There are many problems in mathematics and mathematical physics whose solutions exist in the form of a Fourier integral, an integral having the form Iw= abgx eiwfxdx; Most often one is interested in such integrals when the parameter w approaches a limiting value, say w→infinity . Some methods have been found to approximate Iw by developing a series expansion. One particularly useful method is the so-called method of stationary phase, a technique that may be used to develop an asymptotic expansion. In this paper, two asymptotic expansions are derived for Iw for the case where the region of integration is (a, b) = ( -infinity,infinity ). For the first expansion, Iw has the form Iw= -infinityinfinitygx eiwax2+bx+c dx,a0,w large where g(x) represents a distribution belonging to the space of Schwartz' tempered distributions, S'(R). For the second expansion, Iw has the form Iw= infinityinfinityxse iwfxdx, s≥3,wlarge where the integral is taken in the sense of distributions and f(x) is a function whose derivative has the form f'x=k x-rx+r ,k0,r≠0; Each expansion is derived using the properties and theories of distributions along with those of Fourier transform analysis. Also being presented are examples, results, and observations concerning the geometric conditions placed upon an asymptotic expansion of an n-dimensional Laplace integral having the form Jl= 0infinity&ldots;0 infinityxn11 &ldots;xnnne-l P0+Pn+P0,n dx1&ldots;dxn where, P0=a0xu0,1 1xu0,22&ldots;xu 0,nn;Pn=a nxun,11xun,2 2xun,nn ; P0,n=i=1n-1 aixui,11xu i,22&ldots;xui,n n and ui,j satisfy certain geometric conditions.
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