Legendre polynomials

The Legendre polynomials, sometimes called Legendre functions of the first kind, are defined as solutions of Legendre's differential equation:

Solutions corresponding to the non-negative integer n can be expressed using Rodrigues' formula

or can be constructed using the three term recurrence relation:

Unit description

The recurrence relation given above is the most efficient way to calculate the Legendre polynomial. The LegendreCalculate subroutine uses this relation to calculate P_n (x) for any given x.

The LegendreSum subroutine calculates the sum of Legendre polynomials c₀ P₀ (x) + c₁ P₁ (x) + ... + c_n P_n (x) using Clenshaw's recurrence formula.

The LegendreCoefficients subroutine can represent P_n (x) as a sum of powers of x: c₀  + c₁ x + ... + c_n x ⁿ.

This article is intended for personal use only.

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