Legendre and Related Functions - 14.5 Special Values

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.5.E1 𝖯 Ξ½ ΞΌ ⁑ ( 0 ) = 2 ΞΌ ⁒ Ο€ 1 / 2 Ξ“ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 ) ⁒ Ξ“ ⁑ ( 1 2 - 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 0 superscript 2 πœ‡ superscript πœ‹ 1 2 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 Euler-Gamma 1 2 1 2 𝜈 1 2 πœ‡ {\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)}}}
\FerrersP[\mu]{\nu}@{0} = \frac{2^{\mu}\pi^{1/2}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}-\frac{1}{2}\nu-\frac{1}{2}\mu}}		 β„œ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 ) > 0 , β„œ ⁑ ( 1 2 - 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ ) > 0 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 0 1 2 1 2 𝜈 1 2 πœ‡ 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}\nu-\frac{1}{2}\mu+1)>0,\Re(\frac{1% }{2}-\frac{1}{2}\nu-\frac{1}{2}\mu)>0}} 
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 [Tested: 54]
14.5.E3 𝖰 Ξ½ ΞΌ ⁑ ( 0 ) = - 2 ΞΌ - 1 ⁒ Ο€ 1 / 2 ⁒ sin ⁑ ( 1 2 ⁒ ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 ) Ferrers-Legendre-Q-first-kind πœ‡ 𝜈 0 superscript 2 πœ‡ 1 superscript πœ‹ 1 2 1 2 𝜈 πœ‡ πœ‹ Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 2 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 {\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)}}}
\FerrersQ[\mu]{\nu}@{0} = -\frac{2^{\mu-1}\pi^{1/2}\sin@{\frac{1}{2}(\nu+\mu)\pi}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}}		 β„œ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 2 0 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 0 formulae-sequence 𝜈 πœ‡ 1 0 𝜈 πœ‡ 1 0 {\displaystyle{\displaystyle\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(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}} 
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 [Tested: 45]
14.5.E5 𝖯 0 ⁑ ( x ) = P 0 ⁑ ( x ) shorthand-Ferrers-Legendre-P-first-kind 0 π‘₯ shorthand-Legendre-P-first-kind 0 π‘₯ {\displaystyle{\displaystyle\mathsf{P}_{0}\left(x\right)=P_{0}\left(x\right)}}
\FerrersP[]{0}@{x} = \assLegendreP[]{0}@{x}		 

LegendreP(0, x) = LegendreP(0, x)

LegendreP[0, x] == LegendreP[0, 0, 3, x]
Successful Successful - Successful [Tested: 3]
14.5.E5 P 0 ⁑ ( x ) = 1 shorthand-Legendre-P-first-kind 0 π‘₯ 1 {\displaystyle{\displaystyle P_{0}\left(x\right)=1}}
\assLegendreP[]{0}@{x} = 1		 

LegendreP(0, x) = 1

LegendreP[0, 0, 3, x] == 1
Successful Successful - Successful [Tested: 3]
14.5.E6 𝖯 1 ⁑ ( x ) = P 1 ⁑ ( x ) shorthand-Ferrers-Legendre-P-first-kind 1 π‘₯ shorthand-Legendre-P-first-kind 1 π‘₯ {\displaystyle{\displaystyle\mathsf{P}_{1}\left(x\right)=P_{1}\left(x\right)}}
\FerrersP[]{1}@{x} = \assLegendreP[]{1}@{x}		 

LegendreP(1, x) = LegendreP(1, x)

LegendreP[1, x] == LegendreP[1, 0, 3, x]
Successful Successful - Successful [Tested: 3]
14.5.E6 P 1 ⁑ ( x ) = x shorthand-Legendre-P-first-kind 1 π‘₯ π‘₯ {\displaystyle{\displaystyle P_{1}\left(x\right)=x}}
\assLegendreP[]{1}@{x} = x		 

