Legendre functions have an important property that is, they are orthogonal on the interval −1 ≤ *x* ≤ 1:

(where δ_{mn} 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

PhD researcher at Friedrich-Schiller University Jena, Germany. I'm a physicist specializing in theoretical, computational and experimental condensed matter physics. I like to develop Physics related apps and softwares from time to time. Can code in most of the popular languages. Like to share my knowledge in Physics and applications using this Blog and a YouTube channel.

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