Evaluating the Second Term of Integration by Parts for $\delta (x)$

In summary, using integration by parts and properties of the Dirac delta function, it can be shown that x \frac{d(\delta (x))}{dx} = -\delta (x) where \delta (x) is a Dirac delta function. This can be verified through proper symbol manipulation and evaluating the integral.
  • #1
Reshma
749
6
Show that [tex]x \frac{d(\delta (x))}{dx} = -\delta (x)[/tex]
where [itex]\delta (x)[/itex] is a Dirac delta function.

My work:

Let f(x) be a arbitrary function. Using integration by parts:
[tex]\int_{-\infty}^{+\infty}f(x)\left (x \frac{d(\delta (x))}{dx}\right)dx = xf(x)\delta (x)\vert _{-\infty}^{+\infty} - \int_{-\infty}^{+\infty}d\left (\frac{xf(x) \delta (x)}{dx}\right)dx[/tex]

The first term is zero, since [itex]\delta (x) = 0 [/itex]
at [itex]-\infty, +\infty[/itex].
How is the second term evaluated?
 
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  • #2
Well, if we are to do this symbol manipulation properly, you should have:
[tex]\int_{-\infty}^{\infty}xf(x)\delta{'}dx=xf(x)\delta\mid_{-\infty}^{\infty}-\int_{-\infty}^{\infty}(f(x)+xf'(x))\delta(x)dx=-(f(0)+0*f'(0))=-f(0)[/tex]
 
  • #3
Wow, thanks Arildno! I got it!
 

1. What is the purpose of evaluating the second term of integration by parts for $\delta (x)$?

Evaluating the second term of integration by parts for $\delta (x)$ allows us to solve integrals involving the Dirac delta function. It is a useful tool in many areas of physics and engineering.

2. How do you evaluate the second term of integration by parts for $\delta (x)$?

To evaluate the second term of integration by parts for $\delta (x)$, we use the formula: $\int f(x) \delta (x) dx = f(0)$. This means that the value of the function $f(x)$ at $x=0$ is the integral of $f(x)$ multiplied by the Dirac delta function.

3. Can the second term of integration by parts for $\delta (x)$ be applied to any function?

No, the second term of integration by parts for $\delta (x)$ can only be applied to functions that are continuous at $x=0$. If a function is discontinuous at $x=0$, the second term cannot be evaluated.

4. Is there a specific technique for evaluating the second term of integration by parts for $\delta (x)$?

Yes, there is a specific technique for evaluating the second term of integration by parts for $\delta (x)$. This involves finding the derivative of the function in the integral, evaluating it at $x=0$, and then multiplying it by the integral of the function.

5. What are some applications of evaluating the second term of integration by parts for $\delta (x)$?

Evaluating the second term of integration by parts for $\delta (x)$ has many applications in physics and engineering. It is commonly used in solving differential equations, calculating moments of inertia, and determining the impulse response of a system.

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