Mathematical Methods for Physics and Engineering - Matematica.NET

13.1 FOURIER TRANSFORMSThe derivative of the delta function, δ ′ (t), is defined by∫ ∞−∞[ ] ∞∫ ∞f(t)δ ′ (t) dt = f(t)δ(t) − f ′ (t)δ(t) dt−∞ −∞= −f ′ (0), (13.19)and similarly for higher derivatives.For many practical purposes, effects that are not strictly described by a δ-function may be analysed as such, if they take place in an interval much shorterthan the response interval of the system on which they act. For example, theidealised notion of an impulse of magnitude J applied at time t 0 can be representedbyj(t) =Jδ(t − t 0 ). (13.20)Many physical situations are described by a δ-function in space rather than intime. Moreover, we often require the δ-function to be defined in more than onedimension. For example, the charge density of a point charge q at a point r 0 maybe expressed as a three-dimensional δ-functionρ(r) =qδ(r − r 0 )=qδ(x − x 0 )δ(y − y 0 )δ(z − z 0 ), (13.21)so that a discrete ‘quantum’ is expressed as if it were a continuous distribution.From (13.21) we see that (as expected) the total charge enclosed in a volume Vis given by∫∫{q if r0 lies in V,ρ(r) dV = qδ(r − r 0 ) dV =VV0 otherwise.Closely related to the Dirac δ-function is the Heaviside or unit step functionH(t), for which{1 for t>0,H(t) =0 for t