Built-in Mathematica Symbol

HeavisideTheta

HeavisideTheta[x]
represents the Heaviside theta function

, equal to 0 for

and 1 for

.

HeavisideTheta[x₁, x₂, ...]
represents the multidimensional Heaviside theta function which is 1 only if none of the x_i are not positive.

MORE INFORMATION

HeavisideTheta[x] returns 0 or 1 for all numeric x other than 0.

HeavisideTheta can be used in integrals, integral transforms and differential equations.

HeavisideTheta has attribute Orderless.

For exact numeric quantities, HeavisideTheta internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.

EXAMPLES

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Basic Examples (2)

Differentiate to obtain DiracDelta:

In[1]:=

Out[1]=

Differentiate to obtain DiracDelta:

In[1]:=

Out[1]=

Scope (6)

Generate HeavisideTheta from an integral:

HeavisideTheta threads element-wise over lists:

Integrate over finite and infinite domains:

Numerical integration:

Integrate expressions containing symbolic derivatives of HeavisideTheta:

Indefinite integral interpretation for real arguments:

TraditionalForm typesetting:

Generalizations & Extensions (3)

Multivariate HeavisideTheta:

Differentiate the multivariate HeavisideTheta:

Applications (6)

Use DSolve with DiracDelta source term to find Green's function:

Solve the inhomogeneous ODE through convolution with Green's function:

Compare with the direct result from DSolve:

Model a uniform probability distribution:

Calculate the probability distribution for the sum of two uniformly distributed variables:

Plot the distributions for the sum:

Fundamental solution (Green's function) of the 1D wave equation:

Solution for a given source term:

Plot the solution:

Fundamental solution of the Klein-Gordon

operator:

Visualize the fundamental solution (it is nonvanishing only in the forward light cone):

A cusp-containing peakon solution of the Camassa-Holm equation:

Check the solution:

Plot the solution:

Differentiate and integrate a piecewise defined function in a lossless manner:

Differentiating and integrating recovers the original function:

Using Piecewise does not recover the original function:

Properties & Relations (5)

Expand HeavisideTheta into HeavisideTheta with simpler arguments:

Simplify expressions containing HeavisideTheta:

Use in integrals:

Use in Fourier transforms:

Use in Laplace transforms:

Possible Issues (10)

HeavisideTheta stays unevaluated for vanishing argument:

PiecewiseExpand does not operate on HeavisideTheta because it is a distribution and not a piecewise-defined function:

The precision of the output does not track the precision of the input:

HeavisideTheta can stay unevaluated for numeric arguments:

Machine-precision numericalization of HeavisideTheta can give wrong results:

Use arbitrary-precision arithmetic to obtain the correct result:

A larger setting for $MaxExtraPrecision will not avoid the N::meprec message because the result is exact:

The functions UnitStep and HeavisideTheta are not mathematically equivalent:

Products of distributions with coincident singular support cannot be defined (no Colombeau algebra interpretation):

HeavisideTheta cannot be uniquely defined with complex arguments (no Sato hyperfunction interpretation):

Numerical routines can have problems with discontinuous functions:

Limit does not give HeavisideTheta as a limit of smooth functions:

Neat Examples (1)

Form repeated convolution integrals starting with a product: