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# Questions tagged [dirac-delta]

This tag is for questions involving the Dirac delta function, either in the informal sense, or in the distribution sense. The Dirac delta function is a mathematical construct which is called a generalized function or a distribution and was originally introduced by the British theoretical physicist Paul Dirac.

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### Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation

$y'' + y =\mu_\pi \big(t\big)$ $y''+y= \delta (x- \pi )$ wih initial conditions: $y \big(0\big) =0$ $y' \big(0\big) =0$ It is obvious to me that the first equation is a Heaviside distribution ...
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### Is $\int^{\infty}_{-\infty}\delta(x-x_0)f(x) \, dx = f(x_0)$ sufficient to define delta distributions?

Most of the sources start introductory section of the delta distributions by defining \begin{eqnarray} \delta(x-x_0)&=&\begin{cases} \infty, & \text{if $x=x_0$}.\\ 0, & \text{...
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### Proof with Dirac delta function

I have to prove that Dirac delta function $\delta$ is equal to this limit: $$\delta(x) = \lim_{a\to \infty}\frac{sin(ax)}{\pi x}.$$ We can say, show that the result of the right side of equality is ...
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### Any way to rewrite/simplify $\int dp ~f(p)~\delta[c-g(p)-E(p)]$?

Im thinking of using $$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$ but in my case $$\int dp ~f(p)~\delta[c-g(p)-E(p)]$$ the functions $f$ and $E$ are basically unknown, while $g$ is a ...
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### proof of Delta dirac sifting [closed]

I need to proof this expression : $\int_{-\infty}^\infty \delta(x-a)f(x) \, dx=f(a)$ Starting with this one: $\int_{-\infty}^\infty \delta(x)f(x) \, dx=f(0)$ Thanks in advance
The wikipedia page on the Dirac delta function offers the following heuristic definition of the Dirac delta function: $$\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$$...