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Integer Functions > Signature[{n1,n2,...,nd}] > Introduction to the tensor functions





The best-known properties and formulas of the tensor functions


The tensor functions , , , and can have unit values at infinity:

The tensor functions , , , , and have the following values for some specialized variables:

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 .

The tensor functions , , , , and do not have periodicity.

The tensor functions , , , and are even functions:

The tensor functions , , and have permutation symmetry, for example:

The discrete delta function and Kronecker delta function have the following integral representations along the interval and unit circle :

The tensor functions , , , , and satisfy various identities, for example:

The tensor functions , , , , and have the following complex characteristics:

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 of the tensor functions and can be provided by the following formulas:

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:





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