How Does Using Dirac in a Double Integral Simplify the Integration Process?

In summary, the conversation discusses the use of the Dirac function in a double integral and whether it simplifies to a single integral. The answer depends on the limits of integration, but in general, it does simplify to a single integral over the Dirac function. However, the possibility of non-convergence should be taken into consideration.
  • #1
ooga59
2
0
Hi,

I have a question about using the dirac function in a double integral.

Lets say you have the double integral over the two values x1 and x2:

int( int( sin(x1) * dirac(x1-x2) * sin(x2) ))

Does this just simplify to a single integral:

int( (sin(x1))^2 )



thanks!
 
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  • #2
Well, that depends. You did not write any limits of integration.
[tex]\int_a^b \delta(x- x1)f(x)dx[/tex]
is equal to f(x1) IF [itex]a\le x1\le b[/itex] and 0 otherwise.
 
  • #3
The limits would be from -inf to inf. Thanks.
 
  • #4
AFAIK the answer is yes. But of course that non converging integral might produce some surprises.
 

Related to How Does Using Dirac in a Double Integral Simplify the Integration Process?

What is Dirac in a double integral?

Dirac in a double integral refers to the use of the Dirac delta function in a double integral. The Dirac delta function is a mathematical function that is used to represent a point mass or spike at a specific point in a function. In a double integral, the Dirac delta function is used to represent a point in a two-dimensional function.

What is the purpose of using Dirac in a double integral?

The purpose of using Dirac in a double integral is to simplify the integration process. The Dirac delta function allows us to replace a two-dimensional integral with a one-dimensional integral, which is often easier to solve. It also helps us to solve problems involving point masses or spikes in a function.

What are the properties of the Dirac delta function?

The Dirac delta function has several properties, including:

  • It is zero everywhere except at x = 0, where it is infinite.
  • The integral of the Dirac delta function over its entire domain is equal to 1.
  • It is an even function, meaning that δ(-x) = δ(x).
  • It has the sifting property, which states that the integral of the product of the Dirac delta function and another function is equal to the value of the other function at the point where δ(x) is located.

How is Dirac in a double integral used in physics?

Dirac in a double integral is commonly used in physics to solve problems involving point charges or point masses. It is also frequently used in quantum mechanics to represent the position of a particle in a wave function. In general, the Dirac delta function is a useful tool for representing and solving problems involving point-like objects in a function or system.

What are some limitations of using Dirac in a double integral?

One limitation of using Dirac in a double integral is that it assumes the point mass or spike in the function is infinitely small and infinitely sharp. In reality, all physical objects have a finite size and cannot be represented as a single point. Additionally, the Dirac delta function is not defined at x = 0, which can sometimes cause issues in calculations.

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