# Dirac Delta Function

Gold Member
Homework Statement:
Dirac Delta Properties
Relevant Equations:
$$I = \int_{∞}^{∞}dxf(x)δ((x - x_1)(x-x_2)) = ?$$
If the question was
$$\int_{∞}^{∞}dxf(x)δ((x - x_1)) = ?$$ The answer would be ##f(x_1)##

So the delta function has two roots, I searched the web and some books but I am not sure what approach should I use here. I guess theres sometihng happens when ##x_1 = -x_2##. So I am not sure what to do at this point.

Last edited:

## Answers and Replies

Gold Member
Notice that I is not the delta function ( which is usually considered a distribution and not a function) but an expression involving the delta function or distribution. What are the arguments , if any, given for the expression I?

Gold Member
Notice that I is not the delta function ( which is usually considered a distribution and not a function) but an expression involving the delta function or distribution. What are the arguments , if any, given for the expression I?
There arent any. We need to find I

Staff Emeritus
Gold Member
$$\int_{-∞}^{∞}dxf(x)δ((x - x_1)(x-x_2)) = ?$$

This is a special case of
$$\int_{-\infty}^{\infty} dx f \left(x\right) \delta \left( g \left(x \right) \right).$$

Have you seen this before?

Gold Member
This is a special case of
$$\int_{-\infty}^{\infty} dx f \left(x\right) \delta \left( g \left(x \right) \right).$$

Have you seen this before?
Yes I have seen it.
$$\int_{-\infty}^{\infty} dx f \left(x\right) \delta \left( g \left(x \right) \right) = \Sigma f(x_i)/g'(x_i)$$ ? But I am not sure why this is the case

Staff Emeritus
Gold Member
Yes I have seen it.
$$\int_{-\infty}^{\infty} dx f \left(x\right) \delta \left( g \left(x \right) \right) = \Sigma f(x_i)/g'(x_i)$$ ? But I am not sure why this is the case

Then, it might be good to do your original question as an illustrative example. Shift your coordinates such that the new coordinate system has its zero halfway between ##a_1## and ##a_2##.

Homework Helper
Gold Member
2021 Award
Yes I have seen it.
$$\int_{-\infty}^{\infty} dx f \left(x\right) \delta \left( g \left(x \right) \right) = \Sigma f(x_i)/g'(x_i)$$ ? But I am not sure why this is the case

Perhaps that should be ##|g'(x_i)|##?

Gold Member
Perhaps that should be ##|g'(x_i)|##?
Yes I was being lazy to put those sign.