Introductions to KroneckerDelta

Introduction to the tensor functions

General

The tensor functions discrete delta and Kronecker delta first appeared in the works L. Kronecker (1866, 1903) and T. Levi–Civita (1896).

Definitions of the tensor functions

For all possible values of their arguments, the discrete delta functions δ(n) and δ((Subscript[n, 1]),(Subscript[n, 2]),…), Kronecker delta functions  (Subscript[δ, n]) and  (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]), and signature (Levi–Civita symbol) (Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) are defined by the formulas:

In other words, the Kronecker delta function is equal to 1 if all its arguments are equal.

In the case of one variable, the discrete delta function δ(n) coincides with the Kronecker delta function (Subscript[δ, n]). In the case of several variables, the discrete delta function δ((Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, m])) coincides with Kronecker delta function  (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, m]),0]):

where t is the number of permutations needed to go from the sorted version of {(Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, d])} to {(Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, d])}.

Connections within the group of tensor functions and with other function groups

Representations through equivalent functions

The tensor functions δ(n), δ((Subscript[n, 1]),(Subscript[n, 2]),…),  (Subscript[δ, n]), and (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]) have the following representations through equivalent functions:

The best-known properties and formulas of the tensor functions

Simple values at zero and infinity

The tensor functions δ(n), δ((Subscript[n, 1]),(Subscript[n, 2]),…), (Subscript[δ, n]), and (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]) can have unit values at infinity:

Specific values for specialized variables

The tensor functions  δ(n), δ((Subscript[n, 1]),(Subscript[n, 2]),…),  (Subscript[δ, n]), (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]), and (Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) have the following values for some specialized variables:

Analyticity

δ(n) and (Subscript[δ, n]) are nonanalytical functions defined over ℂ. Their possible values are 0 and 1.

δ((Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, m])) and  (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, m])]) are nonanalytical functions defined over (ℂ^m). Their possible values are 0 and 1.

(Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) is a nonanalytical function, defined over the set of tuples of complex numbers with possible values 0 and±1.

Periodicity

The tensor functions  δ(n), δ((Subscript[n, 1]),(Subscript[n, 2]),…), (Subscript[δ, n]), (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]), and (Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) do not have periodicity.

Parity and symmetry quasi-permutation symmetry

The tensor functions δ(n), δ((Subscript[n, 1]),(Subscript[n, 2]),…), (Subscript[δ, n]), and (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]) are even functions:

The tensor functions δ((Subscript[n, 1]),(Subscript[n, 2]),…), (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]), and (Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) have permutation symmetry, for example:

Integral representations

The discrete delta function δ(n) and Kronecker delta function (Subscript[δ, n,m]) have the following integral representations along the interval (0,2π) and unit circle |z|==1:

Transformations

The tensor functions δ(n), δ((Subscript[n, 1]),(Subscript[n, 2]),…), (Subscript[δ, n]), (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]), and (Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) satisfy various identities, for example:

Complex characteristics

The tensor functions δ(n), δ((Subscript[n, 1]),(Subscript[n, 2]),…),  (Subscript[δ, n]), (Subscript[δ, (Subscript[n, 1]),(Subscript[n, 2]),…]), and (Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]) have the following complex characteristics:

Differentiation

Differentiation of the tensor functions δ(n) and (Subscript[δ, n]) can be provided by the following formulas:

Fractional integro-differentiation of the tensor functions δ(n) and (Subscript[δ, n]) can be provided by the following formulas:

Indefinite integration

Indefinite integration of the tensor functions δ(n) and (Subscript[δ, n]) can be provided by the following formulas:

Summation

The following relations represent the sifting properties of the Kronecker and discrete delta functions:

There exist various formulas including finite summation of signature (Subscript[ε, (Subscript[n, 1]), (Subscript[n, 2]),… , (Subscript[n, d])]), for example:

Applications of the tensor functions

The tensor functions have numerous applications throughout mathematics, number theory, analysis, and other fields.

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