# HeavisideLambda

represents the triangle distribution which is nonzero for .

HeavisideLambda[x1,x2,]

represents the multidimensional triangle distribution which is nonzero for .

# Details # Examples

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## Basic Examples(4)

Evaluate numerically:

Plot in one dimension:

Plot in two dimensions:

Higher derivatives involve DiracDelta distributions:

## Scope(36)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate efficiently at high precision:

### Specific Values(4)

Values of HeavisideLambda at fixed points:

Value at zero:

Evaluate symbolically:

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:

### 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:

Numerical integration:

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(1)

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: