# A very weird improper integral on ^n

1. Jun 22, 2013

### raopeng

1. The problem statement, all variables and given/known data
Verify that $\int_{ℝ^n}exp(-\frac{λ}{2} \langle Ax, x \rangle-i \langle x,ζ \rangle )dx=(\frac{2\pi}{λ})^{\frac{1}{2}}(detA)^{-\frac{1}{2}}exp(-\frac{1}{2λ} \langle A^{-1}ζ, ζ \rangle )$ 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 $exp(-i\langle Cy, ζ \rangle)$ is still left to be dealt with and it doesn't come any where close that it could be of the form $exp(-\frac{1}{2λ} \langle A^{-1}ζ, ζ \rangle)$ after integration as the answer suggests.. right now I'm trully stuck here.. Thanks for any help in advance!

2. Jun 22, 2013

### raopeng

det A here means $|det A| exp(i \sum_0^n{arg w_i})$ where w is the eigenvalue of A.
This question even takes 20 minutes to type.. or I really suck at latex..