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 and , Kronecker delta functions and , and signature (Levi–Civita symbol) 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 coincides with the Kronecker delta function . In the case of several variables, the discrete delta function coincides with Kronecker delta function :
where is the number of permutations needed to go from the sorted version of to .
Connections within the group of tensor functions and with other function groups
Representations through equivalent functions
The tensor functions , , , and have the following representations through equivalent functions:
The bestknown properties and formulas of the tensor functions
Simple values at zero and infinity
The tensor functions , , , and can have unit values at infinity:
Specific values for specialized variables
The tensor functions , , , , and have the following values for some specialized variables:
Analyticity
and are nonanalytical functions defined over . Their possible values are and .
and are nonanalytical functions defined over . Their possible values are and .
is a nonanalytical function, defined over the set of tuples of complex numbers with possible values .
Periodicity
The tensor functions , , , , and do not have periodicity.
Parity and symmetry quasipermutation symmetry
The tensor functions , , , and are even functions:
The tensor functions , , and have permutation symmetry, for example:
Integral representations
The discrete delta function and Kronecker delta function have the following integral representations along the interval and unit circle :
Transformations
The tensor functions , , , , and satisfy various identities, for example:
Complex characteristics
The tensor functions , , , , and have the following complex characteristics:
Differentiation
Differentiation of the tensor functions and can be provided by the following formulas:
Fractional integro‐differentiation of the tensor functions and can be provided by the following formulas:
Indefinite integration
Indefinite integration of the tensor functions and 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 , for example:
Applications of the tensor functions
The tensor functions have numerous applications throughout mathematics, number theory, analysis, and other fields.
