Distribution questions

ej · **Joined:** Wed, 29 Dec 2004 21:23:09 UTC **Posts:** 120

Hi, have two questions

I wanna calculate $f\delta'$ where

is a function of x and
$\delta$ is the dirac function. $\varphi$ is a testfunction

$f\delta' = \ <f\delta',\varphi> \ = \ <\delta',f\varphi> \ = \ -<\delta,(f\varphi)'> \ = \ \int_{-\infty}^{\infty} \delta(x)(f(x)\varphi(x))',dx \\$

Im not really sure how to proceed from here, should I use partial integration? Ive tried it but not sure.

Next question

I want to derivate f(x) = abs(x)

as a distribution

$<f',\varphi> \ = \ -<f,\varphi'> \ = \ -\int_{-\infty}^{\infty} f(x)\varphi(x)',dx \ = \ (Partial int.) \ = \ \int_{-\infty}^{\infty} f(x)'\varphi(x),dx$

Then I simply put in the values of f(x)

and I get

$\int_{0}^{\infty}\varphi(x),dx - \int_{-\infty}^{0}\varphi(x),dx \ = \ 2H$ where H is the heavieside function

But the answer says 2H - 1

ej · **Joined:** Wed, 29 Dec 2004 21:23:09 UTC **Posts:** 120

Question 1 solved

nirax · **Joined:** Mon, 24 Aug 2009 11:40:23 UTC **Posts:** 3

ej wrote:

Hi, have two questions

I wanna calculate $f\delta'$ where

is a function of x and
$\delta$ is the dirac function. $\varphi$ is a testfunction

$f\delta' = \ <f\delta',\varphi> \ = \ <\delta',f\varphi> \ = \ -<\delta,(f\varphi)'> \ = \ \int_{-\infty}^{\infty} \delta(x)(f(x)\varphi(x))',dx \\$

Im not really sure how to proceed from here, should I use partial integration? Ive tried it but not sure.

Next question

I want to derivate f(x) = abs(x)

as a distribution

$<f',\varphi> \ = \ -<f,\varphi'> \ = \ -\int_{-\infty}^{\infty} f(x)\varphi(x)',dx \ = \ (Partial int.) \ = \ \int_{-\infty}^{\infty} f(x)'\varphi(x),dx$

Then I simply put in the values of f(x)

and I get

$\int_{0}^{\infty}\varphi(x),dx - \int_{-\infty}^{0}\varphi(x),dx \ = \ 2H$ where H is the heavieside function

But the answer says 2H - 1

for the second part.

see carefully what is the value of $- \int_{-\infty}^{0}\varphi(x),dx$ .. this integral is same as $\int_{-\infty}^{+\infty}(H-1).\varphi(x),dx$ and not $\int_{-\infty}^{+\infty}H.\varphi(x),dx$ as you are claiming. in fact you have done all the work ... just you have to look carefully.

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