This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 Calculus`DiracDelta` This package defines the function UnitStep and its generalized derivative DiracDelta. The Dirac delta function is not a true function in the usual mathematical sense since it is not defined for every value of x. However, it is commonly used to represent ideal distributions in problems in physics and engineering. The unit step function may be used to represent piecewise continuous functions.This package is automatically loaded by the transform packages in the Calculus subdirectory. UnitStep and DiracDelta, the generalized derivative of UnitStep. This loads the package. In[1]:= < Infinity] -Limit[int, x -> 0, Direction -> 1]) Out[8]= Multidimensional UnitStep and DiracDelta. A multidimensional unit step may be expressed as or as the product . In[9]:= Integrate[Sin[x y] UnitStep[x, y], x, y] Out[9]= Here the integral in x does not evaluate because is not continuous at . The integral in y evaluates only for the real solution of . In[10]:= Integrate[Cot[x] g[y] DiracDelta[x, y^3 + y], x, y] Syntax::newl: The newline character after "Integrate[Cot[x] g[y]" is understood as a multiplication operator. Out[10]= Since the unit step is separable, the value of at is . In[11]:= UnitStep[0, y] Out[11]= This is a plot of the characteristic or indicator function for the set x,y :x^2+y^2<=1,x+y>=1. In[12]:= Plot3D[UnitStep[1 - x^2 - y^2, x + y - 1], {x, 0, 1}, {y, 0, 1}, PlotPoints -> 30] Option for UnitStep. A unit step that is continuous at from the left may be expressed using the ZeroValue option. In[13]:= UnitStep[0, ZeroValue -> 0] Out[13]= The default value of at can be changed by redefining Options[UnitStep]. Now evaluates to . In[14]:= (SetOptions[UnitStep, ZeroValue -> 1/2];UnitStep[0, 0]) Out[14]= Simplifying expressions involving UnitStep. Complicated expressions involving UnitStep can sometimes be simplified using SimplifyUnitStep. In[15]:= UnitStep[x - 2] (DiracDelta[x - 3] + UnitStep[x, y] UnitStep[y - 1]) // SimplifyUnitStep Out[15]= Simplifying expressions involving DiracDelta. Expressions containing terms equivalent to the zero distribution can sometimes be simplified using SimplifyDiracDelta. In[16]:= (1 + x^2 DiracDelta[x, y])/x // SimplifyDiracDelta Out[16]=