Asymptotic Expansions of Double and Multiple Integrals Occurring in Diffraction Theory

At present at I.N.S.T.N., Saclay and Faculté des Sciences, Paris, France Asymptotic expansions of double integrals of the type have been derived in terms of thereal parameter k by the method of stationary phase. The resultscan easily be extended to multi-dimensional integrals. In the first part of this paper a rigorous proof of the applicationof the method of stationary phase to double and multiple integralsis established with the aid of neutralizer or unitary functions.It is shown that the principal contributions to U(k) come fromsmall but otherwise arbitrary neighbourhoods of critical pointsof the integral, which may be located in the interior or onthe boundary of the domain of integration. These points areassociated with the phase or amplitude function. An explicitasymptotic series in the parameter k of the principal contributionis exhibited when the amplitude and the phase functions havein the neighbourhood of a critical point (x₁,y₁) a developmentof the form g(x,y) = (x–x₁)^0–1 (y–y₁)^µ0–1g₁(x,y), (x,y) = (x₁,y₁) + a _,0 (x–x₁ [1 + P(x,y) + b_0,(y–y₁[1+Q(x,y)]. The function g₁ is a regular function and P,Q can be developedin power series in the vicinity of the critical point and vanishat this point. The above expansion we shall call normal or canonicaland the critical point a normal or canonical critical pointof the integral. Although the assumption of the normal form expansion of theamplitude and phase functions is too restrictive for the generalcase, nevertheless it is found to be sufficiently broad to includemost of the important and interesting cases which occur in diffraction,scattering and other problems of mathematical physics. In Part II the principal contribution arising from a criticalpoint of normal type has been calculated in the form of a descendingpower series in the parameter k. It is shown, with the use ofmajorant functions, that the contribution due to the remainderpart of the series is of higher order in the parameter thanthat of the last term of the finite part, which proves the asymptoticcharacter of the series in the sense of Poincaré. Theresults derived here are in agreement with that of Part I. However,the new series has a decided advantage over that given in PartI if calculations are desired for even a few terms of the series,since the coefficients entering in the asymptotic expansionof the principal contribution are expressed directly in termsof the original functions g(x,y) and (x,y) and their derivatives,which is not the case in the formulas derived in Part I. In Part III explicit asymptotic expansions of the double integralare derived for several typical critical points associated withthe phase function. These are important in connection with thetheory of diffraction of optical instruments with large aberrationsand scattering problems. On account of their importance, eachcase has been treated in detail. In the appendices we have given an alternative proof of thetheorem announced in Part I and the derivation of the leadingterm due to a boundary stationary point. There will be foundalso a discussion of the more general integral where the parameterk appears implicitly in the phase function and not explicitlyas considered in the text. Integrals of this kind occur in manybranches of physics, especially when dealing with wave propagationin dispersive and absorbing media. Finally, we have concludedon the basis of our results that the Rubinowicz approach todiffraction and the stationary phase application to diffractionintegrals lead to similar mathematical results, although differentphysical interpretations, in diffraction phenomena, the formerleading to Young diffraction phenomena and the latter to Fresneldiffraction phenomena.