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:
