Starting with the definition of the Dirac delta function,

[skrtic]

Homework Statement

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

Show that \delta( \sqrt{x}) 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]

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

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

 

[skrtic]

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

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

 

[skrtic]

is this a valid assumption

treat \delta(\sqrt{x}) as \delta(\sqrt{x}-0)

 

[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 <br /> \delta(\sqrt{x}) <br /> as <br /> \delta(\sqrt{x}-0) <br />
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 \delta 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 g and integration to generate a distribution G.

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

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

4. Relate G \circ f to G.

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

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