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Delta Function

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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 S or the space of all smooth functions of compact support D) of test functions f. The action of delta on f, commonly denoted delta[f] or <delta,f>, then gives the value at 0 of f for any function f. 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)
(1)

(Bracewell 1999, p. 94).

The delta function has the fundamental property that

int_(-infty)^inftyf(x)delta(x-a)dx=f(a)
(2)

and, in fact,

int_(a-epsilon)^(a+epsilon)f(x)delta(x-a)dx=f(a)
(3)

for epsilon>0.

Additional identities include

delta(x-a)=0
(4)

for x!=a, as well as

delta(ax)=1/(|a|)delta(x)
(5)
delta(x^2-a^2)=1/(2|a|)[delta(x+a)+delta(x-a)]
(6)

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

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

where the x_is are the roots of g. For example, examine

delta(x^2+x-2)=delta[(x-1)(x+2)].
(8)

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

delta(x^2+x-2)=1/3delta(x-1)+1/3delta(x+2).
(9)

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

intf(x)delta^((n))(x)dx=-int(partialf)/(partialx)delta^((n-1))(x)dx.
(10)

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

intxg(x)delta^'(x)dx=-intdelta(x)partial/(partialx)[xg(x)]dx
(11)
=-intdelta(x)[g(x)+xg^'(x)]dx
(12)
=-intg(x)delta(x)dx,
(13)

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

xdelta^'(x)=-delta(x).
(14)

In general, the same procedure gives

int[x^nf(x)]delta^((n))(x)dx=(-1)^nint(partial^n[x^nf(x)])/(partialx^n)delta(x)dx,
(15)

but since any power of x 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

int[x^nf(x)]delta^((n))(x)dx=(-1)^nn!intf(x)delta(x)dx,
(16)

which implies

x^ndelta^((n))(x)=(-1)^nn!delta(x).
(17)

Other identities involving the derivative of the delta function include

delta^'(-x)=-delta^'(x)
(18)
int_(-infty)^inftyf(x)delta^'(x-a)dx=-f^'(a)
(19)
(delta^'*f)(a)=int_(-infty)^inftydelta^'(a-x)f(x)dx=f^'(a)
(20)

where * denotes convolution,

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

and

x^2delta^'(x)=0.
(22)

An integral identity involving delta(1/x) is given by

int_(-1)^1delta(1/x)dx=0.
(23)

The delta function also obeys the so-called sifting property

intf(x)delta(x-x_0)dx=f(x_0)
(24)

(Bracewell 1999, pp. 74-75).

A Fourier series expansion of delta(x-a) gives

a_n=1/piint_(-pi)^pidelta(x-a)cos(nx)dx
(25)
=1/picos(na)
(26)
b_n=1/piint_(-pi)^pidelta(x-a)sin(nx)dx
(27)
=1/pisin(na),
(28)

so

delta(x-a)=1/(2pi)+1/pisum_(n=1)^(infty)[cos(na)cos(nx)+sin(na)sin(nx)]
(29)
=1/(2pi)+1/pisum_(n=1)^(infty)cos[n(x-a)].
(30)

The delta function is given as a Fourier transform as

delta(x)=F_k[1](x)=int_(-infty)^inftye^(-2piikx)dk.
(31)

Similarly,

F_x^(-1)[delta(x)](k)=int_(-infty)^inftydelta(x)e^(2piikx)dx=1
(32)

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

F_x[delta(x-x_0)](k)=int_(-infty)^inftye^(-2piikx)delta(x-x_0)dx=e^(-2piikx_0).
(33)
DeltaFunctionEpsilon

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

delta(x)=1/pilim_(epsilon->0)epsilon/(x^2+epsilon^2),
(34)
=lim_(epsilon->0)1/2epsilon|x|^(epsilon-1)
(35)
=lim_(epsilon->0^+)1/(2sqrt(piepsilon))e^(-x^2/(4epsilon))
(36)
=lim_(epsilon->0)1/(pix)sin(x/epsilon)
(37)
=lim_(epsilon->0)1/epsilonAi(x/epsilon)
(38)
=lim_(epsilon->0)1/epsilonJ_(1/epsilon)((x+1)/epsilon)
(39)
=lim_(epsilon->0)|1/epsilone^(-x^2/epsilon)L_n((2x)/epsilon)|,
(40)

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

delta(x)=lim_(n->infty)1/(2pi)(sin[(n+1/2)x])/(sin(1/2x)).
(41)

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

delta^2(x,y)={0 x^2+y^2!=0; infty x^2+y^2=0,
(42)
int_(-infty)^inftyint_(-infty)^inftydelta^2(x,y)dxdy=1
(43)
delta^2(ax,by)=1/(|ab|)delta^2(x,y),
(44)

and

delta^2(x,y)=delta(x)delta(y).
(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)
int_(-infty)^inftyint_(-infty)^inftyint_(-infty)^inftydelta^3(x,y,z)dxdydz=1
(48)

and

delta^3(x,y,z)=delta(x)delta(y)delta(z).
(49)

in cylindrical coordinates (r,theta,z),

delta^3(r,theta,z)=(delta(r)delta(z))/(pir).
(50)

In spherical coordinates (r,theta,phi),

delta^3(r,theta,phi)=(delta(r))/(2pir^2)
(51)

(Bracewell 1999, p. 85).

A series expansion in cylindrical coordinates gives

delta^3(r_1-r_2)=1/(r_1)delta(r_1-r_2)delta(theta_1-theta_2)delta(z_1-z_2)
(52)
=1/(r_1)delta(r_1-r_2)1/(2pi)sum_(m=-infty)^(infty)e^(im(theta_1-theta_2))1/(2pi)int_(-infty)^inftye^(ik(z_1-z_2))dk.
(53)

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

x(1-x)y^('')+(4-6x)y^'-6y=0
(54)

has classical solution

y(x)=(C_1)/(x^3)+(x^2-x-1+2(x-2)ln(x-1))/(x^3(x-1))C_2,
(55)

and distributional solution

y(x)=C_1delta^('')(x)
(56)

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

See also

Delta Sequence, Doublet Function, Fourier Transform--Delta Function, Generalized Function, Impulse Symbol, Poincaré-Bertrand Theorem, Shah Function, Sokhotsky's Formula Explore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/GeneralizedFunctions/DiracDelta/, http://functions.wolfram.com/GeneralizedFunctions/DiracDelta2/

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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.

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Delta Function

Cite this as:

Weisstein, Eric W. "Delta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeltaFunction.html

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