The simplest interpretation of the Kronecker delta is as the discrete version of the delta function defined by
(1)

The Kronecker delta is implemented in the Wolfram Language as KroneckerDelta[i, j], as well as in a generalized form KroneckerDelta[i, j, ...] that returns 1 iff all arguments are equal and 0 otherwise.
It has the contour integral representation
(2)

where is a contour corresponding to the unit circle and and are integers.
In threespace, the Kronecker delta satisfies the identities
(3)
 
(4)
 
(5)
 
(6)

where Einstein summation is implicitly assumed, , 2, 3, and is the permutation symbol.
Technically, the Kronecker delta is a tensor defined by the relationship
(7)

Since, by definition, the coordinates and are independent for ,
(8)

so
(9)

and is really a mixed secondrank tensor. It satisfies
(10)
 
(11)
 
(12)
 
(13)
 
(14)
