Legendre differential equation

The equation $(1-x^2)y\prime \prime -2xy\prime +\ell (\ell +1)y=0$ has polynomial solutions for nonnegative integer $\ell$.

Associated Legendre functions solve $(1-x^2)y^{\prime \prime }-2x{y}^{\prime }+[\nu (\nu +1)-{\mu }^2/(1-x^2)]y=0$. For integers $\ell \ge 0$ and $0\le m\le \ell$, the regular solution is $P_{\ell }^m(x)=(-1{)}^m(1-x^2{)}^{m/2}(d/dx{)}^m{P}_{\ell }(x)$, using a Legendre polynomial. It enters normalized spherical harmonics. Despite the traditional term associated Legendre polynomial, the factor $(1-x^2{)}^{m/2}$ is not a polynomial when $m$ is odd. Negative integer orders satisfy $P_{\ell }^{-m}=(-1{)}^m(\ell -m)!P_{\ell }^m/(\ell +m)!$.

The polynomial solution of degree l normalized by P_l(1)=1 is the Legendre polynomial P_l.

The Legendre polynomials are orthogonal on $[-1,1]$ with unit weight:

The three-term recurrenceturns multiplication by $x$ into coupling between adjacent Legendre multipoles.

For a contour enclosing $t$,

For fixed $0<\theta <\pi$,

For fixed $\xi >0$,