LegendreP(1, x) = x

LegendreP[1, 0, 3, x] == x
Successful Successful - Successful [Tested: 3]
14.5.E7 𝖰 0 ⁑ ( x ) = 1 2 ⁒ ln ⁑ ( 1 + x 1 - x ) shorthand-Ferrers-Legendre-Q-first-kind 0 π‘₯ 1 2 1 π‘₯ 1 π‘₯ {\displaystyle{\displaystyle\mathsf{Q}_{0}\left(x\right)=\frac{1}{2}\ln\left(% \frac{1+x}{1-x}\right)}}
\FerrersQ[]{0}@{x} = \frac{1}{2}\ln@{\frac{1+x}{1-x}}		 

LegendreQ(0, x) = (1)/(2)*ln((1 + x)/(1 - x))

LegendreQ[0, x] == Divide[1,2]*Log[Divide[1 + x,1 - x]]
Failure Failure
Failed [2 / 3]
Result: .2e-9-3.141592654*I
Test Values: {x = 3/2}


Result: -.2e-9-3.141592654*I
Test Values: {x = 2}

Failed [2 / 3]
Result: Complex[1.1102230246251565*^-16, -3.141592653589793]
Test Values: {Rule[x, 1.5]}


Result: Complex[0.0, -3.141592653589793]
Test Values: {Rule[x, 2]}

14.5.E8 𝖰 1 ⁑ ( x ) = x 2 ⁒ ln ⁑ ( 1 + x 1 - x ) - 1 shorthand-Ferrers-Legendre-Q-first-kind 1 π‘₯ π‘₯ 2 1 π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle\mathsf{Q}_{1}\left(x\right)=\frac{x}{2}\ln\left(% \frac{1+x}{1-x}\right)-1}}
\FerrersQ[]{1}@{x} = \frac{x}{2}\ln@{\frac{1+x}{1-x}}-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
Failed [2 / 3]
Result: .3e-9-4.712388980*I
Test Values: {x = 3/2}


Result: 0.-6.283185308*I
Test Values: {x = 2}

Failed [2 / 3]
Result: Complex[2.220446049250313*^-16, -4.71238898038469]
Test Values: {Rule[x, 1.5]}


Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[x, 2]}

14.5.E9 𝑸 0 ⁑ ( x ) = 1 2 ⁒ ln ⁑ ( x + 1 x - 1 ) shorthand-associated-Legendre-black-Q 0 π‘₯ 1 2 π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle\boldsymbol{Q}_{0}\left(x\right)=\frac{1}{2}\ln% \left(\frac{x+1}{x-1}\right)}}
\assLegendreOlverQ[]{0}@{x} = \frac{1}{2}\ln@{\frac{x+1}{x-1}}		 

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]]
Failure Failure
Failed [1 / 3]
Result: -.2e-9-3.141592654*I
Test Values: {x = 1/2}

Failed [3 / 3]
Result: Complex[0.3952810437829498, -2.9391523179536476*^-16]
Test Values: {Rule[x, 1.5]}


Result: Complex[-1.2159728110007215, -1.5707963267948966]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
14.5.E10 𝑸 1 ⁑ ( x ) = x 2 ⁒ ln ⁑ ( x + 1 x - 1 ) - 1 shorthand-associated-Legendre-black-Q 1 π‘₯ π‘₯ 2 π‘₯ 1 π‘₯ 1 1 {\displaystyle{\displaystyle\boldsymbol{Q}_{1}\left(x\right)=\frac{x}{2}\ln% \left(\frac{x+1}{x-1}\right)-1}}
\assLegendreOlverQ[]{1}@{x} = \frac{x}{2}\ln@{\frac{x+1}{x-1}}-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
Failed [1 / 3]
Result: 0.-1.570796327*I
Test Values: {x = 1/2}

Failed [3 / 3]
Result: Complex[-0.47374510099224176, 6.531449595452549*^-17]
Test Values: {Rule[x, 1.5]}


