Proving properties of the Dirac delta function δ(y)

[deah]

please help me how to
Establish the ff. properties of δ(y).

(a) δ(y) = δ(-y)
(b) δ(y) = δ'(y)
(c) yδ(y)= 0
(d) δ(ay)= 1/a δ(y)
(e) δ(y²-a²) = [1/(2a)] [δ(y-a)+δ(y+a)]
(g) δ(y) δ(y-a) = f(a) δ(y-a)
(h) yδ'(y) = -δ(y)

thanks..

 

[cristo]

You need to show some work for this sort of question and should, in future, post in the homework and coursework questions forum.

What are you denoting by delta? Do you have a definition to work from?

 

[nicktacik]

I may be wrong, but according to a) and b) the only possibility is $$\delta (y) = 0$$. (Which is consistent with c-h).

 

[George Jones]

I may be wrong, but according to a) and b) the only possibility is $$\delta (y) = 0$$. (Which is consistent with c-h).

No, $\delta$ is the Dirac delta "function" (distribution, actually), and this thread is in the wrong forum. It should be in either Advanced Physics or Calculus & Beyond, depending on the course for which deah received this as an assigned question.

deah, how would you start a demsonstration of any of these properties?

 

[nicktacik]

No, $\delta$ is the Dirac delta "function" (distribution, actually), and this thread is in the wrong forum. It should be in either Advanced Physics or Calculus & Beyond, depending on the course for which deah received this as an assigned question.

deah, how would you start a demsonstration of any of these properties?

I was not aware that $$\delta (y) = \delta '(y)$$

 

[George Jones]

I was not aware that $$\delta (y) = \delta '(y)$$

There must be a typo in (b); note also the typo in (g).

 

[ChaoticOrder]

First you have to choose a sequence of functions to work with whose limit is the dirac delta. For instance the sequence

y(x,n) = n^2 x + n for -1/n < x < 0
-n^2 x + n for 0 < x < 1/n
0 otherwise

Then delta(x) = lim(y(x,n),n->infinity)

Use this sequence to prove the properties.