The best-known 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 quasi-permutation 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: