.2537238668211277]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.4194361914217857, -4.2620174636767665]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.23.E4 𝖯 ν μ ⁡ ( x ) = e + μ ⁢ π ⁢ i / 2 ⁢ P ν μ ⁡ ( x + i ⁢ 0 ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 2 Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑖 0 {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=e^{+\mu\pi i/% 2}P^{\mu}_{\nu}\left(x+i0\right)}}
\FerrersP[\mu]{\nu}@{x} = e^{+\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x+ i0}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x) = exp(+ mu*Pi*I/2)*LegendreP(nu, mu, x + I*0)

LegendreP[\[Nu], \[Mu], x] == Exp[+ \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x + I*0]
Failure Failure
Failed [295 / 300]
Result: -.9092249665-2.300467118*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -1.143434975-1.422772544*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [159 / 300]
Result: Complex[0.02990691582525623, -2.924977300264846]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.067091398010022, -0.8210135056644176]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.23.E4 𝖯 ν μ ⁡ ( x ) = e - μ ⁢ π ⁢ i / 2 ⁢ P ν μ ⁡ ( x - i ⁢ 0 ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 2 Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑖 0 {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i/% 2}P^{\mu}_{\nu}\left(x-i0\right)}}
\FerrersP[\mu]{\nu}@{x} = e^{-\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x- i0}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x) = exp(- mu*Pi*I/2)*LegendreP(nu, mu, x - I*0)

LegendreP[\[Nu], \[Mu], x] == Exp[- \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x - I*0]
Failure Failure
Failed [295 / 300]
Result: 5.350830664-.896185152*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: 3.575579140-1.800672871*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [79 / 300]
Result: Complex[1.351552463852863, -10.294914164956062]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[7.255468107198464, -2.190256047354226]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.23.E5 𝖰 ν μ ⁡ ( x ) = 1 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ ( e - μ ⁢ π ⁢ i / 2 ⁢ 𝑸 ν μ ⁡ ( x + i ⁢ 0 ) + e μ ⁢ π ⁢ i / 2 ⁢ 𝑸 ν μ ⁡ ( x - i ⁢ 0 ) ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 Euler-Gamma 𝜈 𝜇 1 superscript 𝑒 𝜇 𝜋 𝑖 2 associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 superscript 𝑒 𝜇 𝜋 𝑖 2 associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\tfrac{1}{2}% \Gamma\left(\nu+\mu+1\right)\left(e^{-\mu\pi i/2}\boldsymbol{Q}^{\mu}_{\nu}% \left(x+i0\right)+e^{\mu\pi i/2}\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)% \right)}}
\FerrersQ[\mu]{\nu}@{x} = \tfrac{1}{2}\EulerGamma@{\nu+\mu+1}\left(e^{-\mu\pi i/2}\assLegendreOlverQ[\mu]{\nu}@{x+i0}+e^{\mu\pi i/2}\assLegendreOlverQ[\mu]{\nu}@{x-i0}\right)		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreQ(nu, mu, x) = (1)/(2)*GAMMA(nu + mu + 1)*(exp(- mu*Pi*I/2)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1)+ exp(mu*Pi*I/2)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1))

