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



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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:

The traditional notation is used in both StandardForm and TraditionalForm:

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
Updated in 2017