HeavisideLambda
✖
HeavisideLambda
represents the multidimensional triangle distribution which is nonzero for
.
Details

- HeavisideLambda[x] is equivalent to Convolve[HeavisidePi[t],HeavisidePi[t],t,x].
- HeavisideLambda can be used in derivatives, integrals, integral transforms, and differential equations.
- HeavisideLambda has attribute Orderless.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0ywpat8ux55u-dgmklp


https://wolfram.com/xid/0ywpat8ux55u-qe2fd0


https://wolfram.com/xid/0ywpat8ux55u-6xivku

Higher derivatives involve DiracDelta distributions:

https://wolfram.com/xid/0ywpat8ux55u-by22tk

Scope (38)Survey of the scope of standard use cases
Numerical Evaluation (7)

https://wolfram.com/xid/0ywpat8ux55u-l274ju


https://wolfram.com/xid/0ywpat8ux55u-q2v2ko


https://wolfram.com/xid/0ywpat8ux55u-wlv0g


https://wolfram.com/xid/0ywpat8ux55u-b0wt9

The precision of the output tracks the precision of the input:

https://wolfram.com/xid/0ywpat8ux55u-y7k4a


https://wolfram.com/xid/0ywpat8ux55u-lspoul

Evaluate efficiently at high precision:

https://wolfram.com/xid/0ywpat8ux55u-di5gcr


https://wolfram.com/xid/0ywpat8ux55u-bq2c6r

HeavisideLambda threads over lists:

https://wolfram.com/xid/0ywpat8ux55u-bnjcdp

Compute average-case statistical intervals using Around:

https://wolfram.com/xid/0ywpat8ux55u-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0ywpat8ux55u-thgd2

Or compute the matrix HeavisideLambda function using MatrixFunction:

https://wolfram.com/xid/0ywpat8ux55u-o5jpo

Specific Values (4)
Values of HeavisideLambda at fixed points:

https://wolfram.com/xid/0ywpat8ux55u-nww7l


https://wolfram.com/xid/0ywpat8ux55u-bmqd0y


https://wolfram.com/xid/0ywpat8ux55u-d5sj8

Find a value of x for which the HeavisideLambda[x]=0.6:

https://wolfram.com/xid/0ywpat8ux55u-f2hrld


https://wolfram.com/xid/0ywpat8ux55u-e4g58p

Visualization (4)
Plot the HeavisideLambda function:

https://wolfram.com/xid/0ywpat8ux55u-ecj8m7

Visualize scaled HeavisideLambda functions:

https://wolfram.com/xid/0ywpat8ux55u-urg5gs

Visualize the composition of HeavisideLambda with a periodic function:

https://wolfram.com/xid/0ywpat8ux55u-lwo8te

Plot HeavisideLambda in three dimensions:

https://wolfram.com/xid/0ywpat8ux55u-s7kqu3

Function Properties (11)
Function domain of HeavisideLambda:

https://wolfram.com/xid/0ywpat8ux55u-cl7ele

It is restricted to real inputs:

https://wolfram.com/xid/0ywpat8ux55u-shglmn

Function range of HeavisideLambda:

https://wolfram.com/xid/0ywpat8ux55u-bavt6m

HeavisideLambda is an even function:

https://wolfram.com/xid/0ywpat8ux55u-twcdez

The area of HeavisideLambda is 1:

https://wolfram.com/xid/0ywpat8ux55u-czklqo

HeavisideLambda has singularities:

https://wolfram.com/xid/0ywpat8ux55u-mdtl3h

However, it is continuous everywhere:

https://wolfram.com/xid/0ywpat8ux55u-mn5jws

Verify the claim at one of its singular points:

https://wolfram.com/xid/0ywpat8ux55u-miy4kz

HeavisideLambda is neither nonincreasing nor nondecreasing:

https://wolfram.com/xid/0ywpat8ux55u-nlz7s

HeavisideLambda is not injective:

https://wolfram.com/xid/0ywpat8ux55u-poz8g


https://wolfram.com/xid/0ywpat8ux55u-ctca0g

HeavisideLambda is not surjective:

https://wolfram.com/xid/0ywpat8ux55u-cxk3a6


https://wolfram.com/xid/0ywpat8ux55u-frlnsr

HeavisideLambda is non-negative:

https://wolfram.com/xid/0ywpat8ux55u-84dui

HeavisideLambda is neither convex nor concave:

