Dirac delta function

In mathematics, the Dirac delta function (δ function) is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function. δ [ φ ] = φ ( 0 ) {displaystyle delta =varphi (0)}     (1) δ ( x ) = δ ( x 1 ) δ ( x 2 ) ⋯ δ ( x n ) . {displaystyle delta (mathbf {x} )=delta (x_{1})delta (x_{2})cdots delta (x_{n}).}     (2) δ x 0 [ φ ] = φ ( x 0 ) {displaystyle delta _{x_{0}}=varphi (x_{0})}     (3) δ ( α x ) = δ ( x ) | α | . {displaystyle delta (alpha x)={frac {delta (x)}{|alpha |}}.}     (4) lim ε → 0 + ∫ − ∞ ∞ η ε ( x ) f ( x ) d x = f ( 0 ) {displaystyle lim _{varepsilon o 0^{+}}int _{-infty }^{infty }eta _{varepsilon }(x)f(x),dx=f(0)}     (5) In mathematics, the Dirac delta function (δ function) is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function. In engineering and signal processing, the delta function, also known as the unit impulse symbol, may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable. The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus 'approximate' or 'nascent' delta functions. The graph of the delta function 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 delta function. 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, in kg m / s {displaystyle { ext{kg m}}/{ ext{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 ) {displaystyle Pdelta (t)} . (The units of δ ( t ) {displaystyle delta (t)} are s − 1 {displaystyle s^{-1}} .) 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, Then the momentum at any time t is found by integration: Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δ t → 0 {displaystyle Delta t o 0} , giving 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 ordinary calculus) lim Δ t → 0 F Δ t {displaystyle lim _{Delta t o 0}F_{Delta t}} is zero everywhere but a single point, where it is infinite. To make proper sense of the delta function, we should instead insist that the property