Aside on the Dirac Delta Function
Aside on the Dirac Delta Function
Plot[DiracDelta[t],{t,-3,3},
PlotStyle->{RGBColor[0,1,0]}];
Plot::plnr: CompiledFunction[{t}, <<1>>, -CompiledCode-][t]
is not a machine-size real number at t = 0..
Mathematica does not like DiracDelta[0]...that should not surprise you. Let's have some fun:
In[35]:=
DiracDelta[.000000001]
Out[35]=
0
In[36]:=
DiracDelta[0]
Out[36]=
DiracDelta[0]
Can Mathematica recreate the properties of the Dirac Delta function?
In[37]:=
Integrate[DiracDelta[t],{t,-Infinity,Infinity}]
Out[37]=
1
In[38]:=
Clear[f]
Integrate[f[t] DiracDelta[t],{t,-Infinity,Infinity}]
Out[38]=
f[0]
Wait until you see this.
In[39]:=
Integrate[DiracDelta[t],t]
Out[39]=
UnitStep[t]
Is this our Heaviside Step Function? Let's see.
In[40]:=
Plot[UnitStep[t],{t,-3,3},
PlotStyle->{RGBColor[0,1,0]}];
In[41]:=
Integrate[UnitStep[t],t]
Out[41]=
t UnitStep[t]
Is this our ramp function?
In[42]:=
Plot[t UnitStep[t],{t,-3,3},
PlotStyle->{RGBColor[0,1,0]}];
I'd say Mathematica handles this pretty well. Wolfram Research, Inc. must know what they are doing. They must also have some people who know their theory.