Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.1 FOURIER TRANSFORMS

The 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 aδ-

function may be analysed as such, if they take place in an interval much shorter

than the response interval of the system on which they act. For example, the

idealised notion of an impulse of magnitudeJapplied at timet 0 can be represented

by

j(t)=Jδ(t−t 0 ). (13.20)

Many physical situations are described by aδ-function in space rather than in

time. Moreover, we often require theδ-function to be defined in more than one

dimension. For example, the charge density of a point chargeqat a pointr 0 may

be expressed as a three-dimensionalδ-function

ρ(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 volumeV

is given by

∫

V

ρ(r)dV=

∫

V

qδ(r−r 0 )dV=

{

q ifr 0 lies inV,

0otherwise.

Closely related to the Diracδ-function is theHeavisideorunit step function

H(t), for which

H(t)=

{

1fort> 0 ,

0fort< 0.

(13.22)

This function is clearly discontinuous att= 0 and it is usual to takeH(0) = 1/2.

The Heaviside function is related to the delta function by

H′(t)=δ(t). (13.23)