View source for 14.6

{{DISPLAYTITLE:Legendre and Related Functions - 14.6 Integer Order}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/14.6.E1 14.6.E1] || <math qid="Q4743">\FerrersP[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersP[]{\nu}@{x}}{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersP[]{\nu}@{x}}{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreP(nu, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreP(nu, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[\[Nu], m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreP[\[Nu], x], {x, m}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 3/2, m = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 1/2, m = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 90]
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| [https://dlmf.nist.gov/14.6.E2 14.6.E2] || <math qid="Q4744">\FerrersQ[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersQ[]{\nu}@{x}}{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[m]{\nu}@{x} = (-1)^{m}\left(1-x^{2}\right)^{m/2}\deriv[m]{\FerrersQ[]{\nu}@{x}}{x}</syntaxhighlight> || <math>\realpart@@{(\nu+\mu+1)} > 0, \realpart@@{(\nu+m+1)} > 0, \realpart@@{(\nu-\mu+1)} > 0, \realpart@@{(\nu-m+1)} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>LegendreQ(nu, m, x) = (- 1)^(m)*(1 - (x)^(2))^(m/2)* diff(LegendreQ(nu, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[\[Nu], m, x] == (- 1)^(m)*(1 - (x)^(2))^(m/2)* D[LegendreQ[\[Nu], x], {x, m}]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| [https://dlmf.nist.gov/14.6.E3 14.6.E3] || <math qid="Q4745">\assLegendreP[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreP[]{\nu}@{x}}{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreP[]{\nu}@{x}}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(nu, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreP(nu, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[\[Nu], m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreP[\[Nu], 0, 3, x], {x, m}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 3/2, m = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 1/2, m = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 90]
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| [https://dlmf.nist.gov/14.6.E4 14.6.E4] || <math qid="Q4746">\assLegendreQ[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreQ[]{\nu}@{x}}{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreQ[m]{\nu}@{x} = \left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreQ[]{\nu}@{x}}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreQ(nu, m, x) = ((x)^(2)- 1)^(m/2)* diff(LegendreQ(nu, x), [x$(m)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[\[Nu], m, 3, x] == ((x)^(2)- 1)^(m/2)* D[LegendreQ[\[Nu], 0, 3, x], {x, m}]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [75 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.6.E5 14.6.E5] || <math qid="Q4747">\Pochhammersym{\nu+1}{m}\assLegendreOlverQ[m]{\nu}@{x} = (-1)^{m}\left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreOlverQ[]{\nu}@{x}}{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Pochhammersym{\nu+1}{m}\assLegendreOlverQ[m]{\nu}@{x} = (-1)^{m}\left(x^{2}-1\right)^{m/2}\deriv[m]{\assLegendreOlverQ[]{\nu}@{x}}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>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}]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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