# Integration involving two DiracDelta with variable limit

I want to write a code to evaluate

$$\int_0^{t-1}\frac{\delta(z-1)}{z}\frac{\delta(t-z-1)}{t-z}dz$$

which should gives the answer $$\frac{\delta(t-2)}{t-1}.$$

I tried to write the code

Integrate[DiracDelta[z - 1]/z * DiracDelta[t - z - 1]/(t - z), {z, 0, t - 1}]


but it didn't give the answer.

How should I write the code to give the expected answer?

• Why should (DiracDelta[-2 + t] HeavisideTheta[-2 + t])/(-1 + t) be true? Can you kindly give a reference? Sep 9 at 14:42
• Are you asking why mathematically should we get that answer? Sep 9 at 17:09
• I asked for a reference for it. What is the reason you think that the right answer should be $\frac{\delta(t-2)}{t-1}$? Hope I am clear. Sep 9 at 17:27
• Just calculate it by hand ✋ Sep 9 at 17:41
• This assumes t>2? Sep 9 at 19:12

Workaround:

FourierTransform[Integrate[InverseFourierTransform[
DiracDelta[z - 1]/z*DiracDelta[t - q z - 1]/(t - z), q, w], {z, 0,
t - 1}, Assumptions -> t \[Element] Reals], w, q] /. q -> 1

(*(DiracDelta[-2 + t] HeavisideTheta[-2 + t])/(-1 + t)*)


Solution with Maple 2023.1 : • Fourier transforms of generalized functions are hard matter. It's unclear to me how to justify FourierTransform[Integrate[InverseFourierTransform.... Also InverseFourierTransform[ DiracDelta[z - 1]/z*DiracDelta[t - q z - 1]/(t - z), q, w] is running without any response for a long time. All that is built on the sand. Last, but not least: why is (DiracDelta[-2 + t] HeavisideTheta[-2 + t])/(-1 + t) true? Sep 9 at 14:25
• @user64494 .My sand is GOLD. Sep 9 at 14:30
• MariuszIwaniuk (@ does not work.): I prefer arguments over emotional words. Sep 9 at 14:43
• InverseFourierTransform[ DiracDelta[z - 1]/z*DiracDelta[t - q z - 1]/(t - z), q, w] results in (E^(-I (-1 + t) w) DiracDelta[-1 + z])/(Sqrt[2 \[Pi]] (-1 + t)). Can somebody give a reference to this result? It would be kind of that person. Sep 9 at 15:41
• It's undefined. The integral cannot have a member of the singular set as an endpoint of the integration. Sep 10 at 21:21

Approximate the DiracDelta by the normal distribution

   Integrate[DiracDelta[z-1 ]/z 1/Sqrt[2 \[Pi] s] E^(-(t-z-1)^2/(2s^2)),
{z,0,t-1}]


$$\fbox{\frac{\theta (t-2) e^{-\frac{(t-2)^2}{2 s^2}}}{\sqrt{2 \pi } \sqrt{s}}\text{ if }t\in \mathbb{R}}$$

Now let $$s\to 0$$, evaluating $$\theta$$ at its jump. Good example to show that products of singular distributions are undefined.

• All that is built on the sand. If you look in W. Rudin, Functional analysis, Chap. 6,7, you will see that $\frac{e^{-\frac{(t-2)^2}{2 s^2}}}{\sqrt{2 \pi } \sqrt{s}}$ converges to the $\delta$-distribution in the weak topology. The pointwise convergence makes no sense. Sep 9 at 14:46
• The product DiracDelta[-2 + t] HeavisideTheta[-2 + t] is not defined because both distributions have $\{2\}$ as their support.See "Problem of multiplying distributions" in the Wiki article as a first reading. Sep 9 at 17:48
• Also the documentation says "Products of distributions with coinciding singular support cannot be defined" and presents as an example Integrate[HeavisideTheta[x] DiracDelta[x], {x, -1, 1}]. Sep 9 at 18:33

Mathematica needs the info t \[Element] Reals !

Using DiracDelta as a limit (see @RolandF's approach) we get

dirac = Function[x, Exp[-(x^2/(2 eps))]/Sqrt[2 Pi eps]]

Integrate[DiracDelta [z - 1]/z* dirac[t - z - 1]/(t - z), {z, 0, t - 1},Assumptions ->{ Element[t, Reals] }]
(*(E^(-((-2 + t)^2/(2 eps))) HeavisideTheta[-2 + t])/(Sqrt[eps]Sqrt[2 \[Pi]] (-1 + t))*)


Using the definition of dirac the result follows to

DiracDelta[t-2] HeavisideTheta[-2 + t] / (-1 + t)

which confirms the expectation of QP and the result @MariuszIwaniuk !

• The product DiracDelta[t-2] HeavisideTheta[-2 + t] is not defined because both distributions have $\{2\}$ as their support (see "Problem of multiplying distributions" in the Wiki article). Sep 9 at 17:41
• Knowing that the definitions of both distributions are well defined I see no reason why the product shouldn't be defined too! Sep 9 at 17:43
• Did you look in the linked Wiki article before having replied? Sep 9 at 17:50
• Also the documentation says "Products of distributions with coinciding singular support cannot be defined" and presents as an example Integrate[HeavisideTheta[x] DiracDelta[x], {x, -1, 1}]. Sep 9 at 18:32