Dirac delta function
Motivation and overview

The graph of the Dirac delta is usually thought of as following the whole x -axis and the positive y -axis. The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar abstractions such as a point charge, point mass or electron point. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).

To be specific, suppose that a billiard ball is at rest. At time t = 0 {\displaystyle t=0} it is struck by another ball, imparting it with a momentum P, with units kg ⋅ m ⋅ s. The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The force therefore is P δ (t); the units of δ (t) are s.

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval Δ t = {\displaystyle \Delta t=}. That is,

F Δ t (t) = { P / Δ t 0 < t ≤ T, 0 otherwise. {\displaystyle F_{\Delta t}(t)={\begin{cases}P/\Delta t&0<t\leq T,\\0&{\text{otherwise}}.\end{cases}}}

Then the momentum at any time t is found by integration:

p (t) = ∫ 0 t F Δ t (τ) d τ = { P t ≥ T P t / Δ t 0 ≤ t ≤ T 0 otherwise. {\displaystyle p(t)=\int _{0}^{t}F_{\Delta t}(\tau)\,d\tau ={\begin{cases}P&t\geq T\\P\,t/\Delta t&0\leq t\leq T\\0&{\text{otherwise.}}\end{cases}}}

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δ t → 0, giving a result everywhere except at 0:

p (t) = { P t > 0 0 t < 0. {\displaystyle p(t)={\begin{cases}P&t>0\\0&t<0.\end{cases}}}

Here the functions F Δ t {\displaystyle F_{\Delta t}} are thought of as useful approximations to the idea of instantaneous transfer of momentum.

The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of pointwise convergence) lim Δ t → 0 + F Δ t {\textstyle \lim _{\Delta t\to 0^{+}}F_{\Delta t}} is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property

∫ − ∞ ∞ F Δ t (t) d t = P, {\displaystyle \int _{-\infty }^{\infty }F_{\Delta t}(t)\,dt=P,}

which holds for all Δ t > 0 {\displaystyle \Delta t>0}, should continue to hold in the limit. So, in the equation F (t) = P δ (t) = lim Δ t → 0 F Δ t (t) {\textstyle F(t)=P\,\delta (t)=\lim _{\Delta t\to 0}F_{\Delta t}(t)}, it is understood that the limit is always taken outside the integral.

In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers. For example, the objects f (x) = δ (x) and g (x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions.
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