Prove a delta function identity

In summary, to prove f(x)delta(g(x)) = f(x) delta (x-x0)/abs(g'(x)), we use the definition of the Dirac delta function and the substitution u = g(x) to rewrite the integral and arrive at the desired result.
  • #1
Yaelcita
14
0
Hi,

I'm stuck with the last proof I need to do

Homework Statement



I need to prove that f(x)delta(g(x)) = f(x) delta (x-x0)/abs(g'(x))
By delta I mean the Dirac delta function here. (I'm new to this forum, so i don't know how to write it all nicely like so many of you do!)

Homework Equations





The Attempt at a Solution



First of all, of course, I put all of that in an integral. And what I've done so far is just replace g(x) = u so that dx=du/g'(x) It looks promising because I have the g'(x) dividing everything, and if I could somehow use the identity that says that delta(kx) = delta/abs(k) I could almost have it but it seems to me that that identity only works for k a constant and g'(x) is not. Also, I have no idea what to do with my f(x). Can I express it as f(g^-1 (x))? It seems just very weird to me. So, basically I'm stuck there. Any help would be great!
 
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  • #2
To prove this, we will use the definition of the Dirac delta function, which states that for any continuous function f(x):\int_{-\infty}^{\infty} f(x) \delta(g(x)) \ dx = f(x_0)where x_0 is the only point where g(x) = 0. We can rewrite the integral as:\int_{-\infty}^{\infty} f(x) \delta(g(x)) \ dx = \int_{-\infty}^{\infty} f(x) \frac{\delta(x-x_0)}{|g'(x)|} \ dxUsing the substitution u = g(x), we have \int_{-\infty}^{\infty} f(x) \frac{\delta(x-x_0)}{|g'(x)|} \ dx = \int_{-\infty}^{\infty} f(x) \frac{\delta(u)}{|g'(x)|} \frac{du}{g'(x)} By the definition of the Dirac delta function, we have \int_{-\infty}^{\infty} f(x) \frac{\delta(u)}{|g'(x)|} \frac{du}{g'(x)} = f(x_0)Thus, f(x) \delta(g(x)) = f(x) \frac{\delta(x-x_0)}{|g'(x)|} as required.
 

Related to Prove a delta function identity

1. What is a delta function identity?

A delta function identity is a mathematical expression that relates the Dirac delta function (also known as the delta function) to other mathematical functions. It is used to simplify calculations and solve certain problems in various fields of science and engineering.

2. How do you prove a delta function identity?

To prove a delta function identity, you must use the properties of the delta function, such as linearity, scaling, and shifting, to manipulate the expression into a form that is equivalent to the desired identity. This often involves using integration techniques and algebraic manipulation.

3. What are some common delta function identities?

Some common delta function identities include the sifting property, which states that the delta function picks out the value of a function at a specific point, and the derivative property, which relates the delta function to the derivative of a function.

4. Why are delta function identities important in science?

Delta function identities are important in science because they allow us to solve complex problems and model physical phenomena using mathematical tools. They are particularly useful in fields such as physics, engineering, and signal processing.

5. Are there any limitations to using delta function identities?

Yes, there are limitations to using delta function identities. They are only applicable in certain situations where the delta function can accurately represent a physical system or phenomenon. Additionally, care must be taken when using delta function identities, as they can sometimes lead to incorrect or nonsensical results if used improperly.

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