Result: Complex[1.1697913722774167, -0.7853981633974483]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
14.5.E11 𝖯 Ξ½ 1 / 2 ⁑ ( cos ⁑ ΞΈ ) = ( 2 Ο€ ⁒ sin ⁑ ΞΈ ) 1 / 2 ⁒ cos ⁑ ( ( Ξ½ + 1 2 ) ⁒ ΞΈ ) Ferrers-Legendre-P-first-kind 1 2 𝜈 πœƒ superscript 2 πœ‹ πœƒ 1 2 𝜈 1 2 πœƒ {\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)}}
\FerrersP[1/2]{\nu}@{\cos@@{\theta}} = \left(\frac{2}{\pi\sin@@{\theta}}\right)^{1/2}\cos@{\left(\nu+\tfrac{1}{2}\right)\theta}		 

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
Failed [50 / 100]
Result: -.7596743150+.9986452891*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)}


Result: -.3969265290-1.700808098*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [50 / 100]
Result: Complex[-0.7596743150203076, 0.9986452891592468]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.5932078691227823, 0.7119534787783219]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.5.E12 𝖯 Ξ½ - 1 / 2 ⁑ ( cos ⁑ ΞΈ ) = ( 2 Ο€ ⁒ sin ⁑ ΞΈ ) 1 / 2 ⁒ sin ⁑ ( ( Ξ½ + 1 2 ) ⁒ ΞΈ ) Ξ½ + 1 2 Ferrers-Legendre-P-first-kind 1 2 𝜈 πœƒ superscript 2 πœ‹ πœƒ 1 2 𝜈 1 2 πœƒ 𝜈 1 2 {\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}}}}
\FerrersP[-1/2]{\nu}@{\cos@@{\theta}} = \left(\frac{2}{\pi\sin@@{\theta}}\right)^{1/2}\frac{\sin@{\left(\nu+\frac{1}{2}\right)\theta}}{\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
Failed [55 / 100]
Result: .5392263657-.8901760048*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)}


Result: .9027151592+.9035040024*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [55 / 100]
Result: Indeterminate
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, -0.5]}


Result: Complex[0.5392263655684584, -0.8901760046482097]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
14.5.E13 𝖰 Ξ½ 1 / 2 ⁑ ( cos ⁑ ΞΈ ) = - ( Ο€ 2 ⁒ sin ⁑ ΞΈ ) 1 / 2 ⁒ sin ⁑ ( ( Ξ½ + 1 2 ) ⁒ ΞΈ ) Ferrers-Legendre-Q-first-kind 1 2 𝜈 πœƒ superscript πœ‹ 2 πœƒ 1 2 𝜈 1 2 πœƒ {\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)}}
\FerrersQ[1/2]{\nu}@{\cos@@{\theta}} = -\left(\frac{\pi}{2\sin@@{\theta}}\right)^{1/2}\sin@{\left(\nu+\tfrac{1}{2}\right)\theta}		 

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
Failed [25 / 50]
Result: -1.856186326+1.486585706*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)}


Result: -1.227388580-2.647682452*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [25 / 50]
Result: Complex[-1.8561863256089288, 1.4865857054438434]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.690848965325271, 2.3698178156702956]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.5.E14 𝖰 Ξ½ - 1 / 2 ⁑ ( cos ⁑ ΞΈ ) = ( Ο€ 2 ⁒ sin ⁑ ΞΈ ) 1 / 2 ⁒ cos ⁑ ( ( Ξ½ + 1 2 ) ⁒ ΞΈ ) Ξ½ + 1 2 Ferrers-Legendre-Q-first-kind 1 2 𝜈 πœƒ superscript πœ‹ 2 πœƒ 1 2 𝜈 1 2 πœƒ 𝜈 1 2 {\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}}}}
\FerrersQ[-1/2]{\nu}@{\cos@@{\theta}} = \left(\frac{\pi}{2\sin@@{\theta}}\right)^{1/2}\frac{\cos@{\left(\nu+\frac{1}{2}\right)\theta}}{\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 Error
Failed [25 / 50]
Result: Complex[-0.3996810371463801, 1.2946383468829223]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.41345894273326, 2.4734286705879205]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.5.E15 P Ξ½ 1 / 2 ⁑ ( cosh ⁑ ΞΎ ) = ( 2 Ο€ ⁒ sinh ⁑ ΞΎ ) 1 / 2 ⁒ cosh ⁑ ( ( Ξ½ + 1 2 ) ⁒ ΞΎ ) Legendre-P-first-kind 1 2 𝜈 πœ‰ superscript 2 πœ‹ πœ‰ 1 2 𝜈 1 2 πœ‰ {\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)}}
\assLegendreP[1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{2}{\pi\sinh@@{\xi}}\right)^{1/2}\cosh@{\left(\nu+\tfrac{1}{2}\right)\xi}		 

