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I'm working on integrals involving DiracDelta functions at the boundary of the integrals. I'm confused by the output Mathematica v10.1.0 gives me in this case:

f[x_] := DiracDelta[x] + Exp[-x];

Integrate[f[x], {x, 0, ∞}]
(* 2 - HeavisideTheta[0] *)

Limit[Integrate[f[x], {x, 0, b}], b -> ∞]
(* 2 *)

Limit[Integrate[f[x], {x, a, ∞}], a -> 0]
(* 1 *)

I understand the first case, where it basically allows me to pick my favorite definition and choose whether I want to include the DiracDelta in the integration. However, in the second line (where I evaluate a limit involving the other boundary) Mathematica seems to be sure that the boundary has to be included. Lastly, when I let the left integration boundary go to zero Mathematica correctly concludes that the DiracDelta should not be included, which is of course only correct if we approach a=0 from above, but this is consistent with the documentation of Limit.

My question is why Mathematica gives different results for the first two examples and how I can ensure that the DiracDelta is always included at the boundary.

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I can't speak for what's happening with your second example, but to answer your second question, you can ensure DiracDelta is included by integrating over a larger interval then taking a limit.

Limit[Integrate[f[x], {x, -ε, ∞}, Assumptions -> 0 < ε < 1/10^10], ε -> 0]
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I can break the calculations down, but I cannot really explain the first one.


This seems wrong:

Integrate[DiracDelta[x] + Exp[-x], {x, 0, b}]
(*  ConditionalExpression[-Cosh[b] + 2 HeavisideTheta[b] + Sinh[b], b ∈ Reals]  *)

Its limit is 2. The ConditionalExpression suggests that Integrate is using assumptions to make choices. Let's help it out below.


This seems right, and equivalent to the first result, 2 - HeavisideTheta[0], if Heaviside[0] is equal to 1/2, which it sometimes is:

Integrate[DiracDelta[x], {x, 0, b}, Assumptions -> b > 0]
(*  1 - E^-b + HeavisideTheta[0]  *)

Its limit as b -> Infinity is 1 + HeavisideTheta[0].


This seems right, too:

Limit[Integrate[DiracDelta[x] + Exp[-x], {x, a, ∞}], a -> 0, Direction -> 1]
(*  2  *)

Since for all a (a < 0), the integral of the DiracDelta part is 1, we do not get the HeavisideTheta in the answer.

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  • $\begingroup$ (at) Michael E2: (at least) in Version 8 things are even stranger: In[168]:= Integrate[DiracDelta[x] + Exp[-x], {x, 0, b}, Assumptions -> b > 0] Out[168]= 1 - E^-b + HeavisideTheta[0] (in agreement with your result) but now: In[169]:= Limit[%, b -> [Infinity]] Out[169]= 2, miraculously HeavisideTheta[0] has been replaced by the vaule 1. $\endgroup$ – Dr. Wolfgang Hintze May 13 '15 at 23:47

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