Is this Dirac delta function integral correct?

In summary: Jason!In summary, the problem is to integrate the expression $\int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx$ with the given conditions and the correct result is $\sum_{N=0}^n (2N+1)$. Although most of the steps in the conversation are correct, the final step should use the fact that $δ(sin(x))= \sum_{zeros}^. \frac{1}{|Cos(nπ)|} δ(x-nπ)$ and the fact that the integral of $δ(x)$ is equal to 1. It is also mentioned that this problem is unrelated to the problem mentioned in the first post
  • #1
JorgeM
30
6
I have to integrate this expression so I started to solve the delta part from the fact that when n=0 it results equals to 1.
And the graph is continuous in segments I thought as the sumation of integers
$$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx $$

From the fact that actually
$$ δ(sin(x))= \sum_{zeros}^. \frac{1}{|Cos(nπ)|} δ(x-nπ)$$
An in the integral

$$ \int_{-(n+1/2)π}^{(n+1/2)π} \sum_{zeros}^. \frac{1}{|Cos(nπ)|} δ(x-nπ)$$

For n=0
$$ \int_{-(1/2)π}^{(1/2)π} \sum_{zeros}^. \frac{1}{|Cos(nπ)|} δ(x-nπ) = 1 $$

But taking a look on the graph of 1/|Cos(x)| and the fact that it is continuous in segments
for n=0 ,1 segment
n=1, 3 segments
n=2, 5 segments
and each one of these is just like the first whose integral is equal to one.
So the value it just a sumation that depends for the n value.
And the result is
$$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx= \sum_{N=0}^n (2N+1)$$

Has anyone solved it before? Is this a correct way of doing it?
Thanks for your advise.
Jorge M
 
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  • #2
JorgeM said:
$$\int x^2e^x \, dx $$
What does this have to do with the problem in post #1?
 
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  • #3
Mark44 said:
What does this have to do with the problem in post #1?
Nothing really, I figured out how to use Latex, so I started to try to write equations and symbols.
However, I have finally written the equations as good as possible in order to make people easy to understand.
But I do not know if what I have made is well-solved.
Thanks.
 
  • #4
JorgeM said:
$$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx= \sum_{N=0}^n (2N+1)$$

Has anyone solved it before? Is this a correct way of doing it?
Thanks for your advise.
Jorge M
Not quite, although most of your steps are correct. I get,
$$ \begin{eqnarray*}
\int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx & = & \int_{-(n+1/2)π}^{(n+1/2)π} \sum_{m=-\infty}^{\infty} \delta(x-m\pi) \, dx \\
& = & \int_{-(n+1/2)π}^{(n+1/2)π} \sum_{m=-n}^n \delta(x-m\pi) \, dx \\
& = & \sum_{m=-n}^n 1\\
& = & 2n + 1.
\end{eqnarray*}
$$

jason
 
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  • #5
jasonRF said:
Not quite, although most of your steps are correct. I get,
$$ \begin{eqnarray*}
\int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx & = & \int_{-(n+1/2)π}^{(n+1/2)π} \sum_{m=-\infty}^{\infty} \delta(x-m\pi) \, dx \\
& = & \int_{-(n+1/2)π}^{(n+1/2)π} \sum_{m=-n}^n \delta(x-m\pi) \, dx \\
& = & \sum_{m=-n}^n 1\\
& = & 2n + 1.
\end{eqnarray*}
$$

jason

Actually, that is the correct result.
Thanks a lot for your reply
 

Related to Is this Dirac delta function integral correct?

1. What is a Dirac delta function integral?

A Dirac delta function integral is a mathematical tool used to represent a point mass or impulse in a continuous function. It is often used in physics and engineering to describe phenomena such as point charges or point masses.

2. How do you know if a Dirac delta function integral is correct?

A Dirac delta function integral is considered correct if it satisfies the properties of a Dirac delta function. These properties include being zero everywhere except at the point of integration, having an integral of 1, and having a value of infinity at the point of integration.

3. Can a Dirac delta function integral be negative?

No, a Dirac delta function integral cannot be negative. It is defined as being zero everywhere except at the point of integration, and it has a value of infinity at that point. This means that it cannot have a negative value.

4. How is a Dirac delta function integral used in scientific research?

A Dirac delta function integral is used in a variety of scientific research, particularly in physics and engineering. It is used to model point masses or point charges, and it can also be used to solve differential equations. It is a powerful tool for representing and analyzing phenomena that involve point masses or impulses.

5. Are there any limitations to using a Dirac delta function integral?

Yes, there are limitations to using a Dirac delta function integral. It is a mathematical tool and does not have a physical representation, so it may not accurately model real-world phenomena. Additionally, it is often used in idealized situations and may not accurately represent complex systems. It is important to use it carefully and with an understanding of its limitations.

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