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
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
Other identities involving the derivative of the delta function include
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
The delta function is given as a Fourier transform as
(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
(Bracewell 1999, p. 85).
(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.
More things to try:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 481-485, 1985. Bracewell, R. "The Impulse Symbol." Ch. 5 in The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 74-104, 2000. Dirac, P. A. M. Quantum Mechanics, 4th ed. London: Oxford University Press, 1958. Gasiorowicz, S. Quantum Physics. New York: Wiley, pp. 491-494, 1974. Kanwal, R. P. "Applications to Ordinary Differential Equations." Ch. 6 in Generalized Functions, Theory and Technique, 2nd ed. Boston, MA: Birkhäuser, pp. 291-255, 1998. Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 97-98, 1984. Spanier, J. and Oldham, K. B. "The Dirac Delta Function ." Ch. 10 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 79-82, 1987. van der Pol, B. and Bremmer, H. Operational Calculus Based on the Two-Sided Laplace Integral. Cambridge, England: Cambridge University Press, 1955.
