# 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(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:

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, https://reference.wolfram.com/language/ref/KroneckerDelta.html (updated 2017).

#### Text

Wolfram Research (1999), KroneckerDelta, Wolfram Language function, https://reference.wolfram.com/language/ref/KroneckerDelta.html (updated 2017).

#### CMS

Wolfram Language. 1999. "KroneckerDelta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/KroneckerDelta.html.

#### APA

Wolfram Language. (1999). KroneckerDelta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KroneckerDelta.html

#### BibTeX

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

#### BibLaTeX

@online{reference.wolfram_2024_kroneckerdelta, organization={Wolfram Research}, title={KroneckerDelta}, year={2017}, url={https://reference.wolfram.com/language/ref/KroneckerDelta.html}, note=[Accessed: 21-July-2024 ]}