Integral of Dirac function from 0 to a.... value

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Discussion Overview

The discussion revolves around the integration of the Dirac delta function within the context of Laplace transforms and convolution formulas. Participants are exploring the challenges associated with evaluating integrals that involve the Dirac delta function, particularly in relation to specific functions and numerical results.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about integrating the Dirac delta function and seeks assistance, indicating uncertainty about their approach.
  • Another participant asserts that the integral of the Dirac delta function should yield the value of the integrand at the point where the delta function is centered, specifically at t' = 0.
  • A different participant questions the previous assertion by suggesting that the numerical results obtained through partial fraction expansion differ from the expected outcome, indicating a potential discrepancy in understanding.
  • Another reply introduces the idea that the value of the integral may need to be halved depending on the placement of the integration limits relative to the Dirac delta function.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple competing views regarding the integration of the Dirac delta function and its implications in the context of Laplace transforms.

Contextual Notes

The discussion highlights potential limitations in understanding the behavior of the Dirac delta function within integrals, particularly concerning the placement of integration limits and the specific forms of the functions involved.

maistral
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Hi. So I'm trying to use Laplace transforms in inverting a particular s-function via the convolution formula.

I ended up with this terrifying-looking thing:

convo.png


So distributing, I ended up with:
convo2.png


Evaluating the second integral poses no problem for me (although I think the integration will definitely be 'hairy'). I have a problem with the first integral though. How on Earth do I integrate the dirac delta? Help! I am totally at a loss here. Or am I doing something wrong?
 

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What is your problem? The integral should be the value of the integrand (without the delta function) at t' = 0.
 
mathman said:
What is your problem? The integral should be the value of the integrand (without the delta function) at t' = 0.
sin(wt) right?

But I did partial fraction expansion using algebra software (mathcad) and the answer was different numerically (I mean, I numerically integrated that function, taking into account that the first integral is sin(w*t)
 
You might want half that value if your integration border is right where the delta is.
It depends on how the delta appears in the Laplace transform.
 
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