We consider, in this work, the asymptotic behaviour for large Ξ», of a Fourier integral
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00002467/resource/name/S0008414X00002467_eqn1.gif?pub-status=live)
where π(x) is in general a Cβ function and a(x) a Cβ function with compact support. It is well known that the asymptotic behaviour of this integral is controlled by the behaviour of π at its critical points (i.e., points where ππ/πxj(x) = 0) and is given by local contributions at these points ([1], [3], [7], [9]).
In general, one assumes the hypothesis of non degenerate isolated critical point, namely that the determinant of the second derivative at the critical point is non zero.