14.28: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/14.28.E1 14.28.E1] || [[Item:Q4960|<math>\assLegendreP[]{\nu}@{z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos@@{\phi}} = \assLegendreP[]{\nu}@{z_{1}}\assLegendreP[]{\nu}@{z_{2}}+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\*\assLegendreP[m]{\nu}@{z_{1}}\assLegendreP[m]{\nu}(z_{2})\cos@{m\phi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[]{\nu}@{z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos@@{\phi}} = \assLegendreP[]{\nu}@{z_{1}}\assLegendreP[]{\nu}@{z_{2}}+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\*\assLegendreP[m]{\nu}@{z_{1}}\assLegendreP[m]{\nu}(z_{2})\cos@{m\phi}</syntaxhighlight> || <math>\realpart@@{(\nu-m+1)} > 0, \realpart@@{(\nu+m+1)} > 0</math> || <syntaxhighlight lang=mathematica>LegendreP(nu, z[1]*z[2]-((z[1])^(2)- 1)^(1/2)*((z[2])^(2)- 1)^(1/2)* cos(phi)) = LegendreP(nu, z[1])*LegendreP(nu, z[2])+ 2*sum((- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))* LegendreP(nu, m, z[1])*LegendreP(nu, m, z[2])*cos(m*phi), m = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[\[Nu], 0, 3, Subscript[z, 1]*Subscript[z, 2]-((Subscript[z, 1])^(2)- 1)^(1/2)*((Subscript[z, 2])^(2)- 1)^(1/2)* Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Subscript[z, 1]]*LegendreP[\[Nu], 0, 3, Subscript[z, 2]]+ 2*Sum[(- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]* LegendreP[\[Nu], m, 3, Subscript[z, 1]]*LegendreP[\[Nu], m, 3, Subscript[z, 2]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/14.28.E1 14.28.E1] || <math qid="Q4960">\assLegendreP[]{\nu}@{z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos@@{\phi}} = \assLegendreP[]{\nu}@{z_{1}}\assLegendreP[]{\nu}@{z_{2}}+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\*\assLegendreP[m]{\nu}@{z_{1}}\assLegendreP[m]{\nu}(z_{2})\cos@{m\phi}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\assLegendreP[]{\nu}@{z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos@@{\phi}} = \assLegendreP[]{\nu}@{z_{1}}\assLegendreP[]{\nu}@{z_{2}}+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\*\assLegendreP[m]{\nu}@{z_{1}}\assLegendreP[m]{\nu}(z_{2})\cos@{m\phi}</syntaxhighlight> || <math>\realpart@@{(\nu-m+1)} > 0, \realpart@@{(\nu+m+1)} > 0</math> || <syntaxhighlight lang=mathematica>LegendreP(nu, z[1]*z[2]-((z[1])^(2)- 1)^(1/2)*((z[2])^(2)- 1)^(1/2)* cos(phi)) = LegendreP(nu, z[1])*LegendreP(nu, z[2])+ 2*sum((- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))* LegendreP(nu, m, z[1])*LegendreP(nu, m, z[2])*cos(m*phi), m = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[\[Nu], 0, 3, Subscript[z, 1]*Subscript[z, 2]-((Subscript[z, 1])^(2)- 1)^(1/2)*((Subscript[z, 2])^(2)- 1)^(1/2)* Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Subscript[z, 1]]*LegendreP[\[Nu], 0, 3, Subscript[z, 2]]+ 2*Sum[(- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]* LegendreP[\[Nu], m, 3, Subscript[z, 1]]*LegendreP[\[Nu], m, 3, Subscript[z, 2]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/14.28.E2 14.28.E2] || [[Item:Q4961|<math>\sum_{n=0}^{\infty}(2n+1)\assLegendreQ[]{n}@{z_{1}}\assLegendreP[]{n}@{z_{2}} = \frac{1}{z_{1}-z_{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}(2n+1)\assLegendreQ[]{n}@{z_{1}}\assLegendreP[]{n}@{z_{2}} = \frac{1}{z_{1}-z_{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((2*n + 1)*LegendreQ(n, z[1])*LegendreP(n, z[2]), n = 0..infinity) = (1)/(z[1]- z[2])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(2*n + 1)*LegendreQ[n, 0, 3, Subscript[z, 1]]*LegendreP[n, 0, 3, Subscript[z, 2]], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Subscript[z, 1]- Subscript[z, 2]]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[DirectedInfinity[], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
| [https://dlmf.nist.gov/14.28.E2 14.28.E2] || <math qid="Q4961">\sum_{n=0}^{\infty}(2n+1)\assLegendreQ[]{n}@{z_{1}}\assLegendreP[]{n}@{z_{2}} = \frac{1}{z_{1}-z_{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}(2n+1)\assLegendreQ[]{n}@{z_{1}}\assLegendreP[]{n}@{z_{2}} = \frac{1}{z_{1}-z_{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((2*n + 1)*LegendreQ(n, z[1])*LegendreP(n, z[2]), n = 0..infinity) = (1)/(z[1]- z[2])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(2*n + 1)*LegendreQ[n, 0, 3, Subscript[z, 1]]*LegendreP[n, 0, 3, Subscript[z, 2]], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Subscript[z, 1]- Subscript[z, 2]]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[DirectedInfinity[], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.6830127018922194, -0.18301270189221946], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.6830127018922194, -0.18301270189221946], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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Latest revision as of 12:38, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.28.E1 P ν ⁡ ( z 1 ⁢ z 2 - ( z 1 2 - 1 ) 1 / 2 ⁢ ( z 2 2 - 1 ) 1 / 2 ⁢ cos ⁡ ϕ ) = P ν ⁡ ( z 1 ) ⁢ P ν ⁡ ( z 2 ) + 2 ⁢ ∑ m = 1 ∞ ( - 1 ) m ⁢ Γ ⁡ ( ν - m + 1 ) Γ ⁡ ( ν + m + 1 ) ⁢ P ν m ⁡ ( z 1 ) ⁢ P ν m ⁡ ( z 2 ) ⁢ cos ⁡ ( m ⁢ ϕ ) shorthand-Legendre-P-first-kind 𝜈 subscript 𝑧 1 subscript 𝑧 2 superscript superscript subscript 𝑧 1 2 1 1 2 superscript superscript subscript 𝑧 2 2 1 1 2 italic-ϕ shorthand-Legendre-P-first-kind 𝜈 subscript 𝑧 1 shorthand-Legendre-P-first-kind 𝜈 subscript 𝑧 2 2 superscript subscript 𝑚 1 superscript 1 𝑚 Euler-Gamma 𝜈 𝑚 1 Euler-Gamma 𝜈 𝑚 1 Legendre-P-first-kind 𝑚 𝜈 subscript 𝑧 1 Legendre-P-first-kind 𝑚 𝜈 subscript 𝑧 2 𝑚 italic-ϕ {\displaystyle{\displaystyle P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^% {1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_% {\nu}\left(z_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1% \right)}{\Gamma\left(\nu+m+1\right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}% (z_{2})\cos\left(m\phi\right)}}
\assLegendreP[]{\nu}@{z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos@@{\phi}} = \assLegendreP[]{\nu}@{z_{1}}\assLegendreP[]{\nu}@{z_{2}}+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\*\assLegendreP[m]{\nu}@{z_{1}}\assLegendreP[m]{\nu}(z_{2})\cos@{m\phi}		 ℜ ⁡ ( ν - m + 1 ) > 0 , ℜ ⁡ ( ν + m + 1 ) > 0 formulae-sequence 𝜈 𝑚 1 0 𝜈 𝑚 1 0 {\displaystyle{\displaystyle\Re(\nu-m+1)>0,\Re(\nu+m+1)>0}} 
LegendreP(nu, z[1]*z[2]-((z[1])^(2)- 1)^(1/2)*((z[2])^(2)- 1)^(1/2)* cos(phi)) = LegendreP(nu, z[1])*LegendreP(nu, z[2])+ 2*sum((- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))* LegendreP(nu, m, z[1])*LegendreP(nu, m, z[2])*cos(m*phi), m = 1..infinity)

