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A very weird improper integral on ^n

  1. Jun 22, 2013 #1
    1. The problem statement, all variables and given/known data
    Verify that [itex]\int_{ℝ^n}exp(-\frac{λ}{2} \langle Ax, x \rangle-i \langle x,ζ \rangle )dx=(\frac{2\pi}{λ})^{\frac{1}{2}}(detA)^{-\frac{1}{2}}exp(-\frac{1}{2λ} \langle A^{-1}ζ, ζ \rangle )[/itex] where A is a symmetric matrix of complex numbers and <ReA x, x> is positive definite, and λ is a positive constant. ζ is a vector in ℝ^n


    2. Relevant equations
    Fubini's Theorem?


    3. The attempt at a solution
    The question is a lot easier if A is brought to diagonal form, so it is reasonable to make a change of variable that x= C y where C belongs to SO(n) such that C^-1 A C = B is diagonal. Since this change of variables means only geometrically a rotation of the R^n plane it should not change the range of values for integrating(still from -∞ to ∞). After this transformation we should be able to apply Fubini's Theorem and perform an iterated integration. But in the exponential function [itex]exp(-i\langle Cy, ζ \rangle)[/itex] is still left to be dealt with and it doesn't come any where close that it could be of the form [itex]exp(-\frac{1}{2λ} \langle A^{-1}ζ, ζ \rangle)[/itex] after integration as the answer suggests.. right now I'm trully stuck here.. Thanks for any help in advance!
     
  2. jcsd
  3. Jun 22, 2013 #2
    det A here means [itex]|det A| exp(i \sum_0^n{arg w_i})[/itex] where w is the eigenvalue of A.
    This question even takes 20 minutes to type.. or I really suck at latex..
     
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