Associated Legendre Function Calculator

Enter degree n, order m, and x to evaluate the associated Legendre function Pₙᵐ(x) via stable recurrence. A building block of spherical harmonics.

Input

Enter the degree n, the order m (0≤m≤n), and the argument x (-1≤x≤1) to evaluate the associated Legendre function Pₙᵐ(x) by recurrence.

Degree n

Integer, 0 or greater

Order m

Integer from 0 to n

Argument x

From -1 to 1

P3^1(x)

-0.3247595264

x = 0.5

Degree n

Order m

Curve Pₙᵐ(x) for degree 3, order 1 (-1 ≤ x ≤ 1)

Values by order at degree 3

This tool evaluates the associated Legendre function Pₙᵐ(x) from degree n (integer 0 or greater), order m (integer with 0≤m≤n), and argument x (-1≤x≤1).
It uses the standard recurrence. It first builds Pₘᵐ(x) from the double factorial and (1-x²)^(m/2), then steps up one degree and applies the three-term recurrence up to the target degree.
The phase follows the Condon-Shortley convention (including the factor (-1)ᵐ). For m≥1 the value is 0 at the endpoints x=±1.
Pₙᵐ(x) is a building block of the spherical harmonics Yₙᵐ(θ,φ), carrying the latitudinal (θ) dependence with x=cosθ.
When m=0 it reduces to the ordinary Legendre polynomial Pₙ(x).

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