HeavisideLambda
represents the triangle distribution which is nonzero for .
HeavisideLambda[x1,x2,…]
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)
Higher derivatives involve DiracDelta distributions:
Scope (38)
Numerical Evaluation (7)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
HeavisideLambda threads over lists:
Compute average-case statistical intervals using Around:
Compute the elementwise values of an array:
Or compute the matrix HeavisideLambda function using MatrixFunction:
Specific Values (4)
Values of HeavisideLambda at fixed points:
Find a value of x for which the HeavisideLambda[x]=0.6:
Visualization (4)
Plot the HeavisideLambda function:
Visualize scaled HeavisideLambda functions:
Visualize the composition of HeavisideLambda with a periodic function:
Plot HeavisideLambda in three dimensions:
Function Properties (11)
Function domain of HeavisideLambda:
It is restricted to real inputs:
Function range of HeavisideLambda:
HeavisideLambda is an even function:
The area of HeavisideLambda is 1:
HeavisideLambda has singularities:
However, it is continuous everywhere:
Verify the claim at one of its singular points:
HeavisideLambda is neither nonincreasing nor nondecreasing:
HeavisideLambda is not injective:
HeavisideLambda is not surjective:
HeavisideLambda is non-negative:
HeavisideLambda is neither convex nor concave:
TraditionalForm typesetting:
Differentiation (4)
Differentiate the univariate HeavisideLambda:
Higher derivatives with respect to x:
Differentiate the multivariate HeavisideLambda:
Differentiate a composition involving HeavisideLambda:
Integration (4)
Integrate over finite domains:
Integrate over infinite domains:
Integrate expressions containing symbolic derivatives of HeavisideLambda:
Integral Transforms (4)
FourierTransform of HeavisideLambda is a squared Sinc function:
Find the LaplaceTransform of HeavisideLambda:
The convolution of HeavisideLambda with HeavisidePi:
Applications (2)
Integrate a function involving HeavisideLambda symbolically and numerically:
Visualize discontinuities in the wavelet domain:
Detail coefficients in the region of discontinuities have larger values:
Properties & Relations (2)
The derivative of HeavisideLambda is a distribution:
At higher orders, the DiracDelta distribution appears:
The derivative of UnitTriangle is a piecewise function:
HeavisideLambda can be expressed in terms of HeavisideTheta:
Text
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.
APA
Wolfram Language. (2008). HeavisideLambda. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HeavisideLambda.html