Mathematical Methods for Physics and Engineering - Matematica.NET

13.1 FOURIER TRANSFORMSf(y)1−a−a − b −a + ba − baa + bxFigure 13.3The aperture function f(y) for two wide slits.After some manipulation we obtain4cosqa sin qb˜f(q) =q √ .2πNow applying (13.10), and remembering that q =(2π sin θ)/λ, we findI(θ) = 16 cos2 qa sin 2 qb,q 2 r2 0where r 0 is the distance from the centre of the aperture. ◭13.1.3 The Dirac δ-functionBefore going on to consider further properties of Fourier transforms we make adigression to discuss the Dirac δ-function and its relation to Fourier transforms.The δ-function is different from most functions encountered in the physicalsciences but we will see that a rigorous mathematical definition exists; the utilityof the δ-function will be demonstrated throughout the remainder of this chapter.It can be visualised as a very sharp narrow pulse (in space, time, density, etc.)which produces an integrated effect having a definite magnitude. The formalproperties of the δ-function may be summarised as follows.The Dirac δ-function has the property thatδ(t) =0 fort ≠0, (13.11)but its fundamental defining property is∫f(t)δ(t − a) dt = f(a), (13.12)provided the range of integration includes the point t = a; otherwise the integral439