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
Failed [100 / 100]
Result: -.5866633690+.3419889424*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I}


Result: .9326102256+.153785626*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [50 / 100]
Result: Complex[1.483322380543576, 0.9219835006286831]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.2433197156086089, -0.16897799632039867]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
14.5.E16 P Ξ½ - 1 / 2 ⁑ ( cosh ⁑ ΞΎ ) = ( 2 Ο€ ⁒ sinh ⁑ ΞΎ ) 1 / 2 ⁒ sinh ⁑ ( ( Ξ½ + 1 2 ) ⁒ ΞΎ ) Ξ½ + 1 2 Legendre-P-first-kind 1 2 𝜈 πœ‰ superscript 2 πœ‹ πœ‰ 1 2 𝜈 1 2 πœ‰ 𝜈 1 2 {\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}}}}
\assLegendreP[-1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{2}{\pi\sinh@@{\xi}}\right)^{1/2}\frac{\sinh@{\left(\nu+\frac{1}{2}\right)\xi}}{\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
Failed [100 / 100]
Result: .852516959e-1-.5567654394*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I}


Result: .2647935712-.6384793854*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [55 / 100]
Result: Complex[5.577974291320897*^-4, -1.2771898182050043]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.2481588696482635, 1.0107401090243302]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
14.5.E17 𝑸 Ξ½ + 1 / 2 ⁑ ( cosh ⁑ ΞΎ ) = ( Ο€ 2 ⁒ sinh ⁑ ΞΎ ) 1 / 2 ⁒ exp ⁑ ( - ( Ξ½ + 1 2 ) ⁒ ΞΎ ) Ξ“ ⁑ ( Ξ½ + 3 2 ) associated-Legendre-black-Q 1 2 𝜈 πœ‰ superscript πœ‹ 2 πœ‰ 1 2 𝜈 1 2 πœ‰ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\boldsymbol{Q}^{+1/2}_{\nu}\left(\cosh\xi\right)=% \left(\frac{\pi}{2\sinh\xi}\right)^{1/2}\frac{\exp\left(-\left(\nu+\frac{1}{2}% \right)\xi\right)}{\Gamma\left(\nu+\frac{3}{2}\right)}}}
\assLegendreOlverQ[+ 1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{\pi}{2\sinh@@{\xi}}\right)^{1/2}\frac{\exp@{-\left(\nu+\frac{1}{2}\right)\xi}}{\EulerGamma@{\nu+\frac{3}{2}}}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\frac{3}{2})>0}} 
exp(-(+ 1/2)*Pi*I)*LegendreQ(nu,+ 1/2,cosh(xi))/GAMMA(nu++ 1/2+1) = ((Pi)/(2*sinh(xi)))^(1/2)*(exp(-(nu +(1)/(2))*xi))/(GAMMA(nu +(3)/(2)))

