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Stuck on a proof by induction that involves the dirac delta function

I'm taking Advanced Mathematics Physics without having any previous experience in physics, which is fine since my math is advanced enough. However I got stock on this problem. I'm not asking you guys to do my homework, just maybe some help to get me going and finishing this problem, that so far is the only one I can't do.

I have to prove by induction that:

Integral from negative infinity to infinity of [(d^{m}/dx^{m})( δ(x))(f(x))] = (-1)^{m} f^{m}(0)

Now I know that:

δ(x-a) = 0 when x isn't equals to a

the integral of δ(x-a)dx = 1

and the integral of f(x) δ(x-a)dx = f(a)

but I'm a little confused...I think it might be me overthinking the problem, or just not being familiar with this sort of problem.

I'm new to the forums and it might be a lot of me to ask for help without previously posting anything, but I'll do my best to help others in problems I do know.

I attached a picture if that helps

Thank you!

Re: Stuck on a proof by induction that involves the dirac delta function

Quote:

Originally Posted by

**aNxello** I'm taking Advanced Mathematics Physics without having any previous experience in physics, which is fine since my math is advanced enough. However I got stock on this problem. I'm not asking you guys to do my homework, just maybe some help to get me going and finishing this problem, that so far is the only one I can't do.

I have to prove by induction that:

Integral from negative infinity to infinity of [(d^{m}/dx^{m})( δ(x))(f(x))] = (-1)^{m} f^{m}(0)

Now I know that:

δ(x-a) = 0 when x isn't equals to a

the integral of δ(x-a)dx = 1

and the integral of f(x) δ(x-a)dx = f(a)

but I'm a little confused...I think it might be me overthinking the problem, or just not being familiar with this sort of problem.

I'm new to the forums and it might be a lot of me to ask for help without previously posting anything, but I'll do my best to help others in problems I do know.

I attached a picture if that helps

Thank you!

Hi aNxello! :)

Can you apply integration by parts to your formula?

Let's start by applying it once.

You can use $\displaystyle u = f(x)$ and $\displaystyle dv = \frac{d^m}{dx^m} \delta(x) dx$.

Re: Stuck on a proof by induction that involves the dirac delta function

Ok let me try, I'll put up what I got in a second!

Re: Stuck on a proof by induction that involves the dirac delta function

would this mean that v = 1. since δ(x-a)dx = 1?

or do I have to do a second integration by part there?

Re: Stuck on a proof by induction that involves the dirac delta function

Quote:

Originally Posted by

**aNxello** would this mean that v = 1. since δ(x-a)dx = 1?

No, since we're not talking about the integral of δ(x)dx, but we're talking about the integral of $\displaystyle \frac{d^m}{dx^m}\delta(x)$.

It is $\displaystyle v = \frac{d^{m-1}}{dx^{m-1}}\delta(x)$.

Generally, if you differentiate first and then integrate, you get the original function (plus an optional integration constant).

Re: Stuck on a proof by induction that involves the dirac delta function

Alright! it's making sense to me now, I had some notes from last class that didn't click, but now I'm linking them with this. Thank you so much for your help!