Solve $$\int_{∞}^{∞}dxf(x)\delta((x-x_1))$$: Dirac Delta Function

In summary, the conversation discusses the use of the delta function in integrals, particularly in the case of two roots. The formula for this case is mentioned and the conversation also touches on shifting coordinates for an example. The question of why the formula works is also raised.
  • #1
Arman777
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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 there's sometihng happens when ##x_1 = -x_2##.

\delta \left(x^{2}-\alpha ^{2}\right)={\frac {1}{2|\alpha |}}{\Big [}\delta \left(x+\alpha \right)+\delta \left(x-\alpha \right){\Big ]}.

So I am not sure what to do at this point.
 
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  • #2
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?
 
  • #3
WWGD said:
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
 
  • #4
Arman777 said:
$$\int_{-∞}^{∞}dxf(x)δ((x - x_1)(x-x_2)) = ?$$
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?
 
  • #5
George Jones said:
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
 
  • #6
Arman777 said:
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##.
 
  • #7
Arman777 said:
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)|##?
 
  • #8
PeroK said:
Perhaps that should be ##|g'(x_i)|##?
Yes I was being lazy to put those sign.
 

Related to Solve $$\int_{∞}^{∞}dxf(x)\delta((x-x_1))$$: Dirac Delta Function

1. What is the Dirac Delta Function?

The Dirac Delta Function, denoted as δ(x), is a mathematical function that is defined as 0 for all values of x except for x = 0, where it is infinity. Its integral from negative infinity to positive infinity is equal to 1.

2. What is the purpose of the Dirac Delta Function?

The Dirac Delta Function is used to represent a point-like source or an impulse in physical systems. It is also used to simplify mathematical calculations in fields such as physics, engineering, and signal processing.

3. How is the Dirac Delta Function related to the integral in the given equation?

The integral in the equation is an example of the "sifting property" of the Dirac Delta Function. It states that the integral of a function multiplied by the Dirac Delta Function is equal to the value of the function at the point where the Dirac Delta Function is located. In this case, the integral is equal to f(x1).

4. Can the Dirac Delta Function be graphically represented?

No, the Dirac Delta Function cannot be graphically represented as it is not a traditional function with a defined shape. However, it is often represented as a spike or a vertical line at x = 0 to illustrate its properties.

5. How is the Dirac Delta Function related to the Kronecker Delta Function?

The Dirac Delta Function and the Kronecker Delta Function are both mathematical functions that represent impulses or point-like sources. The main difference is that the Dirac Delta Function is a continuous function while the Kronecker Delta Function is a discrete function. In other words, the Dirac Delta Function can take on any value while the Kronecker Delta Function can only take on the values 0 or 1.

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