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Mathematics for Physics

223 Delta Function

A mapping that is zero away from one point and has a finite integral concentrated there that is used to represent a point source in an integral.

definition [d] (Dirac Delta Function = Delta Function = Delta Distribution = Generalized Function) A generalized function \(\delta(x)\) characterized informally by

  • \(\delta(x) = 0\) for \(x \neq 0\) .
  • \(\displaystyle \int_{-\infty}^{\infty} \delta(x)\, dx = 1\) .

and, rigorously, by the sifting property

  • \(\displaystyle \int_{-\infty}^{\infty} f(x)\, \delta(x - a)\, dx = f(a)\) .

where

  • \(f\) is a continuous test function; \(a \in \mathbb{R}\).
  • \(\delta\) is a distribution, not an ordinary function.

223.1 Elementary Example

223.1.1 Simple

The delta distribution is concentrated at one point. Informally, it vanishes off \(0\) and has total integral \(1\).

\[ \delta(x) = 0\ \text{for } x \neq 0 \]

\[ \int_{-\infty}^{\infty} \delta(x)\, dx = 1 \]

where

  • \(\delta\) is the Dirac delta.
  • \(x\) is the real variable.

223.1.2 General

Against a continuous test function, delta sifts out the value at the center point \(a\).

\[ \int_{-\infty}^{\infty} f(x)\, \delta(x - a)\, dx = f(a) \]

\[ a = 2,\quad f(x) = x^{2},\quad \int f(x)\, \delta(x-2)\, dx = 4 \]

where

  • \(f\) is a continuous test function.
  • \(a\) is the point where \(\delta\) is centered.

223.2 References

  1. Griffel, D. H. Applied Functional Analysis. Ellis Horwood, 1981. — Example 1.24 (sifting property; \(\delta\) as a distribution).
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013.
  3. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006.