Evaluate numerically:

Plot in one dimension:

Plot in two dimensions:

Differentiate to obtain DiracDelta:

Solve the time‐independent Schrödinger equation with piecewise analytic potential:

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:

The derivative of HeavisideTheta is a distribution:

The derivative of UnitStep is a piecewise function:

Expand HeavisideTheta into HeavisideTheta with simpler arguments:

Simplify expressions containing HeavisideTheta:

Use in integrals:

Use in Fourier transforms:

Use in Laplace transforms:

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:

Form repeated convolution integrals starting with a product: