Delta Function

The delta function is a generalized function that can be defined as the limit of a class of delta sequences. The delta function is sometimes called "Dirac's delta function" or the "impulse symbol" (Bracewell 1999). It is implemented in the Wolfram Language as DiracDelta[x].

Formally, is a linear functional from a space (commonly taken as a Schwartz space or the space of all smooth functions of compact support ) of test functions . The action of on , commonly denoted or , then gives the value at 0 of for any function . In engineering contexts, the functional nature of the delta function is often suppressed.

The delta function can be viewed as the derivative of the Heaviside step function,

(Bracewell 1999, p. 94).

The delta function has the fundamental property that

and, in fact,

for .

Additional identities include

for , as well as

More generally, the delta function of a function of is given by

where the s are the roots of . For example, examine

Then , so and , giving

The fundamental equation that defines derivatives of the delta function is

Letting in this definition, it follows that

where the second term can be dropped since , so (13) implies

In general, the same procedure gives

but since any power of times integrates to 0, it follows that only the constant term contributes. Therefore, all terms multiplied by derivatives of vanish, leaving , so

which implies

Other identities involving the derivative of the delta function include

where denotes convolution,

and

An integral identity involving is given by

The delta function also obeys the so-called sifting property

(Bracewell 1999, pp. 74-75).

A Fourier series expansion of gives

so

The delta function is given as a Fourier transform as

Similarly,

(Bracewell 1999, p. 95). More generally, the Fourier transform of the delta function is

The delta function can be defined as the following limits as ,

where is an Airy function, is a Bessel function of the first kind, and is a Laguerre polynomial of arbitrary positive integer order.

The delta function can also be defined by the limit as

Delta functions can also be defined in two dimensions, so that in two-dimensional Cartesian coordinates

and

Similarly, in polar coordinates,

(Bracewell 1999, p. 85).

In three-dimensional Cartesian coordinates

and

in cylindrical coordinates ,

In spherical coordinates ,

(Bracewell 1999, p. 85).

A series expansion in cylindrical coordinates gives

The solution to some ordinary differential equations can be given in terms of derivatives of (Kanwal 1998). For example, the differential equation

has classical solution

and distributional solution

(M. Trott, pers. comm., Jan. 19, 2006). Note that unlike classical solutions, a distributional solution to an th-order ODE need not contain independent constants.