Saddle-point integration of $C_{\infty }$ "bump" functions

Abstract

This technical note describes the application of saddle-point integration to the asymptotic Fourier analysis of the well-known $C_{\infty }$ "bump" function $\exp [-(1-x^2{)}^{-1}]$, deriving both the asymptotic decay rate $k^{-3/4}\exp (-\sqrt{k})$ of the Fourier transform $F(k)$ and the exact coefficient. The result is checked against brute-force numerical integration and is extended to generalizations of this bump function.

6-page technical report

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