Hi guys,

I would like to ask for some help regarding a problem with Legendre polynomials:

P_n(x) is an arbitrary Legendre polynomial (0<=n and -1<x<1). Prove the following

Thank you in advance!

Printable View

- October 15th 2012, 08:52 AMJohan13Absolute value of Legendre polynomials ??
Hi guys,

I would like to ask for some help regarding a problem with Legendre polynomials:

P_n(x) is an arbitrary Legendre polynomial (0<=n and -1<x<1). Prove the following

Thank you in advance! - October 17th 2012, 02:08 AMVlasevRe: Absolute value of Legendre polynomials ??
What have you tried so far? Have you tried induction?

- October 18th 2012, 01:58 PMJohan13Re: Absolute value of Legendre polynomials ??
Well, I don't think that induction is the right choice, because my professor said that I should start from the Rodrigues-formula (Rodrigues' formula - Wikipedia, the free encyclopedia) and than use some complex Cauchy-integrals. However, my efforts weren't successful yet.

So, I hope someone can give me some ideas how to approach this problem. - October 19th 2012, 12:33 AMVlasevRe: Absolute value of Legendre polynomials ??
Alright, let's see. Here is what would have happened if you tried induction specifically. Suppose that up to and including some , for some function . We have the Bonnet recursion formula

which you can derive from the other formula. Now

all obtained by the triangle inequality and the bounds on . So,

So, technically, you can do an induction but the function you bound with will depend on . I'll take a look at this complex integrals approach and let you know. - October 19th 2012, 12:57 AMVlasevRe: Absolute value of Legendre polynomials ??
Rearrange the formula for to get

The last one we can obtain by the -th Cauchy integral formula

where is a circle with radius, say , centered at the origin. So, we have obtained an integral formula for the Legendre polynomials

Now it might just be an exercise in estimating this integral - October 22nd 2012, 02:18 PMJohan13Re: Absolute value of Legendre polynomials ??
I really appreciate your help Vlasev.

Unfortunately, I think that I am stuck again. I tried to estimate a absolute value of the integral, but what I get that isn't close to the original upper limit. I tried using the fact the absolute value of an integral is smaller than the integral of an absolute value.

The results somehow seems to depend on the value of , however I don't understand why should it. Could you please give me some more help?

Many thanks - October 23rd 2012, 01:31 AMVlasevRe: Absolute value of Legendre polynomials ??
Are you sure that your bound is correct, because in actuality, you can even divide by n or something inside the square root and it'll still be good.