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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 Mathematica 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 three-space, 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 second-rank tensor. It satisfies

			(10)
			(11)
			(12)

RELATED WOLFRAM SITES: http://functions.wolfram.com/IntegerFunctions/KroneckerDelta/