Expansions of Gaussian Integrals Sample Clauses

Expansions of Gaussian Integrals. ‌ We recall that our final goal is to find the asymptotic behaviour for τ large of τn ∫ eiτΦ(z)aτ (z)dz. (4.16) Now we discuss the special case where Φ is a quadratic form, e.g. Φ(z) = ⟨Qz, z⟩/2 for some real symmetric non-singular matrix Q. In this case (4.16) is a Gaussian integral. Using a technique similar to the one used in the standard proof of the stationary phase formula, we are able to obtain an expansion for (4.16). All we need for finding the desired expression is Xxxxxxxx’s formula and the fact that the Fourier transform of Gaussian function is again a Gaussian function. ρ,δ
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