WOLFRAM

represents the triangle distribution which is nonzero for .

HeavisideLambda[x1,x2,]

represents the multidimensional triangle distribution which is nonzero for .

Details

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

Out[1]=1

Plot in one dimension:

Out[1]=1

Plot in two dimensions:

Out[1]=1

Higher derivatives involve DiracDelta distributions:

Out[1]=1

Scope  (38)Survey of the scope of standard use cases

Numerical Evaluation  (7)

Evaluate numerically:

Out[2]=2
Out[3]=3
Out[4]=4

Evaluate to high precision:

Out[1]=1

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

Out[1]=1
Out[2]=2

Evaluate efficiently at high precision:

Out[1]=1
Out[2]=2

HeavisideLambda threads over lists:

Out[1]=1

Compute average-case statistical intervals using Around:

Out[1]=1

Compute the elementwise values of an array:

Out[1]=1

Or compute the matrix HeavisideLambda function using MatrixFunction:

Out[2]=2

Specific Values  (4)

Values of HeavisideLambda at fixed points:

Out[1]=1

Value at zero:

Out[1]=1

Evaluate symbolically:

Out[1]=1

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

Out[1]=1
Out[2]=2

Visualization  (4)

Plot the HeavisideLambda function:

Out[1]=1

Visualize scaled HeavisideLambda functions:

Out[1]=1

Visualize the composition of HeavisideLambda with a periodic function:

Out[1]=1

Plot HeavisideLambda in three dimensions:

Out[1]=1

Function Properties  (11)

Function domain of HeavisideLambda:

Out[1]=1

It is restricted to real inputs:

Out[1]=1

Function range of HeavisideLambda:

Out[1]=1

HeavisideLambda is an even function:

Out[1]=1

The area of HeavisideLambda is 1:

Out[1]=1

HeavisideLambda has singularities:

Out[1]=1

However, it is continuous everywhere:

Out[2]=2

Verify the claim at one of its singular points:

Out[3]=3

HeavisideLambda is neither nonincreasing nor nondecreasing:

Out[1]=1

HeavisideLambda is not injective:

Out[1]=1
Out[2]=2

HeavisideLambda is not surjective:

Out[1]=1
Out[2]=2

HeavisideLambda is non-negative:

Out[1]=1

HeavisideLambda is neither convex nor concave:

Out[1]=1

TraditionalForm typesetting:

Differentiation  (4)

Differentiate the univariate HeavisideLambda:

Out[1]=1

Higher derivatives with respect to x:

Out[1]=1

Differentiate the multivariate HeavisideLambda:

Out[1]=1

Differentiate a composition involving HeavisideLambda:

Out[1]=1

Integration  (4)

Integrate over finite domains:

Out[1]=1
Out[2]=2

Integrate over infinite domains:

Out[1]=1

Numerical integration:

Out[1]=1

Integrate expressions containing symbolic derivatives of HeavisideLambda:

Out[1]=1

Integral Transforms  (4)

FourierTransform of HeavisideLambda is a squared Sinc function:

Out[1]=1
Out[2]=2

FourierSeries:

Out[1]=1
Out[2]=2

Find the LaplaceTransform of HeavisideLambda:

Out[1]=1
Out[2]=2

The convolution of HeavisideLambda with HeavisidePi:

Out[1]=1
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

Integrate a function involving HeavisideLambda symbolically and numerically:

Out[5]=5
Out[6]=6
Out[7]=7
Out[8]=8

Visualize discontinuities in the wavelet domain:

Out[2]=2

Detail coefficients in the region of discontinuities have larger values:

Out[4]=4

Properties & Relations  (2)Properties of the function, and connections to other functions

The derivative of HeavisideLambda is a distribution:

Out[1]=1

At higher orders, the DiracDelta distribution appears:

Out[2]=2

The derivative of UnitTriangle is a piecewise function:

Out[3]=3

HeavisideLambda can be expressed in terms of HeavisideTheta:

Out[1]=1
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.

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 ]}

@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 ]}

@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 ]}