# 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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### Real Analysis question that affects how to think about the Dirac delta function.

Okay, here are the ingredients to this question. Me: 60 years old. 39 years ago I took two semesters of Real Analysis using the Royden textbook. Rusty is an understatement. But I am still quite ...
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### Where is the wild use of the Dirac delta function in physics justfied?

Wikipedia has a wild article about the Dirac delta function. Are the things listed correct? Or is there no proof that they are correct? For my master thesis I want to refer to rigorous proofs of these ...
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### Why do physicists get away with thinking of the Dirac Delta functional as a function?

For instance they use it for finding solutions to things like Poisson's Equation, i.e. the method of Green's functions. Moreover in Quantum Mechanics, it's common practise to think of the delta ...
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### Intuition behind the derivative of dirac delta function

Let me first begin what I mean by saying the intuition behind the " $\delta'(x)$ ". For example the smooth approximations of the delta function looks like the following: (Left:the smooth ...
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### When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to0^+.$$ Obviously, this sequence of functions ...
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### Dirac delta function as a limit of sinc function

I'm looking for a rigorous proof of the statement: $\delta(x) = \lim_{\epsilon->0} \frac{\sin(x/\epsilon)}{\pi x}$ (see (37)). For any non-zero value of x, LHS of the above is by definition zero. ...
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### Dirac delta of a function with zero derivative

It is known that: $$\int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}$$ Where $x_i$ are the roots of $g(x)$. My question is, what happens when $g'(x_i)$ is ...
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### Why is $f(x) \delta(x) = f(0)\delta(x)$ only true when $x=0$?

This is a follow up from a previous question asked by me. I know that $$\delta(x) = \begin{cases} 0 & \space \mathrm{for} \space x \ne 0 \\\infty&\ \mathrm{for} \space x = 0 \end{cases}$$ ...
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### Dirac delta and non-test functions

Normalization of the delta function (distribution) is often informally written as an integral $$\int_{-\infty}^{+\infty} \delta(x) \, dx = 1$$ An attempt to write this formally would be expression ...
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### Dirac delta distribution and sin(x) - what can be a test function?

I read about the Dirac delta distribution some days ago to better understand distributions (or generalized functions), but I've become a bit confused. I used $\delta$ as a "function" ($\delta(x)$) ...
### Formally derive $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$
I have been searching for a derivation of the defining property for the Dirac-delta function: $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$ and found this derivation ...