Starting with the definition of the Dirac delta function,

[skrtic]

Homework Statement

Starting with the definition of the Dirac delta function, show that [itex] \delta( \sqrt{x}) [/itex]um... i have looked in my book and looked online for a problem like this and i really have no clue where to start. the only time i have used the dirac delta function is in an integral with another function and never with it in this form, only like delta(x-a).looking for guidance.

 

[Mark44]

Homework Statement

Starting with the definition of the Dirac delta function, show that [itex] \delta( \sqrt{x}) [/itex]

Show that [itex]\delta( \sqrt{x}) [/itex] is what?

 

[skrtic]

sorry about that.

equal to 0

 

[Mark44]

How is the Dirac delta function defined? The problem asks you to start with this definition.

 

[skrtic]

from what i get out of the text. it is what replaces an inner product that vanishes if x doesn't equal x'.

I also know that it's integral is unity.

 

[Avodyne]

[tex]\int_{-\infty}^{+\infty}dx\,f(x)\delta(x)=f(0)[/tex]

Now, if only the argument of the delta function was [itex]\sqrt{z}[/itex] instead of [itex]x[/itex] ... hmmm, how could we make this happen?

 

[skrtic]

let x= [itex] \sqrt{z} [/itex] then dx=(1/2)z^(-1/2)dz

and we get [itex] \int_{-\infty}^{\infty}dz z^{-1/2} f(z^{-1/2})\delta(z^{-1/2}) [/itex]

 

[skrtic]

is this a valid assumption

treat [itex] \delta(\sqrt{x}) [/itex] as [itex] \delta(\sqrt{x}-0) [/itex]

 

[Hurkyl]

Aside: I know the hand-wavy argument is clear, but does anyone know of a source that actually defines the result of composing a distribution with a function of some sort?

 

[skrtic]

so assuming i made the substitution right and that was what i was supposed to do, i still don't see how i proved that it equals zero.

is that supposed to say if i plug in zero for x i get zero?

and even if i treat [itex]
\delta(\sqrt{x})
[/itex] as [itex]
\delta(\sqrt{x}-0)
[/itex]
i still don't have a function to plug zero into.

i am still quite confused about what i am trying to do.

 

[George Jones]

Aside: I know the hand-wavy argument is clear, but does anyone know of a source that actually defines the result of composing a distribution with a function of some sort?

Section 7.4.d Composition of [itex]\delta[/itex] with a function, from Mathematics for Physics and Physicists by Walter Appel.

The idea, as usual, is to to use the distribution obtained by integrating a locally integrable function against test functions to motivate a more general definition.

1. Use locally integrable [itex]g[/itex] and integration to generate a distribution [itex]G[/itex].

2. Use locally integrable [itex]g \circ f[/itex] and integration to generate a distribution denoted by [itex]G \circ f[/itex].

3. If [itex]x[/itex] is the integration variable in 2., make the substitution [itex]y = f(x)[/itex].

4. Relate [itex]G \circ f[/itex] to [itex]G[/itex].

5. Use 4. to motivate the definition of [itex]T \circ f[/itex] in tems of an arbitrary distribution [itex]T[/itex] and differentiable and bijective function [itex]f[/itex].

If I get time tomorrow, I might type in the details.