Dirac delta integrated from 0 to infinity

In summary, the value of the integral of the Dirac delta function from 0 to infinity, where the function it is integrated with is a constant 1, depends on which direction you approach 0 from. If you approach 0 from the negative side, the answer is 1, while approaching from the positive side gives an answer of 0. This is because the Dirac delta function is symmetric and not a traditional function.
  • #1
Heimisson
44
0
I was wondering if I integrate the dirac delta function from 0 to infinity where the function it's integrated with is the constant 1, will I get 0.5 or 1? And why?

This is not homework so I decided to post this here although I asked this question in class and the teacher (assistant) wasn't too sure. I could ask my professor but he's a scary man and mocks your questions.

So I was hoping someone here could answer this.

thanks
 
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  • #2
Heimisson said:
I was wondering if I integrate the dirac delta function from 0 to infinity where the function it's integrated with is the constant 1, will I get 0.5 or 1? And why?

This is not homework so I decided to post this here although I asked this question in class and the teacher (assistant) wasn't too sure. I could ask my professor but he's a scary man and mocks your questions.

So I was hoping someone here could answer this.

thanks

Depends on which direction you approach 0 from. If your integral is...

[tex]lim_{a\rightarrow{0^-}} \int_a^\infty \delta(x) dx[/tex]

Then the answer is 1. If your integral is...


[tex]lim_{a\rightarrow{0^+}} \int_a^\infty \delta(x) dx[/tex]

Then the answer is 0.
 
  • #3
You get 1/2, Dirac delta is a symmetric function.
 
  • #4
peteratcam said:
You get 1/2, Dirac delta is a symmetric function.

It's not even really a function...
 
  • #5
Char. Limit said:
Depends on which direction you approach 0 from. If your integral is...

[tex]lim_{a\rightarrow{0^-}} \int_a^\infty \delta(x) dx[/tex]

Then the answer is 1. If your integral is...


[tex]lim_{a\rightarrow{0^+}} \int_a^\infty \delta(x) dx[/tex]

Then the answer is 0.

Thanks a lot this makes sense.
 

FAQ: Dirac delta integrated from 0 to infinity

1. What is the Dirac delta function?

The Dirac delta function, denoted as δ(x), is a mathematical construct used in the field of calculus and engineering to represent a point mass or impulse at a specific point. It is defined as zero everywhere except at x = 0, where it is infinitely tall and has an area of 1 under its curve.

2. What does it mean to integrate the Dirac delta function from 0 to infinity?

Integrating the Dirac delta function from 0 to infinity represents the total area under the curve of the function from 0 to infinity. Since the Dirac delta function is zero everywhere except at x = 0, the integral from 0 to infinity is equivalent to the value of the function at x = 0, which is 1.

3. Is the integral of the Dirac delta function from 0 to infinity a finite or infinite value?

The integral of the Dirac delta function from 0 to infinity is a finite value of 1. This is because the Dirac delta function has an area of 1 under its curve at x = 0, and the integral represents the total area under the curve from 0 to infinity.

4. What is the physical significance of integrating the Dirac delta function from 0 to infinity?

The physical significance of integrating the Dirac delta function from 0 to infinity is that it represents the total effect or response of an impulse at x = 0 over an infinite time interval. This is useful in fields such as signal processing and control systems, where the Dirac delta function is used to model sudden changes or impulses in a system.

5. Can the Dirac delta function be integrated from -∞ to ∞?

No, the Dirac delta function cannot be integrated from -∞ to ∞. This is because the Dirac delta function is only defined at x = 0, and it is equal to zero everywhere else. Therefore, the integral from -∞ to ∞ would be equal to 0.

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