Legendre polynomial

The Legendre polynomial P lP_l (for l=0,1,2,l =0,1,2,\ldots) is the polynomial in one variable given by the formula

P l(x)=12 ll!d ldx l(x 21) l P_l(x) = \frac{1}{2^l l!} \frac{d^l}{d x^l}(x^2-1)^l

Alternatively they can be defined via a generating function:

112tx+t 2= n0P n(x)t n \frac{1}{\sqrt{1-2tx+t^2}} = \sum_{n\geq 0} P_n(x) t^n

The Legendre polynomials satisfy:

  • the following differential equation of the second order
    ddx[(1x 2)dP ldx]+l(l+1)P l=0 \frac{d}{d x}[(1-x^2)\frac{d P_l}{d x}] + l(l+1) P_l = 0
  • the recursion relations
    (l+1)P l+1(2l+1)xP l+lP l1=0 (l+1)P_{l+1}-(2l+1)x P_l+l P_{l-1}=0
  • the mixed differential recursion relations
    P l+1P l1=(2l+1)P l P l+1xP l=(l+1)P l (x 21)P llxP l+lP l1=0 \array { P'_{l+1}-P'_{l-1} = (2l+1)P_l \\ P'_{l+1}-x P'_l = (l+1)P_l \\ (x^2-1)P_l'-l x P_l+l P_{l-1} = 0 }

The Legendre polynomials form a complete system of orthogonal polynomials on the interval [1,1][-1,1], that is an unnormalised basis of the Hilbert space L 2[1,1]L^2[-1,1].

The first few Legendre polynomials are P 0(x)=1P_0(x) = 1, P 1(x)=xP_1(x) = x, P 2(x)=12(3x 21)P_2(x)=\frac{1}{2}(3x^2-1), P 3(x)=12(5x 23)P_3(x)=\frac{1}{2}(5x^2-3), P 4(x)=18(35x 430x 2+3)P_4(x)=\frac{1}{8}(35x^4-30x^2+3). Their values at 00 are

P 2n+1(0)=0,P 2n(0)=(1) n(2n1)!!(2n)!=(1) n(2n)!2 2n(n!) 2 P_{2n+1}(0)=0,\,\,\,\,\,P_{2n}(0)=(-1)^n\frac{(2n-1)!!}{(2n)!} = \frac{(-1)^n (2n)!}{2^{2n}(n!)^2}

and P l(±1)=(±1) lP_l(\pm 1)= (\pm 1)^l.

One also has the integral formulas

0 1P 2k+1(x)dx=(1) k(2k)!2 2k+1k!(k+1)! 0 1P 2k(x)dx=δ k0 1 1xP lP k={2(l+1)(2l+1)(2l+3), k=l+1 (2l)(2l1)(2l+1), k=l1 0, otherwise \array { \int^1_0 P_{2k+1}(x) d x = \frac{(-1)^k (2k)!}{2^{2k+1} k! (k+1)!} \\ \int^1_0 P_{2k}(x) d x = \delta_{k0} \\ \int^1_{-1} x P_l P_k = \left\lbrace \array{ \frac{2(l+1)}{(2l+1)(2l+3)},&k=l+1 \\ \frac{(2l)}{(2l-1)(2l+1)},&k=l-1\\ 0,& otherwise} \right. }

A generalization of Legendre polynomials are the Legendre functions P νP_\nu where ν\nu is not necessarily an integer and P l mP^m_l which are given by

P l m(x)=(1) m2 ll!(1x 2) m/2d l+mdx l+m(x 21) l=(1) m(1x 2) m/2d ldx lP l(x) P^m_l(x) = \frac{(-1)^m}{2^l l!} (1-x^2)^{m/2} \frac{d^{l+m}}{d x^{l+m}} (x^2-1)^l = (-1)^m (1-x^2)^{m/2} \frac{d^{l}}{d x^{l}} P_l(x)

for m0m\geq 0 and also

P l m(x)=(1) m(lm)!(l+m)!P l m(x) P^{-m}_l(x)= (-1)^m\frac{(l-m)!}{(l+m)!} P^m_l(x)

