What are Legendre Polynomials and why are they used so often?

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What are Legendre Polynomials and why are they used so often?

In: Mathematics

Legendre polynomials are an infinite number of pretty cool polynomials P_n with nice properties.

One way to define them is recursively : P_0(x) = 1, P_1(x) = x, and then P_ {n+1}(x) = ((2n+1)x*P_n(x) – n*P_{n-1}(x))/(n+1) for n > 0.

Among their properties for every n :

* P_n(x) is a polynomial of degree n
* P_n(1) = 1, and P_n(-1) = (-1)^n (i.e. 1 or -1 depending on the parity of n)
* With x in [-1,1], P_n(x) is in [-1,1]
* All the x such that P_n(x) = 0 (i.e. the “zeroes” of P_n) are in [-1,1], and there are n of them. They also are rather “evenly” spread in [-1,1]
* P_n(-x) = (-1)^n * P_n(x), so each polynomial is “symmetrical” relative to 0 (either the axis or the point, depending on parity)
* the integral on [-1,1] of P_n(x)*P_m(x) dx = 0 for any m != n