Associated Legendre polynomials

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

Associated Legendre polynomials are a family of functions that arise when solving problems with spherical symmetry, such as Laplace’s equation on a sphere. They come from the Legendre equation with an additional "order" parameter m.

What they are
- Denoted P_l^m(x), where l is the degree (a nonnegative integer) and m is the order (an integer with 0 ≤ |m| ≤ l).
- They solve the associated Legendre differential equation:
(1 − x^2) y'' − 2x y' + [l(l+1) − m^2/(1 − x^2)] y = 0.
- When m = 0, they reduce to the ordinary Legendre polynomials P_l(x).

How they’re defined
- A convenient standard definition for nonnegative m is:
P_l^m(x) = (−1)^m (1 − x^2)^{m/2} d^m/dx^m P_l(x),
where P_l(x) are the ordinary Legendre polynomials (Rodrigues’ formula).
- The Condon–Shortley phase (the (−1)^m factor) is a common convention, though some authors omit it.
- For negative m, the negative-order functions are proportional to the positive ones:
P_l^{−m}(x) = (−1)^m (l − m)!/(l + m)! P_l^m(x).

A few quick facts
- For |m| > l, P_l^m = 0.
- Parity: P_l^m(−x) = (−1)^{l−m} P_l^m(x).
- Recurrence: a useful relation is
(l − m + 1) P_{l+1}^m(x) = (2l + 1) x P_l^m(x) − (l + m) P_{l−1}^m(x).

Orthogonality
- For fixed m, the functions P_k^m(x) are orthogonal on [−1, 1]:
∫_{−1}^1 P_k^m(x) P_l^m(x) dx = [2 (l + m)!] / [(2l + 1)(l − m)!] δ_{kℓ}.
- For fixed l, the m-values lead to an orthogonality with a weight:
∫_{−1}^1 P_l^m(x) P_l^n(x) / (1 − x^2) dx equals 0 if m ≠ n; and a finite value when m = n ≠ 0 (infinite for m = n = 0).

From Legendre polynomials to spherical harmonics
- When solving Laplace’s equation on a sphere, the angular part can be written with spherical harmonics:
Y_l^m(θ, φ) = sqrt((2l + 1)(l − m)! / (4π (l + m)!)) P_l^m(cos θ) e^{imφ},
for −l ≤ m ≤ l.
- These functions form a complete, orthonormal set on the sphere and are central to describing angular dependence in many physical problems.

Expressing in θ
- If you rewrite x as cos θ, the associated Legendre functions become P_l^m(cos θ). They can be expressed in terms of powers of sin θ and cos θ, with simple examples:
- P_0^0(cos θ) = 1
- P_1^0(cos θ) = cos θ
- P_1^1(cos θ) = − sin θ
- P_2^0(cos θ) = (1/2)(3 cos^2 θ − 1)
- P_2^1(cos θ) = −3 cos θ sin θ
- P_2^2(cos θ) = 3 sin^2 θ

Generalizations and related functions
- The same differential equation can be extended to non-integer or complex l and m, leading to Legendre functions P_λ^μ(z) and Q_λ^μ(z), which are more general solutions.
- In physics, these functions are intimately tied to the symmetry group SO(3) and to the angular part of problems solved in spherical coordinates.

In short
- Associated Legendre polynomials extend Legendre polynomials to include angular dependence encoded by m.
- They are defined via derivatives of P_l and a phase factor, obey a specific differential equation, have useful recurrence relations, and form the basis of spherical harmonics used in many physical applications.