- #1

binbagsss

- 1,269

- 11

## Homework Statement

I am trying to show that ## \int \delta (x-a) \delta (x-c) dx = \delta (-a-c) ## via integeration by parts, but instead I am getting ##\delta (c-a) ## (or ##\delta (a-c)## depending how I go...).

Can someone please help me out where I've gone wrong: struggling to spot it:

## Homework Equations

above.

## The Attempt at a Solution

I am trying to show that ## \int \delta (x-a) \delta (x-c) dx = \delta (-a-c) ## via integeration by parts, but instead I am getting ##\delta (c-a) ## (or ##\delta (a-c) ## depending how I go...).

Can someone please help me out where I've gone wrong: struggling to spot it:

By parts ## dv/dx= \delta (x-c) ##

## v= H(x-c) ##, ##H(x)## the Heaviside function

##u=\delta(x-a) ##

##du/dx=d/dx ( \delta (x-a) ##

Then

##\delta (x-a) \delta (x-c) dx = ^\infty _{-\infty} [ \delta (x-a) H (x-c) ] - \int H(x-c) d/dx (\delta (x-a)) dx ##

First term vanishes,

=## - ^{\infty}_{c} [\delta (x-a) ] = \delta (c-a) ##, if ##c \geq a##

or ##=0## if ##c<a##

The limit change from ##H(x-c) \neq 0 ## for ##x-c > 0##.

I can't spot what's gone wrong.

thanks alot.