Exp[-(+ 1/2) Pi I] LegendreQ[\[Nu], + 1/2, 3, Cosh[\[Xi]]]/Gamma[\[Nu] + + 1/2 + 1] == (Divide[Pi,2*Sinh[\[Xi]]])^(1/2)*Divide[Exp[-(\[Nu]+Divide[1,2])*\[Xi]],Gamma[\[Nu]+Divide[3,2]]]
Error Failure -
Failed [40 / 80]
Result: Complex[2.271329177520301, 3.117315294925537]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.110539983099107, -2.8061475441370582]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
14.5.E17 𝑸 Ξ½ - 1 / 2 ⁑ ( cosh ⁑ ΞΎ ) = ( Ο€ 2 ⁒ sinh ⁑ ΞΎ ) 1 / 2 ⁒ exp ⁑ ( - ( Ξ½ + 1 2 ) ⁒ ΞΎ ) Ξ“ ⁑ ( Ξ½ + 3 2 ) associated-Legendre-black-Q 1 2 𝜈 πœ‰ superscript πœ‹ 2 πœ‰ 1 2 𝜈 1 2 πœ‰ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\boldsymbol{Q}^{-1/2}_{\nu}\left(\cosh\xi\right)=% \left(\frac{\pi}{2\sinh\xi}\right)^{1/2}\frac{\exp\left(-\left(\nu+\frac{1}{2}% \right)\xi\right)}{\Gamma\left(\nu+\frac{3}{2}\right)}}}
\assLegendreOlverQ[- 1/2]{\nu}@{\cosh@@{\xi}} = \left(\frac{\pi}{2\sinh@@{\xi}}\right)^{1/2}\frac{\exp@{-\left(\nu+\frac{1}{2}\right)\xi}}{\EulerGamma@{\nu+\frac{3}{2}}}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\frac{3}{2})>0}} 
exp(-(- 1/2)*Pi*I)*LegendreQ(nu,- 1/2,cosh(xi))/GAMMA(nu+- 1/2+1) = ((Pi)/(2*sinh(xi)))^(1/2)*(exp(-(nu +(1)/(2))*xi))/(GAMMA(nu +(3)/(2)))

Exp[-(- 1/2) Pi I] LegendreQ[\[Nu], - 1/2, 3, Cosh[\[Xi]]]/Gamma[\[Nu] + - 1/2 + 1] == (Divide[Pi,2*Sinh[\[Xi]]])^(1/2)*Divide[Exp[-(\[Nu]+Divide[1,2])*\[Xi]],Gamma[\[Nu]+Divide[3,2]]]
Error Failure -
Failed [45 / 80]
Result: Complex[2.271329177520301, 3.1173152949255365]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.1105399830991072, -2.806147544137058]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
14.5.E18 𝖯 Ξ½ - Ξ½ ⁑ ( cos ⁑ ΞΈ ) = ( sin ⁑ ΞΈ ) Ξ½ 2 Ξ½ ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) Ferrers-Legendre-P-first-kind 𝜈 𝜈 πœƒ superscript πœƒ 𝜈 superscript 2 𝜈 Euler-Gamma 𝜈 1 {\displaystyle{\displaystyle\mathsf{P}^{-\nu}_{\nu}\left(\cos\theta\right)=% \frac{(\sin\theta)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}}}
\FerrersP[-\nu]{\nu}@{\cos@@{\theta}} = \frac{(\sin@@{\theta})^{\nu}}{2^{\nu}\EulerGamma@{\nu+1}}		 β„œ ⁑ ( Ξ½ + 1 ) > 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0}} 
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
Failed [35 / 80]
Result: .2949209281-1.238111915*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)}


Result: 2.775912070+.3102767417*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, theta = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [35 / 80]
Result: Complex[0.29492092804949727, -1.2381119148256148]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.772257638440087, 3.7251537153578904]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.5.E19 P Ξ½ - Ξ½ ⁑ ( cosh ⁑ ΞΎ ) = ( sinh ⁑ ΞΎ ) Ξ½ 2 Ξ½ ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) Legendre-P-first-kind 𝜈 𝜈 πœ‰ superscript πœ‰ 𝜈 superscript 2 𝜈 Euler-Gamma 𝜈 1 {\displaystyle{\displaystyle P^{-\nu}_{\nu}\left(\cosh\xi\right)=\frac{(\sinh% \xi)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}}}
\assLegendreP[-\nu]{\nu}@{\cosh@@{\xi}} = \frac{(\sinh@@{\xi})^{\nu}}{2^{\nu}\EulerGamma@{\nu+1}}		 β„œ ⁑ ( Ξ½ + 1 ) > 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0}} 
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
Failed [35 / 80]
Result: -.1260431913-1.267273114*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)}

