Dear All,

I would like to know if the following integral converges, and if so, what is the easiest way to prove it?

where is the physicists' Hermite polynomial:

.

Thank you.

Regards.

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- May 16th 2012, 02:11 PMBabouConvergence of an integral involving Hermite polynomials
Dear All,

I would like to know if the following integral converges, and if so, what is the easiest way to prove it?

where is the physicists' Hermite polynomial:

.

Thank you.

Regards. - May 17th 2012, 01:04 AMgirdavRe: Convergence of an integral involving Hermite polynomials
If is a polynomial of degree , the integral is convergent. To see that, write for .

- May 17th 2012, 10:34 AMBabouRe: Convergence of an integral involving Hermite polynomials
Dear girdav,

Thanks a lot for your reply. I appreciate it.

Would you mind detailing your suggestion that uses for ? Indeed, the latter uses positive but is negative, and it seems to me easier to prove that the integral in question is finite by using rather than .

Regards. - May 17th 2012, 10:55 AMgirdavRe: Convergence of an integral involving Hermite polynomials
You get an upper bound after taking the inverse.

- May 17th 2012, 11:16 AMBabouRe: Convergence of an integral involving Hermite polynomials
Thanks girdav. That makes sense.