Distributions and Intergration by Parts

[PBTR3]

"Distributions" and Intergration by Parts

Homework Statement

Has it been proven that it is ok to use Integration by parts on "Distributions" like Dirac Delta functions inside an integration?

Homework Equations

Need to figure out how to write integral signs and Greek alphabet symbols with this Linux system.

The Attempt at a Solution

I do not know enough about "distributions" to even attempt this.

 

[nickjer]

Why would you need to use integration by parts. Taking integrals with dirac delta functions is very easy. There would be no need for it. You can integrate any function like this:

$$\int f(x) \delta(x-a) dx = f(a)$$

assuming the bounds ran through x=a.

 

[PBTR3]

In order to prove various identities like delta prime (x) = - delta prime (-x) the instructions in the book I am using says that most identities are proved by integration by parts. But my question is how do I know that integration by parts is even applicable to a distribution function under an integral sign?

 

[nickjer]

One way to prove things like that is to use the Gaussian representation of the delta function with the limit to make it into a delta function. Then just take the limit outside of the integral and perform all of your integration tricks on the Gaussian function. Once you are satisfied with what you have, bring the limit back inside.

 

[PBTR3]

I will try taking the limit after the integration.
Now if I can just prove all these identities using Gaussian functions and parts, then limits.
Thank you!
PBTR3

 

[nickjer]

Also, http://mathworld.wolfram.com/DeltaFunction.html has an equation (Eq. 10) that looks a lot like integration by parts. Unfortunately, they don't prove it. But at least you know it is possible.

 

[George Jones]

Do either

$$\int^\infty_{-\infty} f \left(x\right) \delta' \left(x\right) dx$$

or

$$\int^\infty_{-\infty} \delta \left(x\right) f' \left(x\right) dx$$

formally by parts, and see what happens.