# Dictionary:Dirac function

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(di rak’) A delta function or impulse (q.v.). Named for Paul Adrien Maurice Dirac (1902–1984), British physicist.

## Properties of the Dirac delta "function"

The original desired properties of the Dirac delta function is unit measure

${\displaystyle \int _{-\infty }^{\infty }\delta (t)\;dt=1}$

and the sifting property

${\displaystyle \int _{-\infty }^{\infty }f(t)\delta (t)\;dt=f(0).}$

The support, (which is to say, the part of the domain where the function is nonzero), of the Dirac delta function is ${\displaystyle t=0}$, so the limits of integration may be reduced to a neighborhood of ${\displaystyle t=0.}$

## Delta sequences

The integral of a function that has only a single non-zero value is zero. This means that the Dirac delta is not a function in the classical sense (which is to say a rule that maps points of the real number line into a set of real numbers.)[1]

Instead we may think of the Dirac function as being the limit of a sequence of increasingly strongly peaked functions that exhibit the sifting property, have unit measure, and turn on only in an interval of decreasing length as the index ${\displaystyle n}$ increases.

We thus, consider the sequence ${\displaystyle S_{n}(t),}$ called delta sequences or generalized functions such that

${\displaystyle \int _{-\infty }^{\infty }S_{n}(t)\;dt=1}$

and which exhibits the sifting property in the limit

${\displaystyle \lim _{n\rightarrow \infty }\int _{-\infty }^{\infty }f(t)S_{n}(t)\;dt=f(0).}$

The limit in the above expression is not the proper limit of a sequence of functions, but rather defines a limit of a sequence of integrals. There are many possible examples of delta sequence functions. There is no equivalence property in terms of integrability (beyond the existence of the defining integral) or differentiability that will distinguish these functions, and the delta sequence functions need not have a limit the point of support of the delta function.

The delta sequence notion is the basis of the notion of generalized functions. Thus, objects like the Dirac delta are not functions, but may be represented as the limit of a sequence of smooth functions.

## The Schwartz theory of Distributions

In 1950, French mathematician Laurent Schwartz published his Theory of Distributions, which describes a general class of objects under the equivalence class of forming an inner product with members of a class of smooth functions called test functions.[2] [3] The test functions ${\displaystyle \phi (t)}$ are infinitely differentiable, and vanish ${\displaystyle C^{\infty }}$ smoothly as ${\displaystyle |t|\rightarrow \infty }$, meaning that the functions and all of their derivatives vanish at infinity, but are nonzero only on some finite domain. The distributions are defined as being the set of mathematical objects, which when multiplied by a test function, may be integrated on infinite limits to produce a finite result. All of the regular functions, the Dirac delta function, and all of its derivatives are thus defined as being members of the Schwartz class of distributions.

## Derivatives of the delta function

Formally we can consider defining "delta prime" the delta function derivative. We could consider formally defining the delta function derivative by forming the derivative as a limit of the difference of delta sequence functions, yet because not all delta sequence functions need be differentiable at the place where the delta function would act, and because we would be mixing questionable limiting processes, this approach, is, at best a heuristic.

A better approach is to use the Schwartz notion of test functions. We consider formally the action of an hypothetical delta function derivative ${\displaystyle \delta ^{\prime }(t)}$ as the action of the integration with a test function ${\displaystyle \phi }$

${\displaystyle \int _{-\infty }^{\infty }\delta ^{\prime }(t)\phi (t)\;dt}$

where ${\displaystyle \phi (t)\in C_{0}^{\infty }(\mathbf {R} )}$ meaning that ${\displaystyle \phi (t)}$ is continuous, infinitely differentiable, and vanishes at infinity.

Applying integration by parts, we obtain

${\displaystyle \int _{-\infty }^{\infty }\delta ^{\prime }(t)\phi (t)\;dt={\biggl .}\phi (t)\delta (t){\biggr |}_{-\infty }^{\infty }-\int _{-\infty }^{\infty }\delta (t)\phi (t)\;dt=-\phi ^{\prime }(t).}$

noting that ${\displaystyle \delta (t)}$ vanishes at ${\displaystyle \pm \infty }$.

This result may be extended to the ${\displaystyle m}$-derivative

${\displaystyle \int _{-\infty }^{\infty }\delta ^{(m)}(t)\phi (t)\;dt=(-1)^{m}\phi ^{(m)}(t).}$

## References

1. Lighthill, M. J. (1962). Fourier analysis and generalised functions. Cambridge University Press, p.46.
2. Schwartz, Laurent. "Theory des distributions." Hermann, Paris (1950).
3. Friedlander, F. G., & Joshi, M. S. (1998). Introduction to the Theory of Distributions. Cambridge University Press.