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Linear Differential Operator/Inner Product

  1. May 19, 2009 #1
    Can anyone help me?

    Q. An inner product is defined on the vector space V of all (real) polynomials (of arbitrary
    degree) by

    <f,g>=Int(x*e^(-x)*f(x)*g(x),x=o..infinity

    Show that the linear differential operator

    L=-x*(d^2/dx^2)-(2-x)*(d/dx):V->V

    is symmetric with respect to this inner product.


    Don't necessarily need the exact answer, just need to know how I'm meant to go about it.
     
    Last edited: May 19, 2009
  2. jcsd
  3. May 19, 2009 #2

    HallsofIvy

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    Let f and g be a polynomials. Show that <Lf, g>= <f, Lg>.
     
  4. May 19, 2009 #3
    Ah right, thanks.

    So all I need to do is replace the f(x) in the integral with Lf(x) (and the same with g(x) on the other side)? I'll give it a go anyway. Cheers.
     
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