These P l mP^m_l are satisfying the orthogonality relations

1 1P l m(x)P k m(x)dx=22l+1(l+m)!(lm)!δ lk \int_{-1}^1 P^m_l(x) P^m_k(x) d x = \frac{2}{2l+1}\frac{(l+m)!}{(l-m)!}\delta_{lk}

recursion relation

xP l m(x)=l+|m|2l+1P l1 m(x)+l|m|+12l+1P l+1 m(x) x P^m_l(x) = \frac{l+|m|}{2l+1}P^m_{l-1}(x)+\frac{l-|m|+1}{2l+1} P^m_{l+1}(x)

and the differential equation

ddx[(1x 2)dP l mdx]+[(l(l+1)m 21x 2]P l m=0 \frac{d}{d x}[(1-x^2)\frac{d P^m_l}{d x}]+[(l(l+1)-\frac{m^2}{1-x^2}]P^m_l = 0

P ν(x)P_\nu(x) is a special case of a hypergeometric function, namely

P ν(x)= 2F 1(ν,ν+1;1;1x2) P_\nu(x)={}_2 F_1(-\nu,\nu+1;1;\frac{1-x}{2})

Legendre polynomials enter the expressions for the spherical function?s for sphere S 2S^2 in 3d:

Y lm(θ,ϕ)=2l+14π(lm)!(l+m)!P l m(cosθ)e imϕ Y_{l m}(\theta,\phi) = \sqrt{ \frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!} } P^m_l(cos\theta) e^{im\phi}

(If it is not clear from mathML rendering – both fractions are under square root – including both the numerators and denominators).

If (θ,ϕ)(\theta,\phi) and (θ,ϕ)(\theta',\phi') are two points of the unit sphere in spherical coordinates (polar angle, azimuth), and γ\gamma is the angle between the two corresponding rays from the origin then

P l(cosγ)= m=l l(1) mP l m(cosθ)P l m(cosθ)cos(m(ϕϕ))=4π2l+1 m=l lY lm(θ,ϕ)Y lm *(θ,ϕ) P_l(cos \gamma) = \sum_{m=-l}^l (-1)^m P_l^m(cos \theta)P_l^{-m}(cos \theta') cos(m(\phi-\phi')) = \frac{4\pi}{2l+1}\sum_{m=-l}^l Y_{lm}(\theta,\phi) Y_{lm}^*(\theta',\phi')

what for l=1l=1 reduces to the spherical law of cosine from spherical trigonometry:

cosγ=P 1(cosγ)=cosθcosθ+sinθsinθcos(ϕϕ) cos \gamma = P_1(cos \gamma) = cos\theta cos\theta' + sin\theta sin\theta' cos(\phi-\phi')

The orthogonality relation for Legendre polynomials gives Laplace’s formula

S 2dΩ k^Y l(k^)P l(k^p^)=4π2l+1δ llY l(p^) \int_{S^2} d\Omega_{\hat{k}} Y_l(\hat{k}) P_l(\hat{k}\hat{p}) = \frac{4\pi}{2l+1} \delta_{ll'} Y_l(\hat{p})

where k^,p^\hat{k},\hat{p} are unit vectors and Y l=a mY lmY_l = \sum a_m Y_{lm} is some spherical function.

The following orthogonality integral relation is over product of unit spheres in 3\mathbb{R}^3:

k^ p^dΩ k^dΩ p^P l(k^p^)P l(k^q^)P l(p^q^)=(4π2l+1)δ llδ ll,(*) \int_{\hat{k}}\int_{\hat{p}} d\Omega_{\hat{k}}d\Omega_{\hat{p}} P_l(\hat{k}\hat{p}) P_{l'}(\hat{k}\hat{q}) P_{l''}(\hat{p}\hat{q}) = \left(\frac{4\pi}{2l+1}\right)\delta_{ll'}\delta_{ll''}, \,\,\,\,\,(*)

where the arguments of Legendre polynomials are the inner products of the unit vectors.

Zoran: the last formula (*)(*) is in my own formula notes which I have written as a student many years ago and used hundreds of times, but it now looks to me suspicious; I have no time to check it right now.

Revised on May 12, 2017 05:47:47 by Urs Schreiber (