Delta Function (original) (raw)

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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, delta 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 delta on , commonly denoted delta[f] or <delta,f> , 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 derivativeof the Heaviside step function,

	(1)

(Bracewell 1999, p. 94).

The delta function has the fundamental property that

	(2)

and, in fact,

	(3)

for epsilon>0 .

Additional identities include

	(4)

for x!=a , as well as

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

| $delta[g(x)]=sum_(i)(delta(x-x_i))/(\|g^'(x_i)|),$ | (7) | | ---------------------------------------------------------------------------------------------------------------------------------------- | --- |

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

	(8)

Then g^'(x)=2x+1 , so g^'(x_1)=g^'(1)=3 and g^'(x_2)=g^'(-2)=-3 , giving

	(9)

The fundamental equation that defines derivatives of the delta function delta(x) is

	(10)

Letting f(x)=xg(x) in this definition, it follows that

where the second term can be dropped since intxg^'(x)delta(x)dx=0 , so (13) implies

	(14)

In general, the same procedure gives

	(15)

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

	(16)

which implies

	(17)

Other identities involving the derivative of the delta function include

	(18)

	(19)

	(20)

where denotes convolution,

| $int_(-infty)^infty\|delta^'(x)|dx=infty,$ | (21) | | --------------------------------------------------------------------------------------------------------------------------------- | ---- |

and

	(22)

An integral identity involving delta(1/x) 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 delta(x-a) gives

The delta function is given as a Fourier transformas

	(31)

Similarly,

	(32)

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

	(33)

DeltaFunctionEpsilon

The delta function can be defined as the following limits as epsilon->0 ,

where Ai(x) is an Airy function, J_n(x) is a Bessel function of the first kind, and L_n(x) is a Laguerre polynomial of arbitrary positive integer order.

DeltaFunctionN

The delta function can also be defined by the limit as n->infty

	(41)

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

$delta^2(x,y)={0 x^2+y^2!=0; infty x^2+y^2=0,$	(42)

	(43)

| $delta^2(ax,by)=1/(\|ab|)delta^2(x,y),$ | (44) | | ------------------------------------------------------------------------------------------------------------------------------ | ---- |

and

	(45)

Similarly, in polar coordinates,

| $delta^2(x,y)=(delta(r))/(pi\|r|)$ | (46) | | ------------------------------------------------------------------------------------------------------------------------- | ---- |

(Bracewell 1999, p. 85).

In three-dimensional Cartesian coordinates

$delta^3(x,y,z)=delta^3(x)={0 x^2+y^2+z^2!=0; infty x^2+y^2+z^2=0$	(47)

	(48)

and

	(49)

in cylindrical coordinates (r,theta,z) ,

	(50)

In spherical coordinates (r,theta,phi) ,

	(51)

(Bracewell 1999, p. 85).

A series expansion in cylindrical coordinatesgives

The solution to some ordinary differential equations can be given in terms of derivatives of delta(x) (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.

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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 delta(x-a) ." 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.