Legendre Polynomial

The orthogonal basis set for spherical coordinates

$P_l(x)\equiv \frac{1}{2^{l}l!}{\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^l(x^2-1{)}^l$

• One use it to calculate the associated Legendre polynomial

${\int }_{-1}^1{P}_n(x)P_m(x)\mathrm{d}x=\frac{2}{2n+1}{\delta }_{nm}(x)$

• Orthogonality condition of Legendre polynomials
• ${\delta }_{nm}$ is the Kronecker Delta

$P_l(-x)=(-1{)}^l{P}_l(x)$

• Even $l$ corresponds to even functions, odd $l$ corresponds to odd functions

${\text{d}}_{\theta }(\sin \theta {\text{d}}_{{}_{\theta }}\Theta )=-l(l+1)\sin (\theta )\Theta$

• Differential equation that the Legendre polynomial solves for

$\Theta (\theta )=P_l(\cos \theta )$

Example Values

TableForm[Simplify[Table[D[(x^2 - 1)^l, {x, l}]/(2^l*l!), {l, 0, 4}]]]
TableForm[Simplify[Table[LegendreP[l,x],{l,0,4}]]] (*Same but compact*)

Legendre Polynomial	Value
$P_l(x)$	$1$
$P_1(x)$	$x$
$P_2(x)$	$\frac{1}{2}(3x^2-1)$
$P_3(x)$	$\frac{1}{2}x(5x^2-3)$
$P_4(x)$	$\frac{1}{8}(35x^4-30x^2+3)$