https://wolfram.com/xid/0ywpat8ux55u-8kku21

TraditionalForm typesetting:

https://wolfram.com/xid/0ywpat8ux55u-elu5e0

Differentiation (4)
Differentiate the univariate HeavisideLambda:

https://wolfram.com/xid/0ywpat8ux55u-krpoah

Higher derivatives with respect to x:

https://wolfram.com/xid/0ywpat8ux55u-z33jv

Differentiate the multivariate HeavisideLambda:

https://wolfram.com/xid/0ywpat8ux55u-p3txf

Differentiate a composition involving HeavisideLambda:

https://wolfram.com/xid/0ywpat8ux55u-d9p2kz

Integration (4)
Integrate over finite domains:

https://wolfram.com/xid/0ywpat8ux55u-dcx3ui


https://wolfram.com/xid/0ywpat8ux55u-km76j

Integrate over infinite domains:

https://wolfram.com/xid/0ywpat8ux55u-jgy57y


https://wolfram.com/xid/0ywpat8ux55u-ordeem

Integrate expressions containing symbolic derivatives of HeavisideLambda:

https://wolfram.com/xid/0ywpat8ux55u-h7492c

Integral Transforms (4)
FourierTransform of HeavisideLambda is a squared Sinc function:

https://wolfram.com/xid/0ywpat8ux55u-cxb66w


https://wolfram.com/xid/0ywpat8ux55u-jhqimo


https://wolfram.com/xid/0ywpat8ux55u-f64drv


https://wolfram.com/xid/0ywpat8ux55u-kmfcq4

Find the LaplaceTransform of HeavisideLambda:

https://wolfram.com/xid/0ywpat8ux55u-di03k3


https://wolfram.com/xid/0ywpat8ux55u-bk17w8

The convolution of HeavisideLambda with HeavisidePi:

https://wolfram.com/xid/0ywpat8ux55u-bcycw


https://wolfram.com/xid/0ywpat8ux55u-ic6l4j

Applications (2)Sample problems that can be solved with this function
Integrate a function involving HeavisideLambda symbolically and numerically:

https://wolfram.com/xid/0ywpat8ux55u-l16e8o


https://wolfram.com/xid/0ywpat8ux55u-dquzjs


https://wolfram.com/xid/0ywpat8ux55u-dfr0f3


https://wolfram.com/xid/0ywpat8ux55u-bg7f13

Visualize discontinuities in the wavelet domain:

https://wolfram.com/xid/0ywpat8ux55u-i2926z

https://wolfram.com/xid/0ywpat8ux55u-xndaw6

Detail coefficients in the region of discontinuities have larger values:

https://wolfram.com/xid/0ywpat8ux55u-vobk7q

https://wolfram.com/xid/0ywpat8ux55u-mtt4yw

Properties & Relations (2)Properties of the function, and connections to other functions
The derivative of HeavisideLambda is a distribution:

https://wolfram.com/xid/0ywpat8ux55u-1gxedf

At higher orders, the DiracDelta distribution appears:

https://wolfram.com/xid/0ywpat8ux55u-3u6cl

The derivative of UnitTriangle is a piecewise function:

https://wolfram.com/xid/0ywpat8ux55u-72k7v9

HeavisideLambda can be expressed in terms of HeavisideTheta:

https://wolfram.com/xid/0ywpat8ux55u-c6hg10

Wolfram Research (2008), HeavisideLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/HeavisideLambda.html.
Text
Wolfram Research (2008), HeavisideLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/HeavisideLambda.html.
Wolfram Research (2008), HeavisideLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/HeavisideLambda.html.
CMS
Wolfram Language. 2008. "HeavisideLambda." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HeavisideLambda.html.
Wolfram Language. 2008. "HeavisideLambda." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HeavisideLambda.html.
APA
Wolfram Language. (2008). HeavisideLambda. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HeavisideLambda.html
Wolfram Language. (2008). HeavisideLambda. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HeavisideLambda.html
BibTeX
@misc{reference.wolfram_2025_heavisidelambda, author="Wolfram Research", title="{HeavisideLambda}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/HeavisideLambda.html}", note=[Accessed: 20-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_heavisidelambda, organization={Wolfram Research}, title={HeavisideLambda}, year={2008}, url={https://reference.wolfram.com/language/ref/HeavisideLambda.html}, note=[Accessed: 20-May-2025
]}