LegendreQ[\[Nu], \[Mu], x] == Divide[1,2]*Gamma[\[Nu]+ \[Mu]+ 1]*(Exp[- \[Mu]*Pi*I/2]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1]+ Exp[\[Mu]*Pi*I/2]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1])
Failure Failure
Failed [120 / 120]
Result: -15.30496809+11.59724304*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -10.41616244+10.97902682*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [135 / 135]
Result: Complex[-3.9489024974094016, 0.15503510169416979]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-4.5992221195498555, 6.976681726631964]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.23.E6 𝖰 ν μ ⁡ ( x ) = e - μ ⁢ π ⁢ i / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ 𝑸 ν μ ⁡ ( x + i ⁢ 0 ) + 1 2 ⁢ π ⁢ i ⁢ e + μ ⁢ π ⁢ i / 2 ⁢ P ν μ ⁡ ( x + i ⁢ 0 ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 2 Euler-Gamma 𝜈 𝜇 1 associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 1 2 𝜋 𝑖 superscript 𝑒 𝜇 𝜋 𝑖 2 Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑖 0 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i/% 2}\Gamma\left(\nu+\mu+1\right)\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)+% \tfrac{1}{2}\pi ie^{+\mu\pi i/2}P^{\mu}_{\nu}\left(x+i0\right)}}
\FerrersQ[\mu]{\nu}@{x} = e^{-\mu\pi i/2}\EulerGamma@{\nu+\mu+1}\assLegendreOlverQ[\mu]{\nu}@{x+ i0}+\tfrac{1}{2}\pi ie^{+\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x+ i0}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreQ(nu, mu, x) = exp(- mu*Pi*I/2)*GAMMA(nu + mu + 1)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x + I*0)/GAMMA(nu+mu+1)+(1)/(2)*Pi*I*exp(+ mu*Pi*I/2)*LegendreP(nu, mu, x + I*0)

LegendreQ[\[Nu], \[Mu], x] == Exp[- \[Mu]*Pi*I/2]*Gamma[\[Nu]+ \[Mu]+ 1]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x + I*0]/Gamma[\[Nu] + \[Mu] + 1]+Divide[1,2]*Pi*I*Exp[+ \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x + I*0]
Failure Failure
Failed [120 / 120]
Result: -29.08177200+29.72441292*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -18.94845706+26.98747914*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [90 / 135]
Result: Complex[-3.303261395604329, 0.35704787691241624]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-5.262064714407579, 5.6951304506187865]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.23.E6 𝖰 ν μ ⁡ ( x ) = e + μ ⁢ π ⁢ i / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ 𝑸 ν μ ⁡ ( x - i ⁢ 0 ) - 1 2 ⁢ π ⁢ i ⁢ e - μ ⁢ π ⁢ i / 2 ⁢ P ν μ ⁡ ( x - i ⁢ 0 ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 2 Euler-Gamma 𝜈 𝜇 1 associated-Legendre-black-Q 𝜇 𝜈 𝑥 𝑖 0 1 2 𝜋 𝑖 superscript 𝑒 𝜇 𝜋 𝑖 2 Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑖 0 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{+\mu\pi i/% 2}\Gamma\left(\nu+\mu+1\right)\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)-% \tfrac{1}{2}\pi ie^{-\mu\pi i/2}P^{\mu}_{\nu}\left(x-i0\right)}}
\FerrersQ[\mu]{\nu}@{x} = e^{+\mu\pi i/2}\EulerGamma@{\nu+\mu+1}\assLegendreOlverQ[\mu]{\nu}@{x- i0}-\tfrac{1}{2}\pi ie^{-\mu\pi i/2}\assLegendreP[\mu]{\nu}@{x- i0}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreQ(nu, mu, x) = exp(+ mu*Pi*I/2)*GAMMA(nu + mu + 1)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x - I*0)/GAMMA(nu+mu+1)-(1)/(2)*Pi*I*exp(- mu*Pi*I/2)*LegendreP(nu, mu, x - I*0)

LegendreQ[\[Nu], \[Mu], x] == Exp[+ \[Mu]*Pi*I/2]*Gamma[\[Nu]+ \[Mu]+ 1]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x - I*0]/Gamma[\[Nu] + \[Mu] + 1]-Divide[1,2]*Pi*I*Exp[- \[Mu]*Pi*I/2]*LegendreP[\[Nu], \[Mu], 3, x - I*0]
Failure Failure
Failed [120 / 120]
Result: .677676788-16.36319923*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -2.477472256-12.44203554*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [45 / 135]
Result: Complex[-17.39472965859494, -1.6880401639683693]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.8057990956489687, 0.19849176253311906]
Test Values: {Rule[x, 0.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
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