Legendre and Related Functions - 14.6 Integer Order

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.6.E1 𝖯 Ξ½ m ⁑ ( x ) = ( - 1 ) m ⁒ ( 1 - x 2 ) m / 2 ⁒ d m 𝖯 Ξ½ ⁑ ( x ) d x m Ferrers-Legendre-P-first-kind π‘š 𝜈 π‘₯ superscript 1 π‘š superscript 1 superscript π‘₯ 2 π‘š 2 derivative shorthand-Ferrers-Legendre-P-first-kind 𝜈 π‘₯ π‘₯ π‘š {\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1% -x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{P}_{\nu}\left(x\right)}{{% \mathrm{d}x}^{m}}}}
\FerrersP[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersP[]{\nu}@{x}}{x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreP(nu, x), [x$(m)])

LegendreP[\[Nu], m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreP[\[Nu], x], {x, m}]
Failure Failure
Failed [3 / 90]
Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 3/2, m = 1}


Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 1/2, m = 1}

... skip entries to safe data
 Successful [Tested: 90]
14.6.E2 𝖰 Ξ½ m ⁑ ( x ) = ( - 1 ) m ⁒ ( 1 - x 2 ) m / 2 ⁒ d m 𝖰 Ξ½ ⁑ ( x ) d x m Ferrers-Legendre-Q-first-kind π‘š 𝜈 π‘₯ superscript 1 π‘š superscript 1 superscript π‘₯ 2 π‘š 2 derivative shorthand-Ferrers-Legendre-Q-first-kind 𝜈 π‘₯ π‘₯ π‘š {\displaystyle{\displaystyle\mathsf{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1% -x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{Q}_{\nu}\left(x\right)}{{% \mathrm{d}x}^{m}}}}
\FerrersQ[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersQ[]{\nu}@{x}}{x}		 β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ + m + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - m + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 𝜈 π‘š 1 0 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 𝜈 π‘š 1 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu+m+1)>0,\Re(\nu-\mu+1)>0,% \Re(\nu-m+1)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreQ(nu, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreQ(nu, x), [x$(m)])

LegendreQ[\[Nu], m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreQ[\[Nu], x], {x, m}]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
14.6.E3 P Ξ½ m ⁑ ( x ) = ( x 2 - 1 ) m / 2 ⁒ d m P Ξ½ ⁑ ( x ) d x m Legendre-P-first-kind π‘š 𝜈 π‘₯ superscript superscript π‘₯ 2 1 π‘š 2 derivative shorthand-Legendre-P-first-kind 𝜈 π‘₯ π‘₯ π‘š {\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m% /2}\frac{{\mathrm{d}}^{m}P_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}
\assLegendreP[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreP[]{\nu}@{x}}{x}		 

LegendreP(nu, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreP(nu, x), [x$(m)])

LegendreP[\[Nu], m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreP[\[Nu], 0, 3, x], {x, m}]
Failure Failure
Failed [3 / 90]
Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 3/2, m = 1}


Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 1/2, m = 1}

... skip entries to safe data
 Successful [Tested: 90]
14.6.E4 Q Ξ½ m ⁑ ( x ) = ( x 2 - 1 ) m / 2 ⁒ d m Q Ξ½ ⁑ ( x ) d x m Legendre-Q-second-kind π‘š 𝜈 π‘₯ superscript superscript π‘₯ 2 1 π‘š 2 derivative shorthand-Legendre-Q-second-kind 𝜈 π‘₯ π‘₯ π‘š {\displaystyle{\displaystyle Q^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m% /2}\frac{{\mathrm{d}}^{m}Q_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}
\assLegendreQ[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreQ[]{\nu}@{x}}{x}		 

LegendreQ(nu, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreQ(nu, x), [x$(m)])

