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,
|
(1) |
(Bracewell 1999, p. 94).
The delta function has the fundamental property that
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(2) |
and, in fact,
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(3) |
for .
Additional identities include
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(4) |
for , as well as
More generally, the delta function of a function of is given by
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(7) |
where the s are the roots of . For example, examine
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(8) |
Then , so and , giving
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(9) |
The fundamental equation that defines derivatives of the delta function is
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(10) |
Letting in this definition, it follows that
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(11) | |||
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(12) | |||
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(13) |
where the second term can be dropped since , so (13) implies
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(14) |
In general, the same procedure gives
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(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
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(16) |
which implies
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(17) |
Other identities involving the derivative of the delta function include
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(18) |
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(19) |
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(20) |
where denotes convolution,
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(21) |
and
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(22) |
An integral identity involving is given by
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(23) |
The delta function also obeys the so-called sifting property
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(24) |
(Bracewell 1999, pp. 74-75).
A Fourier series expansion of gives
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(25) | |||
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(26) | |||
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(27) | |||
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(28) |
so
The delta function is given as a Fourier transform as
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(31) |
Similarly,
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(32) |
(Bracewell 1999, p. 95). More generally, the Fourier transform of the delta function is
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(33) |
The delta function can be defined as the following limits as ,
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(34) | |||
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(35) | |||
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(36) | |||
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(37) | |||
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(38) | |||
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(39) | |||
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(40) |
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
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(41) |
Delta functions can also be defined in two dimensions, so that in two-dimensional Cartesian coordinates
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(42) |
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(43) |
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(44) |
and
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(45) |
Similarly, in polar coordinates,
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(46) |
(Bracewell 1999, p. 85).
In three-dimensional Cartesian coordinates
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(47) |
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(48) |
and
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(49) |
|
(50) |
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(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
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(54) |
has classical solution
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(55) |
and distributional solution
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(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.
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/
Explore with Wolfram|Alpha
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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Delta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DeltaFunction.html