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

Steven G. Johnson

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

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arxiv 1508.04376 v1 pith:UTXHQQZR submitted 2015-08-18 math.CV

Steven G. Johnson This is my paper

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keywords bumpintegrationasymptoticfourierfunctioninftysaddle-pointanalysis

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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.

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