An example of such a family is formed by the family of Gaussian functions F α( x) = exp(−α x²), where the different values of the single parameter α distinguish the different members. There exist families of regular functions F α( x) of which the family members differ by the value of a single parameter α. Block ("boxcar") function (red) times regular function f( x) (blue). Mathematicians say that the delta function is a linear functional on a space of test functions.įig. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand ("under the integral"). The Dirac delta function is not an ordinary well-behaved map, but a distribution, also known as an improper or generalized function. In analogy, the Dirac delta function δ( x− a) is defined by (replace i by x and the summation over i by an integration over x), Note that the Kronecker delta acts as a "filter" in a summation: Heuristically, the Dirac delta function can be seen as an extension of the Kronecker delta from integral indices (elements of ) to real indices (elements of ). In the limit of a point mass the distribution becomes a Dirac delta function. When the distribution becomes smaller and smaller, while M is constant, the mass distribution shrinks to a point mass, which by definition has zero extent and yet has a finite-valued integral equal to total mass M. A physical model that visualizes a delta function is a mass distribution of finite total mass M-the integral over the mass distribution. Dirac in his seminal book on quantum mechanics. The Dirac delta function is a function introduced in 1930 by P.
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