# KroneckerDelta

KroneckerDelta[n1,n2,]

gives the Kronecker delta , equal to 1 if all the are equal, and 0 otherwise.

# Details • gives 1; gives 0 for other numeric n.
• KroneckerDelta has attribute Orderless.
• An empty template can be entered as kd . Arguments in the subscript should be separated by commas.
• The comma can be made invisible by using the character \[InvisibleComma] or , .

# Examples

open allclose all

## Basic Examples(4)

Evaluate numerically:

Construct an identity matrix:

Use in sums to pick out elements:

Plot over a subset of the integers:

## Scope(18)

### Numerical Evaluation(4)

Evaluate numerically:

Complex number inputs:

KroneckerDelta always returns an exact result irrespective of the precision of the input:

Evaluate efficiently at high precision:

### Specific Values(3)

Value at zero:

The multi-argument form gives 1 when all inputs are equal:

Evaluate symbolically:

### Visualization(3)

Plot the single-argument KroneckerDelta using integer-width bins:

Visualize KroneckerDelta over the reals. Except for a jump at , it is indistinguishable from a zero function:

Plot KroneckerDelta in three dimensions:

### Function Properties(4)

KroneckerDelta is defined for all real and complex inputs:

Function range of KroneckerDelta:

The function range for complex values is the same:

KroneckerDelta accepts list inputs:

### Differentiation and Integration(4)

First derivative with respect to :

Series expansion at a generic point:

Compute the indefinite integral using Integrate:

Verify the antiderivative:

More integrals:

## Applications(3)

Use in sums to pick out terms:

Generate a banded matrix with two superdiagonals:

Pick out elements:

## Properties & Relations(2)

Reduce an equation containing KroneckerDelta:

The support of KroneckerDelta has measure zero:

## Possible Issues(2)

KroneckerDelta can stay unevaluated for numeric arguments:

A larger setting for \$MaxExtraPrecision can be needed: Equality testing of the arguments takes numerical precision into account:

## Neat Examples(1)

Express products of signatures as sums of products of Kronecker deltas:

Special cases for 3D; summation over repeated indices is assumed:

Introduced in 1999
(4.0)
|
Updated in 2017
(11.1)