Problem about Rodrigues' formula and Legendre polynomials

In summary, we used Rodrigues' formula to show that the integral of the product of two Legendre polynomials from -1 to 1 is equal to 2 divided by 2n+1. We also showed that the Legendre polynomial can be expressed as a function of the derivative of (x^2-1)^n. By substituting r=n-1, we can continue to simplify the expression.
  • #1
Another1
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using Rodrigues' formula show that \(\displaystyle \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{2}{2n+1}\)

\(\displaystyle {P}_{n}(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n\)

my thoughts

\(\displaystyle \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{1}{2^{2n}(n!)^2}\int_{-1}^{1} \,\frac{d^n}{dx^n}(x^2-1)^n\frac{d^n}{dx^n}(x^2-1)^ndx\)

let
\(\displaystyle u = \frac{d^n}{dx^n}(x^2-1)^n\) and \(\displaystyle du =\frac{d^{n+1}}{dx^{n+1}}(x^2-1)^ndx\)
\(\displaystyle dv = \frac{d^n}{dx^n}(x^2-1)^ndx\) and \(\displaystyle v = \frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n\)

so
\(\displaystyle uv=\frac{d^n}{dx^n}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n\)
\(\displaystyle -vdu=-\int_{-1}^{1} \,\frac{d^{n+1}}{dx^{n+1}}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^ndx\)

\(\displaystyle uv-vdu = \frac{d^n}{dx^n}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n-\int_{-1}^{1} \,\frac{d^{n+1}}{dx^{n+1}}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^ndx\)

what should I do?
 
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  • #2
Another said:
using Rodrigues' formula show that \(\displaystyle \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{2}{2n+1}\)

\(\displaystyle {P}_{n}(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n\)

my thoughts

\(\displaystyle \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{1}{2^{2n}(n!)^2}\int_{-1}^{1} \,\frac{d^n}{dx^n}(x^2-1)^n\frac{d^n}{dx^n}(x^2-1)^ndx\)

let
\(\displaystyle u = \frac{d^n}{dx^n}(x^2-1)^n\) and \(\displaystyle du =\frac{d^{n+1}}{dx^{n+1}}(x^2-1)^ndx\)
\(\displaystyle dv = \frac{d^n}{dx^n}(x^2-1)^ndx\) and \(\displaystyle v = \frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n\)

so
\(\displaystyle uv=\frac{d^n}{dx^n}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n\)
\(\displaystyle -vdu=-\int_{-1}^{1} \,\frac{d^{n+1}}{dx^{n+1}}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^ndx\)

\(\displaystyle uv-vdu = \frac{d^n}{dx^n}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n-\int_{-1}^{1} \,\frac{d^{n+1}}{dx^{n+1}}(x^2-1)^n\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^ndx\)

what should I do?

Hi Another, :)

Interesting question, thanks. Notice that,

$${P}_{n}(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n=\frac{(2x)n}{2^n n!}\frac{d^{n-1}}{dx^{n-1}}(x^2 - 1)^{n-1}$$

and so on and generally for a differentiation of $r$ times we get,

$${P}_{n}(x) = \frac{(2x)^{r}}{2^n (n-r)!}\frac{d^{n-r}}{dx^{n-r}}(x^2 - 1)^{n-r}$$

No substitute $r=n-1$ and see what you get. I am sure you'll be able to continue from there :)
 

FAQ: Problem about Rodrigues' formula and Legendre polynomials

What is Rodrigues' formula and how is it used?

Rodrigues' formula is a mathematical formula used for expressing Legendre polynomials. It states that the nth order Legendre polynomial can be written as a function of its derivative and a constant. This formula is useful for simplifying calculations involving Legendre polynomials.

What are Legendre polynomials and what are they used for?

Legendre polynomials are a set of orthogonal polynomials that are commonly used in mathematics and physics. They have many applications, including solving differential equations, in numerical analysis, and in quantum mechanics.

How do you derive Rodrigues' formula?

To derive Rodrigues' formula, you start with the generating function for Legendre polynomials and use mathematical techniques such as the chain rule and integration by parts. This results in an expression for the nth order Legendre polynomial in terms of its derivative and a constant, which is known as Rodrigues' formula.

What are the limitations of Rodrigues' formula?

While Rodrigues' formula is a useful tool for simplifying calculations involving Legendre polynomials, it does have some limitations. It is only valid for real values of the variable x and for polynomials of finite order. Also, it can be difficult to apply in some cases, such as when the derivative of the polynomial is not readily available.

Are there any alternative methods for expressing Legendre polynomials?

Yes, besides Rodrigues' formula, there are several other methods for expressing Legendre polynomials, such as through recurrence relations, generating functions, and the Gram-Schmidt process. Each method has its own advantages and limitations, and the choice of which one to use depends on the specific problem at hand.

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