Result: 2.520491622+1.199838208*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, xi = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [35 / 80]
Result: Complex[-0.12604319089926652, -1.2672731138072273]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[2.5204916224127887, 1.1998382094597244]
Test Values: {Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
14.5.E20 𝖯 1 2 ⁑ ( cos ⁑ ΞΈ ) = 2 Ο€ ⁒ ( 2 ⁒ E ⁑ ( sin ⁑ ( 1 2 ⁒ ΞΈ ) ) - K ⁑ ( sin ⁑ ( 1 2 ⁒ ΞΈ ) ) ) shorthand-Ferrers-Legendre-P-first-kind 1 2 πœƒ 2 πœ‹ 2 complete-elliptic-integral-second-kind-E 1 2 πœƒ complete-elliptic-integral-first-kind-K 1 2 πœƒ {\displaystyle{\displaystyle\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=% \frac{2}{\pi}\left(2E\left(\sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(% \sin\left(\tfrac{1}{2}\theta\right)\right)\right)}}
\FerrersP[]{\frac{1}{2}}@{\cos@@{\theta}} = \frac{2}{\pi}\left(2\compellintEk@{\sin@{\tfrac{1}{2}\theta}}-\compellintKk@{\sin@{\tfrac{1}{2}\theta}}\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 [Tested: 10] Successful [Tested: 10]
14.5.E21 𝖯 - 1 2 ⁑ ( cos ⁑ ΞΈ ) = 2 Ο€ ⁒ K ⁑ ( sin ⁑ ( 1 2 ⁒ ΞΈ ) ) shorthand-Ferrers-Legendre-P-first-kind 1 2 πœƒ 2 πœ‹ complete-elliptic-integral-first-kind-K 1 2 πœƒ {\displaystyle{\displaystyle\mathsf{P}_{-\frac{1}{2}}\left(\cos\theta\right)=% \frac{2}{\pi}K\left(\sin\left(\tfrac{1}{2}\theta\right)\right)}}
\FerrersP[]{-\frac{1}{2}}@{\cos@@{\theta}} = \frac{2}{\pi}\compellintKk@{\sin@{\tfrac{1}{2}\theta}}		 

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 [Tested: 10] Successful [Tested: 10]
14.5.E22 𝖰 1 2 ⁑ ( cos ⁑ ΞΈ ) = K ⁑ ( cos ⁑ ( 1 2 ⁒ ΞΈ ) ) - 2 ⁒ E ⁑ ( cos ⁑ ( 1 2 ⁒ ΞΈ ) ) shorthand-Ferrers-Legendre-Q-first-kind 1 2 πœƒ complete-elliptic-integral-first-kind-K 1 2 πœƒ 2 complete-elliptic-integral-second-kind-E 1 2 πœƒ {\displaystyle{\displaystyle\mathsf{Q}_{\frac{1}{2}}\left(\cos\theta\right)=K% \left(\cos\left(\tfrac{1}{2}\theta\right)\right)-2E\left(\cos\left(\tfrac{1}{2% }\theta\right)\right)}}
\FerrersQ[]{\frac{1}{2}}@{\cos@@{\theta}} = \compellintKk@{\cos@{\tfrac{1}{2}\theta}}-2\compellintEk@{\cos@{\tfrac{1}{2}\theta}}		 

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 [Tested: 10] Successful [Tested: 10]
14.5.E23 𝖰 - 1 2 ⁑ ( cos ⁑ ΞΈ ) = K ⁑ ( cos ⁑ ( 1 2 ⁒ ΞΈ ) ) shorthand-Ferrers-Legendre-Q-first-kind 1 2 πœƒ complete-elliptic-integral-first-kind-K 1 2 πœƒ {\displaystyle{\displaystyle\mathsf{Q}_{-\frac{1}{2}}\left(\cos\theta\right)=K% \left(\cos\left(\tfrac{1}{2}\theta\right)\right)}}
\FerrersQ[]{-\frac{1}{2}}@{\cos@@{\theta}} = \compellintKk@{\cos@{\tfrac{1}{2}\theta}}		 

