# (p.225) Appendix 2 THE DIRAC DELTA FUNCTION

# (p.225) Appendix 2 THE DIRAC DELTA FUNCTION

The Dirac delta function is not a function in the usual sense of being defined for values of its argument, but is a generalized function in that it is defined by a limiting procedure and only has meaning within an integral. It is often written and manipulated without reference to an integral but it should be remembered that results so obtained are strictly prescriptions for manipulating integrals containing delta functions.

The delta function may be defined as the limit as of a normalized function of *x* and which is symmetric about its main or only peak at *x* = 0. Such functions tend to zero in the limit for all *non-zero* values of *x*, and hence, since the function is normalized to unit area, the value at *x* = 0 must tend to infinity. Examples of suitable functions are , . A delta function peaked at *x* = *a* is simply obtained by replacing *x* by *x* − *a* in the function and then taking the limit . The integral of the delta function is, for *p* < *q*,

*H*(

*x*–

*a*) can then be defined using (A2.1) as

*f*(

*x*) is a function for which

*f*(

*a*) is defined then

*f*(

*x*) is itself a generalized function then (A2.3) is meaningful if

*f*(

*a*) is meaningful. For example,

*a*or

*b*. Comparing (A2.1) with (A2.3), we see that

The Fourier transform of can be calculated using the sifting property (A2.3) as

The extension of the definition of the delta function into two or more dimensions is achieved by considering the product of individual delta functions, one for each dimension. For example, if and are two vectors then we define the three-dimensional delta function as

*α*-plane.

Consider the integral

*f*(

*x*) for which the first integral exists, then in the limit the right-hand side of (A2.14) becomes , using (A2.3). This gives rise to the symbol which we call the first derivative of the delta function, given by the limit with the property

*(*

^{m}δ*x*)/d

*x*, with the properties

^{m}*g*(0) exists. Further, using the sifting property (A2.3) with

*f*(x) = −

*g*(

*x*), we have