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 2000). 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 test function . In engineering contexts, the functional nature of the delta function is often suppressed.

The limits of delta sequences are understood in the sense of generalized functions. Their integrals against each smooth function of compact support converge to . Pointwise convergence of the defining functions is not required.

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

(Bracewell 2000, 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 2000, pp. 74-75).

For , a Fourier series expansion of on gives

so

On the entire real line, this Fourier series represents the periodic impulse train , a shifted and rescaled Shah function.

With the negative-exponent convention for the Fourier transform, the delta function is given by

The corresponding inverse Fourier transform gives

(Bracewell 2000, p. 95). More generally, the Fourier transform of the delta function is

The delta function has the following representations as limits of generalized functions as ,

where is an Airy function, is a Bessel function of the first kind, and is a Laguerre polynomial of fixed polynomial degree , with a positive integer. The Bessel function of the first kind representation applies on . The Laguerre polynomial representation requires a normalization depending on (R. Scholz, pers. comm., Feb. 9, 2017). The normalization constant is

For , substituting shows that the Laguerre polynomial approximant has unit integral and converges to the delta function as .

On , the delta function is also the distributional limit of a normalized Dirichlet kernel as

On the entire real line, this limit is the periodic impulse train , a rescaled Shah function.

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

and

Similarly, in polar coordinates,

(Bracewell 2000, p. 89).

In three-dimensional Cartesian coordinates

and

in cylindrical coordinates ,

In spherical coordinates ,

(Bracewell 2000, p. 90).

These formulas in polar coordinates, cylindrical coordinates, and spherical coordinates use and interpret through symmetric delta sequences. In this convention,

The factors above give unit normalization when combined with the corresponding volume element. Alternatively, define a unit-normalized radial delta function by

The equivalent formulas are then

The radial normalization convention must therefore be specified when comparing these formulas (Gsponer 2007).

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 ordinary 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.