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 [Tested: 10] Successful [Tested: 10]
14.5.E24 P 1 2 ⁑ ( cosh ⁑ ΞΎ ) = 2 Ο€ ⁒ e ΞΎ / 2 ⁒ E ⁑ ( ( 1 - e - 2 ⁒ ΞΎ ) 1 / 2 ) shorthand-Legendre-P-first-kind 1 2 πœ‰ 2 πœ‹ superscript 𝑒 πœ‰ 2 complete-elliptic-integral-second-kind-E superscript 1 superscript 𝑒 2 πœ‰ 1 2 {\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)}}
\assLegendreP[]{\frac{1}{2}}@{\cosh@@{\xi}} = \frac{2}{\pi}e^{\xi/2}\compellintEk@{\left(1-e^{-2\xi}\right)^{1/2}}		 

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 [Tested: 10] Successful [Tested: 10]
14.5.E25 P - 1 2 ⁑ ( cosh ⁑ ΞΎ ) = 2 Ο€ ⁒ cosh ⁑ ( 1 2 ⁒ ΞΎ ) ⁒ K ⁑ ( tanh ⁑ ( 1 2 ⁒ ΞΎ ) ) shorthand-Legendre-P-first-kind 1 2 πœ‰ 2 πœ‹ 1 2 πœ‰ complete-elliptic-integral-first-kind-K 1 2 πœ‰ {\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)}}
\assLegendreP[]{-\frac{1}{2}}@{\cosh@@{\xi}} = \frac{2}{\pi\cosh@{\frac{1}{2}\xi}}\compellintKk@{\tanh@{\tfrac{1}{2}\xi}}		 

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 [Tested: 10] Successful [Tested: 10]
14.5.E26 𝑸 1 2 ⁑ ( cosh ⁑ ΞΎ ) = 2 ⁒ Ο€ - 1 / 2 ⁒ cosh ⁑ ΞΎ ⁒ sech ⁑ ( 1 2 ⁒ ΞΎ ) ⁒ K ⁑ ( sech ⁑ ( 1 2 ⁒ ΞΎ ) ) - 4 ⁒ Ο€ - 1 / 2 ⁒ cosh ⁑ ( 1 2 ⁒ ΞΎ ) ⁒ E ⁑ ( sech ⁑ ( 1 2 ⁒ ΞΎ ) ) shorthand-associated-Legendre-black-Q 1 2 πœ‰ 2 superscript πœ‹ 1 2 πœ‰ 1 2 πœ‰ complete-elliptic-integral-first-kind-K 1 2 πœ‰ 4 superscript πœ‹ 1 2 1 2 πœ‰ complete-elliptic-integral-second-kind-E 1 2 πœ‰ {\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)}}
\assLegendreOlverQ[]{\frac{1}{2}}@{\cosh@@{\xi}} = 2\pi^{-1/2}\cosh@@{\xi}\sech@{\tfrac{1}{2}\xi}\compellintKk@{\sech@{\tfrac{1}{2}\xi}}-4\pi^{-1/2}\cosh@{\tfrac{1}{2}\xi}\compellintEk@{\sech@{\tfrac{1}{2}\xi}}		 

