Associated Legendre Polynomials
📂Functions Associated Legendre Polynomials Definition Associated Legendre polynomials are defined in the following ways.
As a Solution to a Differential Equation The solutions to the associated Legendre differential equation below are referred to as associated Legendre polynomials .
( 1 − x 2 ) d 2 y d x 2 − 2 x d y d x + [ l ( l + 1 ) − m 2 1 − x 2 ] y = 0 or d d x [ ( 1 − x 2 ) y ′ ] + [ l ( l + 1 ) − m 2 1 − x 2 ] y = 0
\begin{align*}
&& (1-x^{2}) \frac{d^{2}y}{dx^{2}} - 2x \frac{dy}{dx} + \left[l(l+1) - \frac{m^{2}}{1-x^{2}}\right] y &= 0 \\
\text{or} && \frac{d}{dx} \left[(1-x^{2})y^{\prime}\right] + \left[l(l+1) - \frac{m^{2}}{1-x^{2}}\right] y &= 0
\end{align*}
or ( 1 − x 2 ) d x 2 d 2 y − 2 x d x d y + [ l ( l + 1 ) − 1 − x 2 m 2 ] y d x d [ ( 1 − x 2 ) y ′ ] + [ l ( l + 1 ) − 1 − x 2 m 2 ] y = 0 = 0
The polynomial function P l m P_{l}^{m} P l m associated Legendre polynomial .
P l m ( x ) = ( 1 − x 2 ) ∣ m ∣ 2 d ∣ m ∣ d x ∣ m ∣ P l ( x ) = ( 1 − x 2 ) ∣ m ∣ 2 d ∣ m ∣ d x ∣ m ∣ [ 1 2 l l ! d l d x l ( x 2 − 1 ) l ]
\begin{align*}
P_{l}^{m}(x) &= (1-x ^{2})^{\frac{|m|}{2}} \frac{ d^{|m|} }{ dx^{|m|} } P_{l}(x) \\
&=(1-x ^{2})^{\frac{|m|}{2}} \frac{ d^{|m|} }{ dx^{|m|} }\left[ \dfrac{1}{2^l l!} \dfrac{d^l}{dx^l}(x^2-1)^l\right]
\end{align*}
P l m ( x ) = ( 1 − x 2 ) 2 ∣ m ∣ d x ∣ m ∣ d ∣ m ∣ P l ( x ) = ( 1 − x 2 ) 2 ∣ m ∣ d x ∣ m ∣ d ∣ m ∣ [ 2 l l ! 1 d x l d l ( x 2 − 1 ) l ]
Here, P l P_{l} P l Legendre polynomial , and the above formula is known as Rodrigues’ formula.
Description In the case where m = 0 m=0 m = 0 P l 0 ( x ) = P l ( x ) P_{l}^{0}(x) = P_{l}(x) P l 0 ( x ) = P l ( x )
Properties The associated Legendre differential equation expressed using trigonometric functions is as follows.
d 2 y d θ 2 + cot θ d y d θ + ( l ( l + 1 ) − m 2 sin 2 θ ) y = 0 o r 1 sin θ ( sin θ d y d θ ) + ( l ( l + 1 ) − m 2 sin 2 θ ) y = 0
\begin{align*}
\frac{ d^{2} y}{ d \theta^{2} }+\cot \theta \frac{ d y}{ d \theta}+ \left( l(l+1) -\frac{m^{2}}{\sin ^{2 }\theta} \right)y=0 \\
\mathrm{or} \quad\frac{1}{\sin \theta}\left(\sin \theta \frac{dy}{d\theta} \right)+ \left(l(l+1) -\frac{ m^{2}}{\sin ^{2} \theta} \right)y=0
\end{align*}
d θ 2 d 2 y + cot θ d θ d y + ( l ( l + 1 ) − sin 2 θ m 2 ) y = 0 or sin θ 1 ( sin θ d θ d y ) + ( l ( l + 1 ) − sin 2 θ m 2 ) y = 0
Relationship Depending on the Sign of m m m Associated Legendre polynomials have the following relationship depending on the sign of m m m Link )
P l − m ( x ) = ( − 1 ) m ( l − m ) ! ( l + m ) ! P l m ( x )
P_{l}^{-m}(x)=(-1)^{m}\frac{(l-m)!}{(l+m)!}P_{l}^{m}(x)
P l − m ( x ) = ( − 1 ) m ( l + m )! ( l − m )! P l m ( x )
Orthogonality For a fixed m m m [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] orthogonal set . (Link )
∫ − 1 1 P l m ( x ) P k m ( x ) d x = 2 2 l + 1 ( l + m ) ! ( l − m ) ! δ l k
\int_{-1}^{1} P_{l}^{m}(x)P_{k}^{m}(x)dx =\frac{ 2}{ 2l+1 }\frac{(l+m)!}{(l-m)!}\delta_{lk}
∫ − 1 1 P l m ( x ) P k m ( x ) d x = 2 l + 1 2 ( l − m )! ( l + m )! δ l k
When x = cos θ x=\cos \theta x = cos θ
∫ 0 π P l m ( cos θ ) P k m ( cos θ ) sin θ d θ = 2 2 l + 1 ( l + m ) ! ( l − m ) ! δ l k
\int_{0}^{\pi} P_{l}^{m}(\cos \theta)P_ {k}^{m}(\cos\theta)\sin \theta d\theta =\frac{ 2}{ 2l+1 }\frac{(l+m)!}{(l-m)!}\delta_{lk}
∫ 0 π P l m ( cos θ ) P k m ( cos θ ) sin θ d θ = 2 l + 1 2 ( l − m )! ( l + m )! δ l k
Normalization The normalized associated Legendre polynomials are as follows. (Link )
P l m ( x ) = 2 l + 1 2 ( l − m ) ! ( l + m ) ! P l m ( x )
P_{l}^{m}(x) = \sqrt{\frac{2l+1}{2}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(x)
P l m ( x ) = 2 2 l + 1 ( l + m )! ( l − m )! P l m ( x )
