Gold 10

Legendre polynomials

Gold ID

10

Link

https://sigir21.wmflabs.org/wiki/Legendre_polynomials#math.60.57

Formula

$\frac{1}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}=\frac{1}{\sqrt{r^2+{r^{\prime }}^2-2r{r}^{\prime }\cos \gamma }}=\sum _{\ell =0}^{\mathrm{\infty }}\frac{{r^{\prime }}^{\ell }}{r^{\ell +1}}{P}_{\ell }(\cos \gamma )$

TeX Source

\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations
		-

Semantic LaTeX

Translation

\frac{1}{|\mathbf{x} - \mathbf{x} '|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} \LegendrepolyP{\ell}@{\cos \gamma}

Expected (Gold Entry)

\frac{1}{|\mathbf{x} - \mathbf{x} '|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} \LegendrepolyP{\ell}@{\cos \gamma}

Mathematica

Translation

Expected (Gold Entry)

Divide[1, Abs[x - x\[Prime]]] == Divide[1, Sqrt[r^2+(r\[Prime])^(2)-2*r*r\[Prime] Cos[\[Gamma]]]] == Sum[Divide[(r\[Prime])^(\[ScriptL]), r^(\[ScriptL]+1)]*LegendreP[\[ScriptL], Cos[\[Gamma]]], {\[ScriptL], 0, Infinity}]

Maple

Translation

Expected (Gold Entry)