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 [Tested: 10]
Failed [10 / 10]
Result: Complex[-0.8843996963296057, 0.10723567454157107]
Test Values: {Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.4538488510851968, -0.4630204881028235]
Test Values: {Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.5.E27 𝑸 - 1 2 ⁑ ( cosh ⁑ ΞΎ ) = 2 ⁒ Ο€ - 1 / 2 ⁒ e - ΞΎ / 2 ⁒ K ⁑ ( e - ΞΎ ) shorthand-associated-Legendre-black-Q 1 2 πœ‰ 2 superscript πœ‹ 1 2 superscript 𝑒 πœ‰ 2 complete-elliptic-integral-first-kind-K superscript 𝑒 πœ‰ {\displaystyle{\displaystyle\boldsymbol{Q}_{-\frac{1}{2}}\left(\cosh\xi\right)% =2\pi^{-1/2}e^{-\xi/2}K\left(e^{-\xi}\right)}}
\assLegendreOlverQ[]{-\frac{1}{2}}@{\cosh@@{\xi}} = 2\pi^{-1/2}e^{-\xi/2}\compellintKk@{e^{-\xi}}		 

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
Failed [5 / 10]
Result: -.101404509+1.824239856*I
Test Values: {xi = -1/2+1/2*I*3^(1/2)}

Result: -.90465021e-1-1.714290815*I
Test Values: {xi = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[0.16749403535362406, 1.47562407248214]
Test Values: {Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-2.5106529782887232, 0.796583020821415]
Test Values: {Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.5.E28 𝖯 2 ⁑ ( x ) = P 2 ⁑ ( x ) shorthand-Ferrers-Legendre-P-first-kind 2 π‘₯ shorthand-Legendre-P-first-kind 2 π‘₯ {\displaystyle{\displaystyle\mathsf{P}_{2}\left(x\right)=P_{2}\left(x\right)}}
\FerrersP[]{2}@{x} = \assLegendreP[]{2}@{x}		 

LegendreP(2, x) = LegendreP(2, x)
 
LegendreP[2, x] == LegendreP[2, 0, 3, x]
 Successful Successful - Successful [Tested: 3]
14.5.E28 P 2 ⁑ ( x ) = 3 ⁒ x 2 - 1 2 shorthand-Legendre-P-first-kind 2 π‘₯ 3 superscript π‘₯ 2 1 2 {\displaystyle{\displaystyle P_{2}\left(x\right)=\frac{3x^{2}-1}{2}}}
\assLegendreP[]{2}@{x} = \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 - Successful [Tested: 3]
14.5.E29 𝖰 2 ⁑ ( x ) = 3 ⁒ x 2 - 1 4 ⁒ ln ⁑ ( 1 + x 1 - x ) - 3 2 ⁒ x shorthand-Ferrers-Legendre-Q-first-kind 2 π‘₯ 3 superscript π‘₯ 2 1 4 1 π‘₯ 1 π‘₯ 3 2 π‘₯ {\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}}
\FerrersQ[]{2}@{x} = \frac{3x^{2}-1}{4}\ln@{\frac{1+x}{1-x}}-\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
Failed [2 / 3]
Result: .1e-8-9.032078880*I
Test Values: {x = 3/2}

Result: -.1e-8-17.27875960*I
Test Values: {x = 2}
Failed [2 / 3]
Result: Complex[0.0, -9.032078879070655]
Test Values: {Rule[x, 1.5]}

Result: Complex[0.0, -17.27875959474386]
Test Values: {Rule[x, 2]}

14.5.E30 𝑸 2 ⁑ ( x ) = 3 ⁒ x 2 - 1 8 ⁒ ln ⁑ ( x + 1 x - 1 ) - 3 4 ⁒ x shorthand-associated-Legendre-black-Q 2 π‘₯ 3 superscript π‘₯ 2 1 8 π‘₯ 1 π‘₯ 1 3 4 π‘₯ {\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}}
\assLegendreOlverQ[]{2}@{x} = \frac{3x^{2}-1}{8}\ln@{\frac{x+1}{x-1}}-\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
Failed [1 / 3]
Result: 0.+.1963495409*I
Test Values: {x = 1/2}
Failed [2 / 3]
Result: Complex[0.006453837346904523, -9.365446450684121*^-18]
Test Values: {Rule[x, 1.5]}

Result: Complex[0.23977862743400533, 0.2454369260617026]
Test Values: {Rule[x, 0.5]}

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