Help regarding Legendre Rodrigue's formula problem.

1. Jan 30, 2010

yungman

1. The problem statement, all variables and given/known data

Question:
Use Rodrigues' formula and integral by parts to show:

$$\int^1 _{-1}f(x)P_n (x)dx=\frac{(-1)^n}{2^n n!}\int^1_{-1}f^{(n)}(x)(x^2 -1)^n dx$$

(As a convention $$f^{(0)}(x)=f(x)$$

2. Relevant equations

Rodrigues' Formula: $$P_n(x)=\frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n$$

Bonnet's recurrence relation:

$$(n+1)P_{n+1}(x)+nP_{n-1}(x)=(2n+1)xP_n(x)$$

$$P'_{n+1}(x)=P'_{n-1}(x)+(2n+1)P_n(x)$$

3. The attempt at a solution

I tried
1)$$\int^1 _{-1}f(x)P_n (x)dx=\frac{1}{2^n n!}\int^1_{-1}f(x)\frac{d^n}{dx^n}(x^2 -1)^n dx$$

Then use $$U=f(x), dV=\frac{d^n}{dx^n}(x^2 -1)^n dx$$

That did not go far.

2) $$P_n(x)=\frac{1}{2n+1}[P'_{n+1}(x)-P'_{n-1}(x)]$$

substitude into $$\int^1 _{-1}f(x)P_n (x)dx$$

That went no where.

What is $$f^{(n)}(x)$$? is it $$\frac{d^n}{dx^n}f(x)$$?

2. Jan 30, 2010

Pengwuino

What happened when you did integration by parts as in your first attempt? That should immediately pop out the answer with n-integration by parts and you just need to deal with the term you receive from integrating by parts with the limits.

And yes, $$f^{(n)}(x)=\frac{d^n}{dx^n}f(x)$$

Last edited: Jan 30, 2010
3. Jan 30, 2010

yungman

I did not attempt to perform integrate by parts after the first time because for one, n can be any number and the answer I got after the first time don't seems to be any simplier than the original equation. Also something tell me that there is a trickier way to proof than to integrate it out.

Do you mean there is no short cut but to really do it n times? I would go through it if that is the only way.
Thanks.

4. Jan 30, 2010

Pengwuino

Yes, n can be any number, but if you do it n-times, you know all that happens is the f(x) acquires n-derivatives by the fact that the Legendre polynomials explicitly have $$\frac{d}{dx}$$ in them. For example, if you simply had

$$\int f(x)\frac{d^n}{dx^n}g(x)dx$$

doing integration by parts n-times is simple and it is exactly what you're doing here. You're literally pulling n-derivatives off g(x) and putting them onto f(x). Now the actual work here is that m-applications of integration by parts creates m-terms of

$$[f^{(m)} (x)\frac{{dg^{n - m} (x)}}{{dx^{n - m} }}]_{x = - 1}^{x = 1}$$

that need to be evaluated. At this point this is where the recursion relationships should come in handy. Do one integration by parts and look at the constant term and see if it goes away or not.

And actually, after giving it some thought, you may not even need the recursion relations.

Last edited: Jan 30, 2010
5. Jan 30, 2010

yungman

Thanks, I'll try it tomorrow. My grandson is running a fever and he is being difficult!!!

6. Jan 31, 2010

yungman

I want to verify:

$$\int \frac{d^n}{dx^n}(x^2-1)^n dx=\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^{n-1} +C_1$$

$$\int \frac{d^{n-1}}{dx^{n-1}}(x^2-1)^{n-1}dx=\frac{d^{n-2}}{dx^{n-2}}(x^2-1)^{n-2}+C_2$$

Last edited: Jan 31, 2010
7. Jan 31, 2010

Pengwuino

No, those statements are not true. What do you know about $$\int \frac{d}{dx}g(x) dx$$?

8. Jan 31, 2010

yungman

It equal to g(x)+C. I did not write the C out. But that don't apply here.

I worked it out, this is my answer:

1) $$\int^1 _{-1}f(x)P_n (x)dx=\frac{1}{2^n n!}\int^1_{-1}f(x)\frac{d^n}{dx^n}(x^2 -1)^n dx$$

Then use $$U=f(x), dV=\frac{d^n}{dx^n}(x^2 -1)^n dx$$

$$\int^1 _{-1}f(x)P_n (x)dx = \frac{1}{2^n n!}[f(x)\frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n]_{x=-1}^{x=1} - \frac{1}{2^n n!}\int^1_{-1} f^{(1)}(x) \frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n dx$$

$$[f(x)\frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n]_{x=-1}^{x=1} =[f(x)2x\frac{d^{n-2}}{dx^{n-2}}(x^2 -1)^{n-1}]_{x=-1}^{x=1} =0$$

because of the x inside and equal zero when substitude x=1 and subtract x=-1. Therefore:

$$\int^1 _{-1}f(x)P_n (x)dx = - \frac{1}{2^n n!}\int^1_{-1} f^{(1)}(x) \frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n dx$$

$$= -\frac{1}{2^n n!}[f^{(1)}(x)\frac{d^{n-2}}{dx^{n-2}}(x^2 -1)^n]_{x=-1}^{x=1} + \frac{1}{2^n n!}\int^1_{-1} f^{(2)}(x) \frac{d^{n-2}}{dx^{n-2}}(x^2 -1)^n dx$$

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

$$= \frac{1}{2^n n!}[f^{(2)}(x)\frac{d^{n-3}}{dx^{n-3}}(x^2 -1)^n]_{x=-1}^{x=1} - \frac{1}{2^n n!}\int^1_{-1} f^{(3)}(x) \frac{d^{n-3}}{dx^{n-3}}(x^2 -1)^n dx$$

$$= - \frac{1}{2^n n!}\int^1_{-1} f^{(3)}(x) \frac{d^{n-3}}{dx^{n-3}}(x^2 -1)^n dx$$

The step repeat until:

$$= (-1)^{n-1}\frac{1}{2^n n!}[f^{(n)}(x)(x^2 -1)^n]_{x=-1}^{x=1} + (-1)^n \frac{1}{2^n n!}\int^1_{-1}f^{(n)}(x)(x^2 -1)^n dx$$

In this case $$[(x^2 -1)^n]_{x=-1}^{x=1}=0$$ Therefore

$$\int^1 _{-1}f(x)P_n (x)dx=\frac{(-1)^n}{2^n n!}\int^1_{-1}f^{(n)}(x)(x^2 -1)^n dx$$

But it does not work if f(x) is a lower order polynomial than n!!!

Last edited: Jan 31, 2010
9. Jan 31, 2010

Pengwuino

There are no integration constants when you are using definite integrals. Also, in the terms that go like

$$\frac{1}{2^n n!}[f(x)\frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n]_{x=-1}^{x=1}$$

You can't simply plug in x=-1,1. You have to determine how that derivative acts first. Now that might be a problem.... I think this is where the recurrence relationships will come in handy.

10. Jan 31, 2010

yungman

Actually I thought about the constant and I change the post already as you wrote this!!! We crossed.

I'll look at what you say and work it out in a little bit.

Thanks

Last edited: Jan 31, 2010
11. Jan 31, 2010

yungman

Yes I see what happen:

$$[f(x)\frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n]_{x=-1}^{x=1} =[f(x)2x\frac{d^{n-2}}{dx^{n-2}}(x^2 -1)^{n-1}]_{x=-1}^{x=1} =0$$

because of the x inside function equal zero when substitude x=1 and subtract the same function with x=-1.

I updated the last post. Can you take a look whether I got it or not. I still have question on the last line about if F(x) is a polynomial with power less than n. That still would not work.