Delta-function

$\delta$-function, Dirac delta-function, $\delta(x)$

A function which makes it possible to describe the spatial density of a physical magnitude (mass, charge, intensity of a heat source, force, etc.) which is concentrated or applied at a point $a$ of a space $\mathbf R^n$. For instance, using the delta-function the density of a point mass $m$ located at a point $a$ is written as $m\delta(x-a)$. The delta-function may be formally defined by the relation

\[\int\limits_{\mathbf R^n}\delta(x-a)f(x)\,dx=f(a)\]

for any continuous function $f$. The derivatives $\delta^{(k)}$ of the delta-function may be defined in a similar manner:

\[\int\limits_{\mathbf R^n}\delta^k(x-a)f(x)\,dx=(-1)^kf^{(k)}(a)\]

for the class of functions $f$ that are continuous in $\mathbf R^n$ with derivatives $f^{(k)}$ up to the order $k$ inclusive. The formal operator relations, which are frequently employed, and which express the following properties of the delta-function:

\[\delta(-x)=\delta(x);\qquad\delta(cx)=|c|^{-1}\delta(x),\quad c=\mathrm{const},\]

\[x\delta(x)=0;\qquad\delta(x)+x\delta'(x)=0,\]

etc., should be understood in the sense of the above definitions, i.e. these relations become meaningful only after having been integrated against sufficiently smooth functions. Thus, the delta-function is not an ordinary function in the sense of the classical theory of functions, and is defined in the theory of generalized functions as a singular generalized function, i.e. as the continuous linear functional in the space of infinitely-differentiable functions $f$ of compact support, assigning to $f$ its value at zero: $(\delta,f)=f(0)$.

Comments

The Dirac delta-function is the derivative (in the sense of distributions or generalized functions) of the Heaviside function (Heaviside distribution) $h$, defined by $h(x)=0$ for $x<0$, $h(x)=1$ for $x>0$ (the value at zero does not matter; as usual for a distribution it suffices for it to be defined apart from a set of measure zero).

References

[a1]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1966) MR0209834 Zbl 0149.09501
[a2]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1. Properties and operations , Acad. Press (1964) (Translated from Russian) MR435831 Zbl 0115.33101