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



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  (24)

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  (10)

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:

KroneckerDelta is not an analytic function:

It has both singularities and discontinuities:

KroneckerDelta is neither nondecreasing nor nonincreasing:

KroneckerDelta is not injective:

KroneckerDelta is not surjective:

KroneckerDelta is non-negative:

KroneckerDelta is neither convex nor concave:

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  (5)

Use in sums to pick out terms:

Generate a banded matrix with two superdiagonals:

Pick out elements:

Compute MoebiusMu using KroneckerDelta and LiouvilleLambda:

Decompose a spherical harmonic into a sum of products of two spherical harmonics:

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:

Wolfram Research (1999), KroneckerDelta, Wolfram Language function, (updated 2017).


Wolfram Research (1999), KroneckerDelta, Wolfram Language function, (updated 2017).


Wolfram Language. 1999. "KroneckerDelta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017.


Wolfram Language. (1999). KroneckerDelta. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_kroneckerdelta, author="Wolfram Research", title="{KroneckerDelta}", year="2017", howpublished="\url{}", note=[Accessed: 21-July-2024 ]}


@online{reference.wolfram_2024_kroneckerdelta, organization={Wolfram Research}, title={KroneckerDelta}, year={2017}, url={}, note=[Accessed: 21-July-2024 ]}