# What is the Dirac Delta Function 1

## delta function

**δ function**d function, *Dirac's δ function*, δ-*distribution* an expression of the form δ (*x* – *x*_{0}), the one to the point *x* = *x*_{0} concentrated distribution. From a mathematical point of view, the δ-function is a *distribution*.

"To get a picture of δ (x), consider a function of the real variable x that vanishes everywhere except in a small area, say length *ε*, around the origin x = 0 and which is so large within the range that its integral over this range is equal to 1. How the function looks exactly within the area is unimportant, provided it does not take on unnecessarily nonsensical forms (for example it would always be in order *ε*^{-}^{ 1}). Then this function goes for *ε*

0 into the function δ (x). "(From: P.A.M. Dirac," The Principles of Quantum Mechanics ", 4th edition, Oxford 1958, p. 58.)

The use of the δ-function can be extended to *Kirchhoff* Traced back to 1882, but is mostly used with *Dirac* who introduced them to quantum mechanics. The δ-function is expediently imagined as an infinitely high and infinitely narrow tip, the area of which is 1. The most important property of the δ-function for practical computing is that it assigns its function value at one point to any "reasonable" function under an integral:

.

These and some other calculation rules and formulas are compiled in Table 1.

With the help of the δ-function, the particle image can be taken over into field theory: a point charge of strength *q* in point **r**_{0} owns the charge distribution *ρ*(* r*) =

*q*δ (

*–*

**r**

**r**_{0}). The potential of such a point charge results from the solution of

*Poisson's equation*

to

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