-- ----- JOURNAL OF RESEARCH of the Nati ona l Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 69B, No. 3, July-September 1965 On a Relation Between Two-Dimensional Fourier Integrals and Series of Hankel Transforms j. V. Cornacchio* and R. P. Soni* (June I, 1965) Proce dur es ar e develop ed for expr essing two·dimensional Fouri er tr ansforms in te rm s of tabu· lated one·d im ensional transforms. In the th eore ti cal solution r ece ntly obtained for s tationar y spa ti a l- cohere nce functions over radiatin g apertures, [I]' the evaluation of the two-dimensional Fouri er int egral of the fa r-fi eld intensity dis tributi on is r eq uir ed. Sin ce the appearan ce of such int egrals is also quite co mmon in other areas of math emati cal ph ysi cs, it would be useful to re nd er th e ir evaluation amenable to the application of extensively tabulat ed r es ults available in the lit era tur e. (For exa mpl es of such so ur ces see [2] , [4] , [5], and [6] .) For fun ct ions of one variable, compre hensive tables of Fourier tran s- forms exist [2], a nd although it is possible to reduce the k dimensio nal Fourier tran sform of radial func- tions 2 [3] to Hankel tran sforms [3 , p. 69] for which extensive tabl es [5] are available, the re ar e no tabl es giving th e Fouri er transform for k > 1 of arbitrary function s (i. e., nonr a di al) eve n in th e case of k = 2. In thi s pap er, the two-dimensional Fourier tran sform is re du ce d to a form which facilitates its evaluati on by th e u se of exist ing tabl es [4 , 5] and also yiel ds a result which is an exte nsion of that giv en in Bochner and Chandras e kharan [3], for the case k=2, to fun c- tions which are not necessarily radial. It will be shown that if g(a, (3 ) is the two-dime nsional Fouri er tran sfo rm of fix, y), i.e., 00 g(a , (3) = f f I(x , y)e i ( crx+ {3 Y) dxdy (1) then, if I is sufficiently well be haved , it is po ss ible to express g(a, (3) in the following form: where , j, m {g; Z} = Jo '" Vzt g(t) J m(zt)dt; (3) An are ce rtain functions associated with fix, y), and a,.(a, (3) are fun ctions indepe nd e nt of f It may be not ed that . j, rn {g; z} is the Hankel transfor m of order m of g( t). Th e derivation of (2) pro cee ds by rewriting (1) in th e form f "" f 27T g(a, (3) = ,'=0 o= of(r, e) ei(crrcos 9+{3rsi n O) rdrde. (4 ) Assuming that fir, e) can be represented by th e fol- lowing Fouri er se ri es '" f(r,e)= L In(r)e- inO , n=-oo where 1 f27T In(r) = 27T 0 fir, e) e inO de, (5) and that the interchange of summation and integra- tion is valid, (4) can be written as g(a, (3) = i f oo In(r)rdr f27T ei(-nO+ar cos O +{3r sin 0) de . n= - oo 0 0 (6) g(a, (3 ) = f an(a, (3 );, n {An; Va 2 + f3 2} n=O (2) However, using the well-known integral repre se ntat ion for In(Z) , the Bessel function of order n [6, p. 14, eq (2)], it can be shown that *1B1\1 Sys te ms Develo pm e nt Div isio n , Endi cott , N. Y. I Fi gures in br ac kets indicate th e lit e rature references at the end of thi s paper. 2fixl, X2, . ..• xd is a radial function if it can be writte n in the form j{X l, X2, ... • . .. where F(I. ) is defined for a ll l O. 173 f 2 7T o ei(-nO+ crr cos O+ {3r sin 0) de = 27Tein tan- 1 (a / {3) In(Va 2 + (32 r). (7)