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 ![delta[f]](http://mathworld.wolfram.com/images/equations/DeltaFunction/Inline7.gif)
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,
![d/(dx)[H(x)]=delta(x)](http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation1.gif) |
(1)
|
(Bracewell 1999, p. 94).
The delta function has the fundamental property that
 |
(2)
|
and, in fact,
 |
(3)
|
for 
.
Additional identities include
 |
(4)
|
for 
, as well as
More generally, the delta function of a function of 
is given by
 |
(7)
|
where the 
s are the roots of 
. For example, examine
![delta(x^2+x-2)=delta[(x-1)(x+2)].](http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation6.gif) |
(8)
|
Then 
, so 
and 
, giving
 |
(9)
|
The fundamental equation that defines derivatives of the delta function 
is
 |
(10)
|
Letting 
in this definition, it follows that
where the second term can be dropped since 
, so (13) implies
 |
(14)
|
In general, the same procedure gives
 |
(15)
|
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
 |
(16)
|
which implies
 |
(17)
|
Other identities involving the derivative of the delta function include
 |
(18)
|
 |
(19)
|
 |
(20)
|
where 
denotes convolution,
 |
(21)
|
and
 |
(22)
|
An integral identity involving 
is given by
 |
(23)
|
The delta function also obeys the so-called sifting property
 |
(24)
|
(Bracewell 1999, pp. 74-75).
A Fourier series expansion of 
gives
so
The delta function is given as a Fourier transform as
=int_(-infty)^inftye^(-2piikx)dk.](http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation20.gif) |
(31)
|
Similarly,
=int_(-infty)^inftydelta(x)e^(2piikx)dx=1](http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation21.gif) |
(32)
|
(Bracewell 1999, p. 95). More generally, the Fourier transform of the delta function is
=int_(-infty)^inftye^(-2piikx)delta(x-x_0)dx=e^(-2piikx_0).](http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation22.gif) |
(33)
|
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(x)=lim_(n->infty)1/(2pi)(sin[(n+1/2)x])/(sin(1/2x)).](http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation23.gif) |
(41)
|
Delta functions can also be defined in two dimensions, so that in two-dimensional Cartesian coordinates
![delta^2(x,y)=<span style=]() {0 x^2+y^2!=0; infty x^2+y^2=0, " src="http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation24.gif" style="height:46px; width:171px" /> |
(42)
|
 |
(43)
|
 |
(44)
|
and
 |
(45)
|
Similarly, in polar coordinates,
 |
(46)
|
(Bracewell 1999, p. 85).
In three-dimensional Cartesian coordinates
![delta^3(x,y,z)=delta^3(x)=<span style=]() {0 x^2+y^2+z^2!=0; infty x^2+y^2+z^2=0 " src="http://mathworld.wolfram.com/images/equations/DeltaFunction/NumberedEquation29.gif" style="height:46px; width:256px" /> |
(47)
|
 |
(48)
|
and
 |
(49)
|
in cylindrical coordinates 
,
 |
(50)
|
In spherical coordinates 
,
 |
(51)
|
(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
 |
(54)
|
has classical solution
 |
(55)
|
and distributional solution
 |
(56)
|
(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.
REFERENCES:
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.
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