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represents the Heaviside theta function , equal to 0 for and 1 for .
represents the multidimensional Heaviside theta function which is 1 only if none of the are not positive.
  • HeavisideTheta[x] returns 0 or 1 for all real numeric x other than 0.
  • HeavisideTheta can be used in integrals, integral transforms and differential equations.
  • 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.
Differentiate to obtain DiracDelta:
Click for copyable input
Differentiate to obtain DiracDelta:
Click for copyable input
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:
Multivariate HeavisideTheta:
Differentiate the multivariate HeavisideTheta:
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:
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 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:
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