Legendre Polynomial: Understanding the Basics

[asdf1]

can someone explain step-by-step why the legendre polynomial came into being? I'm having one hard time understanding it...

 

[mathphys]

The Legendre Polynomias are related to the Legendre differential equation, and may be obtained by solving it using the Frobenius method.

http://mathworld.wolfram.com/LegendreDifferentialEquation.html

There are more ways to arrive to them also.

 

[Dr Transport]

If memory serves me correctly, you can generate the Legendre polynomials using Gram-Schmidt orthogonalization on the interval $$-1 \leq x \leq 1$$ starting with a constant 1 as $$L_0$$.

The generating function method is used to calculate moments of charge distributions.

You have to master them to get anywhere in physics., they pop up all the time.

 

[rachmaninoff]

If memory serves me correctly, you can generate the Legendre polynomials using Gram-Schmidt orthogonalization on the interval $$-1 \leq x \leq 1$$ starting with a constant 1 as $$L_0$$.

Yup, assuming you start with the linearly independent set $$\{ 1, \, x, \, x^2, \, x^3 \ldots \}$$.

 

[asdf1]

thank you very much! :)