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Math Help - Convergence of an integral involving Hermite polynomials

  1. #1
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    Convergence 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?

     I_n = \int_{- \infty}^{+ \infty} e^{- x^2 / 2} H_n(x) dx, \forall \ n \in \mathbb{N}^{+},

    where  $H_n(x)$ is the physicists' Hermite polynomial:

     H_n(x) = (-1)^n e^{x^2} \frac{d^n e^{-x^2}}{d x^n} .

    Thank you.

    Regards.
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  2. #2
    Super Member girdav's Avatar
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    Re: Convergence of an integral involving Hermite polynomials

    If P is a polynomial of degree d, the integral \int_{-\infty}^{+\infty}e^{-x^2/2}P(x)dx is convergent. To see that, write e^t\geq \frac{t^{d+2}}{(d+2)!} for t\geq 0.
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  3. #3
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    Re: 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  e^{t} \geq \frac{t^{d+2}}{(d+2)!} for  t \geq 0 ? Indeed, the latter uses  t positive but  -x^2/2 is negative, and it seems to me easier to prove that the integral in question is finite by using  \leq rather than  \geq .

    Regards.
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  4. #4
    Super Member girdav's Avatar
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    Re: Convergence of an integral involving Hermite polynomials

    You get an upper bound after taking the inverse.
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  5. #5
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    Re: Convergence of an integral involving Hermite polynomials

    Thanks girdav. That makes sense.
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