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

In summary, the conversation revolved around using Laplace transforms to invert an s-function and encountering difficulties with integrating the dirac delta. The solution was suggested to be finding the value of the integral without the delta function at t=0. There was also a mention of using partial fraction expansion and the possibility of needing to halve the value depending on the Laplace transform.
  • #1
maistral
240
17
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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  • #2
What is your problem? The integral should be the value of the integrand (without the delta function) at t' = 0.
 
  • #3
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)
 
  • #4
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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What is the Dirac function?

The Dirac function, also known as the delta function, is a mathematical function that is zero everywhere except at a single point, where it is infinite. It is often used in theoretical physics and engineering to represent a point-like source or an impulse.

What is the integral of the Dirac function?

The integral of the Dirac function is defined as the area under the curve of the function. However, since the function is zero everywhere except at a single point, the integral is equal to zero for all intervals that do not include the point where the function is non-zero.

What is the significance of the integral of the Dirac function from 0 to a specific value?

The integral of the Dirac function from 0 to a specific value represents the total "strength" or "weight" of the function within that interval. It can be interpreted as the sum of all the "impulses" within that interval, with each impulse having a magnitude of infinity at a single point.

Why is the integral of the Dirac function often used in theoretical physics?

The Dirac function is often used in theoretical physics because it allows for the simplification of complex equations and models. It can be used to represent point-like sources or impulses in a system, making it a useful tool in studying physical phenomena.

Can the integral of the Dirac function be negative?

No, the integral of the Dirac function cannot be negative. Since the function is zero everywhere except at a single point, the integral will always either be zero or a positive value. This is because the "area" under the curve of the function cannot be negative.

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