# Dirac Delta Function

1. Jan 15, 2010

### E92M3

1. The problem statement, all variables and given/known data
Evaluate the following integrals:
$$\int^{+\infty}_{-\infty}\delta[f(x)]dx$$
and
$$\int^{+\infty}_{-\infty}\delta[f(x)]g(x)dx$$

2. Relevant equations
$$\int^{+\infty}_{-\infty}\delta(x)dx=1$$
$$\int^{+\infty}_{-\infty}\delta(x)f(x)dx=f(0)$$
$$\int^{+\infty}_{-\infty}\delta(x-a)f(x)dx=f(a)$$

3. 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?

Last edited: Jan 15, 2010
2. Jan 15, 2010

### Hurkyl

Staff Emeritus
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

3. Jan 16, 2010

### E92M3

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?

4. Jan 17, 2010

### Hurkyl

Staff Emeritus
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.

5. Jan 17, 2010

### 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.

6. Jan 17, 2010

### Hurkyl

Staff Emeritus
(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?

7. Jan 17, 2010

### E92M3

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

8. Jan 17, 2010

### Hurkyl

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

9. Jan 17, 2010

### 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.
$$\int_{f(-\infty)}^{f(+\infty)}\frac{\delta(u)du}{f'[f^{-1}(u)]}=\frac{1}{f'[f^{-1}(u')]}|_{u'=0}$$

Last edited: Jan 17, 2010
10. Jan 17, 2010

### Hurkyl

Staff Emeritus
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)