Asymptotic Expansion of Laplace-Fourier-Type Integrals

2021/04/20 by Konrad, Sara, Bartelmann, Matthias
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2104.10028

Abstract

We study the asymptotic behaviour of integrals of the Laplace-Fourier type P(k) = ∫_Ωe^-|k|sf(x)e^{i kx}d x , with k∈ℝ^d in d≥1 dimensions, with Ω⊂ℝ^d and sufficiently well-behaved functions f:Ω→ℝ. Our main result is P(k) ∼ \frace^-|k|sf(0)|k|^sd/2√(((2π)^d)/(det A)) exp(-\frack^\top A^-1k2|k|^s) for |k|→∞, where A is the Hessian matrix of the function f at its critical point, assumed to be at x₀ = 0. In one dimension, the Hessian is replaced by the second derivative, A = f''(0). We also show that the integration domain Ω can be extended to ℝ^d without changing the asymptotic behaviour.

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