Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:The following equations are satisfied: Using the Iverson bracket:Similarly, in digital signal processing, the same concept is represented as a sequence or discrete function on ℤ (the integers):The Kronecker delta has the so-called sifting property that for j ∈ ℤ:In probability theory and statistics, the Kronecker delta and Dirac delta function can both be used to represent a discrete distribution. If the support of a distribution consists of points x = {x1, ..., xn}, with corresponding probabilities p1, ..., pn, then the probability mass function p(x) of the distribution over x can be written, using the Kronecker delta, asIf it is considered as a type (1,1) tensor, the Kronecker tensor can be writtenδij with a covariant index j and contravariant index i:For any integer n, using a standard residue calculation we can write an integral representation for the Kronecker delta as the integral below, where the contour of the integral goes counterclockwise around zero. This representation is also equivalent to a definite integral by a rotation in the complex plane.The Kronecker comb function with period N is defined (using DSP notation) as:The Kronecker delta is also called degree of mapping of one surface into another. Suppose a mapping takes place from surface Suvw to Sxyz that are boundaries of regions, Ruvw and Rxyz which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for Suvw, and Suvw to Suvw are each oriented by the outer normal n: