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6.30.1  Legendre polynomials: legendre

The Legendre polynomial L(n,x) of degree n is a polynomial solution of the differential equation

(x2−1) y″−2 x y′−n(n+1) y=0

The Legendre polynomials satisfy the recurrence relation:

     
  L(0,x)=1         
L(1,x)=x         
L(n,x)
=
2n−1
n
x L(n−1,x)−
n−1
n
L(n−2,x)
         

These polynomials are orthogonal for the scalar product:

<f,g>=
+1


−1
f(x)g(xdx 

The legendre command finds the Legendre polynomials.


Examples.


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