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Integral of a complex exponential

  1. Nov 21, 2008 #1
    1. The problem statement, all variables and given/known data

    let be [tex] A_{i,j} [/tex] a Hermitian Matrix with only real values then

    [tex] \int_{V} dV e^{iA_{j,k}x^{j}x^{k}}= \delta (DetA) (2\pi)^{n} [/tex] (1)

    2. Relevant equations

    [tex] \int_{V} dV e^{iA_{j,k}x^{j}x^{k}} = \delta (DetA) (2\pi)^{n} [/tex]

    3. The attempt at a solution

    the idea is that the integral (1) is divergent when the Matrix A is not invertible Det=A and the Dirac delta is not defined at x=0 ,for example if detA=0 then at least one of the eigenvalues is 0 so the exponential takes the value 1 and the integral is divergent
     
  2. jcsd
  3. Nov 21, 2008 #2

    Avodyne

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    Science Advisor

    Try working in the basis in which A is diagonal.
     
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