LegendreQ[\[Nu], m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreQ[\[Nu], 0, 3, x], {x, m}]
Failure Failure Error
Failed [75 / 90]
Result: Plus[Complex[-0.4598393885300628, 0.18181080125096066], Times[-1.118033988749895, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Equal[[1], Times[-1, Power[Plus[-1, Power[1.5, 2]], -1], Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[1.5, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 0, 3, 1.5]]]]]}]][1.0]]], {Rule[m, 1], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[1.6909557968522604, -0.413901027514361], Times[-2.5, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, Power[Plus[1, ], 2], 1.5, [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, 1.5], Plus[1, 1.5], [Plus[2, ]]]], 0], Equal[[0], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Equal[[1], Times[-1, Power[Plus[-1, Power[1.5, 2]], -1], Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Times[1.5, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 0, 3, 1.5]], Times[-1, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 0, 3, 1.5]]]]]}]][2.0]]], {Rule[m, 2], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
14.6.E5 ( Ξ½ + 1 ) m ⁒ 𝑸 Ξ½ m ⁑ ( x ) = ( - 1 ) m ⁒ ( x 2 - 1 ) m / 2 ⁒ d m 𝑸 Ξ½ ⁑ ( x ) d x m Pochhammer 𝜈 1 π‘š associated-Legendre-black-Q π‘š 𝜈 π‘₯ superscript 1 π‘š superscript superscript π‘₯ 2 1 π‘š 2 derivative shorthand-associated-Legendre-black-Q 𝜈 π‘₯ π‘₯ π‘š {\displaystyle{\displaystyle{\left(\nu+1\right)_{m}}\boldsymbol{Q}^{m}_{\nu}% \left(x\right)=(-1)^{m}\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}% \boldsymbol{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}}}
\Pochhammersym{\nu+1}{m}\assLegendreOlverQ[m]{\nu}@{x} = (-1)^{m}\left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreOlverQ[]{\nu}@{x}}{x}		 

pochhammer(nu + 1, m)*exp(-(m)*Pi*I)*LegendreQ(nu,m,x)/GAMMA(nu+m+1) = (- 1)^(m)*((x)^(2)- 1)^(m/2)* diff(LegendreQ(nu,x)/GAMMA(nu+1), [x$(m)])

Pochhammer[\[Nu]+ 1, m]*Exp[-(m) Pi I] LegendreQ[\[Nu], m, 3, x]/Gamma[\[Nu] + m + 1] == (- 1)^(m)*((x)^(2)- 1)^(m/2)* D[Exp[-(\[Nu]) Pi I] LegendreQ[\[Nu], 2, 3, x]/Gamma[\[Nu] + 3], {x, m}]
Failure Failure Error
Failed [90 / 90]
Result: Plus[Complex[0.482758812955306, -0.29762130115013324], Times[Complex[-1.0778621920495528, 0.20681719187113978], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, 1.5, Plus[3, Times[5, ], Times[2, Power[, 2]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[Plus[-12, Times[-8, ], Times[-2, Power[, 2]], Times[24, Power[1.5, 2]], Times[24, , Power[1.5, 2]], Times[6, Power[, 2], Power[1.5, 2]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[-1, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Times[-1, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[2, ]]], Times[2, P<syntaxhighlight lang=mathematica>Result: Plus[Complex[1.8263637314445087, -0.806860371328253], Times[Complex[2.4101731317997332, -0.4624572999394857], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[2, 1.5, Plus[3, Times[5, ], Times[2, Power[, 2]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[Plus[-12, Times[-8, ], Times[-2, Power[, 2]], Times[24, Power[1.5, 2]], Times[24, , Power[1.5, 2]], Times[6, Power[, 2], Power[1.5, 2]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[-1, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], Times[-1, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[2, ]]], Times[2, Plus[3, ], Plus[5, Times[2, ]], Plus[-1, 1.5], 1.5, Plus[1, 1.5], [Plus[3, ]]], Times[Plus[3, ], Plus[4, ], Power[Plus[-1, 1.5], 2], Power[Plus[1, 1.5], 2], [Plus[4, ]]]], 0], Equal[[0], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Equal[[1], Times[Power[Plus[-1, Power[1.5, 2]], -1], Plus[Times[-1, 1.5, Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[Plus[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]]]]], Equal[[2], Times[Rational[1, 2], Power[Plus[-1, Power[1.5, 2]], -2], Plus[Times[4, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[2, Power[1.5, 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[3, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[2, 1.5, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-2, 1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]]]]], Equal[[3], Times[Rational[-1, 6], Power[Plus[-1, Power[1.5, 2]], -3], Plus[Times[30, 1.5, LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[6, Power[1.5, 3], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[11, Power[1.5, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-6, 1.5, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[6, Power[1.5, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[-1, 1.5, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[Power[1.5, 3], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2, 3, 1.5]], Times[6, LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[6, Power[1.5, 2], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-7, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-5, Power[1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]], Times[-1, Power[1.5, 2], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 3], LegendreQ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2, 3, 1.5]]]]]}]][2.0]]], {Rule[m, 2], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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