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Mathematics for Physics

238 Sifting Property

A property of an integral that picks out the value of a function at one point that is used to simplify equations that involve a concentrated source.

definition [d] (Sifting Property = Sifting Formula) The defining property of the Dirac delta function \(\delta\): integrating it against a continuous function \(f\) sifts out the value of \(f\) at the point where the delta is centered:

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

where

  • \(\delta\) is the Dirac delta function.
  • \(f\) is a continuous test function.
  • \(a \in \mathbb{R}\) is the point at which \(\delta\) is centered.
  • \(x\) is the variable of integration.

Note:

  • the identity holds provided the interval of integration contains \(a\); otherwise the integral is \(0\).

238.1 Elementary Example

238.1.1 Simple

The sifting property picks out \(f(a)\) from an integral against \(\delta(x-a)\).

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

\[ a = 1,\quad f(x) = 3x,\quad \text{result } = 3 \]

where

  • \(\delta\) is the Dirac delta.
  • \(f\) is a continuous test function.
  • \(a\) is the center point.

238.1.2 General

If the integration interval does not contain \(a\), the integral is \(0\). If it does, the value is \(f(a)\).

\[ \int_{0}^{2} x^{2}\, \delta(x - 1)\, dx = 1 \]

\[ \int_{3}^{5} x^{2}\, \delta(x - 1)\, dx = 0 \]

where

  • the first interval contains \(a = 1\).
  • the second interval does not contain \(a = 1\).

238.2 References

  1. Kreyszig, E. Advanced Engineering Mathematics, 10th ed. Wiley, 2011. — sifting property and sifting formula.
  2. Griffel, D. H. Applied Functional Analysis. Ellis Horwood, 1981. — Example 1.24, sifting property.
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — defining property.