HeavisideLambda

HeavisideLambda[x]

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  (31)

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:

HeavisideLambda threads over lists:

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  (6)

Function domain of HeavisideLambda:

It is restricted to real inputs:

Approximate function range of HeavisideLambda:

HeavisideLambda is an even function:

The area of HeavisideLambda is 1:

HeavisideLambda is continuous everywhere, including its three singular points:

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:

Numerical integration:

Integrate expressions containing symbolic derivatives of HeavisideLambda:

Integral Transforms  (4)

FourierTransform of HeavisideLambda is a squared Sinc function:

FourierSeries:

Find the LaplaceTransform of HeavisideLambda:

The convolution of HeavisideLambda with HeavisidePi:

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:

Introduced in 2008
 (7.0)