Legendre functions have an important property that is, they are orthogonal on the interval −1 ≤ x ≤ 1:
(where δmn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise).
We can verify this result using Scilab.
To work with Legendre Polynomials we use the Scilab function legendre(n,m,x).
Which basically returns the value of the Associated Legendre Polynomial for a given value of m,n and x.
However, since I only wanted Legendre Polynomials so I’ll have to put m=0.
The following code returns the value of the integral, ∫Pm(x)*Pn(x)dx,
//Legendre Polynomials Orthogonality Verification
clc;
n=input("Enter n:");
m=input("Enter m:");
a=integrate('legendre(m,0,x)*legendre(n,0,x)','x',-1,1,0.001);
mprintf('∫P%i(x)*P%f(x)dx= %g\n',m,n,a);
Output:
Enter n:1
Enter m:1
∫P1(x)*P1.000000(x)dx= 0.666667
Enter n:5
Enter m:2
∫P2(x)*P5.000000(x)dx= 0
Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.