LegendreP[\[Nu], 0, 3, Subscript[z, 1]*Subscript[z, 2]-((Subscript[z, 1])^(2)- 1)^(1/2)*((Subscript[z, 2])^(2)- 1)^(1/2)* Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Subscript[z, 1]]*LegendreP[\[Nu], 0, 3, Subscript[z, 2]]+ 2*Sum[(- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]* LegendreP[\[Nu], m, 3, Subscript[z, 1]]*LegendreP[\[Nu], m, 3, Subscript[z, 2]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]
Translation Error Translation Error - -
14.28.E2 ∑ n = 0 ∞ ( 2 ⁢ n + 1 ) ⁢ Q n ⁡ ( z 1 ) ⁢ P n ⁡ ( z 2 ) = 1 z 1 - z 2 superscript subscript 𝑛 0 2 𝑛 1 shorthand-Legendre-Q-second-kind 𝑛 subscript 𝑧 1 shorthand-Legendre-P-first-kind 𝑛 subscript 𝑧 2 1 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}(2n+1)Q_{n}\left(z_{1}\right)P_% {n}\left(z_{2}\right)=\frac{1}{z_{1}-z_{2}}}}
\sum_{n=0}^{\infty}(2n+1)\assLegendreQ[]{n}@{z_{1}}\assLegendreP[]{n}@{z_{2}} = \frac{1}{z_{1}-z_{2}}		 

sum((2*n + 1)*LegendreQ(n, z[1])*LegendreP(n, z[2]), n = 0..infinity) = (1)/(z[1]- z[2])

Sum[(2*n + 1)*LegendreQ[n, 0, 3, Subscript[z, 1]]*LegendreP[n, 0, 3, Subscript[z, 2]], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Subscript[z, 1]- Subscript[z, 2]]
Failure Failure Skipped - Because timed out
Failed [100 / 100]
Result: Plus[DirectedInfinity[], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.6830127018922194, -0.18301270189221946], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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