       Re: Integrating DiracDelta to get UnitStep

• To: mathgroup at smc.vnet.net
• Subject: [mg91260] Re: Integrating DiracDelta to get UnitStep
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Tue, 12 Aug 2008 04:46:14 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <g7p2tm\$arr\$1@smc.vnet.net>

```CRC wrote:

> I am a bit confused by Mathematica 6.0.3 behavior.  I expect that:
>
> In[n]:= Integrate[DiracDelta[x], {x, -\[Infinity], t},
>   Assumptions -> Im[t] == 0]
>
> Will produce:
>
> Out[n]= UnitStep[t]
>
>
> Out[n]= 1
>
> However,
>
> In[n+1]:= Plot[ Integrate[DiracDelta[x], {x, -\[Infinity], t},
>    Assumptions -> Im[t] == 0], {t, -2, 2} ]
>
> produces the expected plot of UnitStep[t].

What you get, indeed, is the plot of HeavisideTheta[t]:

Plot[Integrate[DiracDelta[x], {x, -\[Infinity], t},
Assumptions -> Im[t] == 0], {t, -2, 2}, PlotStyle -> Thick]

Plot[HeavisideTheta[t], {t, -2, 2}, PlotStyle -> Thick]

Plot[UnitStep[t], {t, -2, 2}, PlotStyle -> Thick]

> Why doesn't the integration output the UnitStep function?

Mathematica does not define UnitStep as a primitive of DirectDelta. It
defines DiractDelta as the first derivaitve of HeavisideTheta.

D[HeavisideTheta[t], t] (* === DiracDelta[t] *)

UnitStep[x] represents the unit step function,
equal to 0 for x < 0 and 1 for x >= 0.

HeavisideTheta[x] represents the Heaviside theta function
\[Theta](x), equal to 0 for x < 0 and 1 for x > 0.

Note that HeavisideTheta[x] is not defined for x == 0 and that the value
of the integral is 1/2 for x form -inf to zero. This might explained why
Mathematica does not returned HeavisideTheta[t].

In:= Integrate[DiracDelta[x], {x, -\[Infinity], t},
Assumptions -> Im[t] == 0 && Re[t] < 0]

Out= 0

In:= Integrate[DiracDelta[x], {x, -\[Infinity], t},
Assumptions -> Im[t] == 0 && Re[t] == 0]

Out= 1/2

In:= Integrate[DiracDelta[x], {x, -\[Infinity], t},
Assumptions -> Im[t] == 0 && Re[t] > 0]

Out= 1

In:= UnitStep[{-1, 0, 1}]

Out= {0, 1, 1}

In:= HeavisideTheta[{-1, 0, 1}]

Out= {0, HeavisideTheta, 1}

HTH,
-- Jean-Marc

```

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