Evaluating Dirac Delta Integrals: Homework Statement

[E92M3]

Homework Statement

Evaluate the following integrals:
\int^{+\infty}_{-\infty}\delta[f(x)]dx
and
\int^{+\infty}_{-\infty}\delta[f(x)]g(x)dx

Homework Equations

\int^{+\infty}_{-\infty}\delta(x)dx=1
\int^{+\infty}_{-\infty}\delta(x)f(x)dx=f(0)
\int^{+\infty}_{-\infty}\delta(x-a)f(x)dx=f(a)

The Attempt at a Solution

Part a:
\int^{+\infty}_{-\infty}\delta[f(x)]dx
let:
u=f(x)
du=f'(x)dx
\int^{+\infty}_{-\infty}\delta[f(x)]dx=\int^{+\infty}_{-\infty}\frac{\delta(u)du}{f'(x)}=\int^{+\infty}_{-\infty}\frac{\delta(u)du}{f'[f^{-1}(u)]}=\frac{1}{f'[f^{-1}(0)]}
Is this correct?

 

[Hurkyl]

I don't see

1. How you computed the new limits of integration after substituting
2. How you determined f was invertible
3. How you determined f was differentiable

 

[E92M3]

I don't see

1. How you computed the new limits of integration after substituting
2. How you determined f was invertible
3. How you determined f was differentiable

All three were not specified, I just assumed. Usually when we get questions like these the functions are assumed to be invertible and differentiable. I just copied the question verbatim. Without those assumptions I can't go anywhere... or can I?

 

[Hurkyl]

The first one was specified: the original limits of integration were (-\infty, +\infty). When you did the substitution, it looks like you transformed the integrand but forgot to transform the limits.

What to do if f is not differentiable? I dunno.

What to do if f is not invertible? This one should be manageable -- in fact, the identity you are tasked to derive is often stated for the non-invertible case! (with an extra assumption on f')


It's a shame your text doesn't make some statement someplace about what assumptions it's making on functions, either in the exercise or somewhere in an introductory chapter.

 

[E92M3]

Well... This is the first homework of the intro E&M class, not math class... so I guess I can't really blame anyone for not making statements.

Ok, now for the limits:
If I don't know the function f(x), then how can I set the limits for the integration? Where do you suggest that I look? I looked the the wiki page for the dirac delta but doesn't really help.

 

[Hurkyl]

(assuming of differentiable and invertible...)

You did everything else about the problem without "knowing" the function f(x) -- what is it about the limits that's giving you trouble?

 

[E92M3]

Well I let u=f(x), so how can I know how f(x) behave at at x=infinity?

 

[Hurkyl]

What would you do if you did know f(x)?

 

[E92M3]

The new limits of integration.
\int_{f(-\infty)}^{f(+\infty)}\frac{\delta(u)du}{f'[f^{-1}(u)]}
Now, assuming that u or for that matter f(x) has places which equal zero between the limits of the integral above, the delta function picks out the terms that u=f(x)=0.
\int_{f(-\infty)}^{f(+\infty)}\frac{\delta(u)du}{f'[f^{-1}(u)]}=\frac{1}{f'[f^{-1}(u')]}|_{u'=0}

 

[Hurkyl]

That looks reasonable. Now, we did make use of the assumption f is invertible -- because you used a substitution. (Also, if f is invertible, its graph cannot cross the x-axis twice)

But you forgot one thing (albeit an unfortunately common one) -- while Dirac delta does satisfy

\int_{-3}^{7} \delta(x) f(x) \, dx = f(0)​

it does not satisfy

\int_{4}^{-2} \delta(x) f(x) \, dx = f(0)​

...




If f is not invertible, you still seem to have some idea about what the integrand could be. But substitution only works on domains where f is invertible! Can you think of a way to proceed? (You'll probably have to make an additional assumption on f -- so figure out an idea, and then figure out what you assumed to use your idea)