On a relation between two-dimensional Fourier integrals and series of Hankel transforms

Abstract

In the theore ti cal solution rece ntly obtained for stationary spati al-cohere nce functions over radiating apertures, [I]' the evaluation of the two-dime nsional Fourie r integral of the far-fi eld inte nsity di s tribution is required. Since the appearance of s uc h integrals is also quite common in other areas of mathe matical physics, it would be useful to render their evaluation amenable to the application of ex tensively tabulated result s available in the literature. (For examples of suc h sources see [2] , [4] , [5], and [6].) For fun ctions of one variable, comprehe nsive tables of Fourier tran sforms exis t [2], and although it is possible to reduce the k dimensional Fourier tran sform of radial functions 2 [3] to Hankel transforms [3 , p. 69] for which extensive tables [5] are available , there are no tables giving the Fourier transform for k > 1 of arbitrary functions (i. e., nonradi al) even in the case of k = 2. In this paper, the two-dimensional Fourier tran sform is reduced to a form which facilitates its evaluation by th e use of existing tables [4 , 5] and also yields a result which is an extension of that given in Bochner and Chandrasekharan [3], for the case k=2, to functions which are not necessarily radial. It will be shown that if g(a, (3 ) is the two-dimensional Fourier transform of fix, y), i.e.,