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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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### How to show this property of the delta function?

Let $\mathcal{D}(\mathbb{R})$ be the space of test-functions in $\mathbb{R}$ and let $f$ be a $C^\infty$ function. I want to show that if $f$ has $n$ zeroes $x_1,\dots,x_n$ in the interval where it is ...
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### Laplace Transform of Dirac Delta function

I've seen everywhere that that the Laplace Transform of Dirac Delta function is: $$L[\delta(t-a)] = e^{-sa} \text{ when } a > 0$$ But they never explain what happens when $a < 0$. Can I assume ...
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### Fourier representation of complex Dirac function

I have confusion on the Fourier representation of complex Dirac function, recently. As $t$ is real value, we have \begin{align} \delta(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{iwt}\text{d}w \end{...
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### Identifying $\int_{-\infty}^\infty e^{i k x} dx$ as Dirac delta distribution

The expression $\int_{-\infty}^\infty e^{i k x} dx$ is sometimes identified as the Dirac delta function. This identification is said "formal" or "symbolic", and some physics texts say that the theory ...
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### Interchanging differentiation and expectation

I have a nonnegative random variable $X$ with $E[X] < \infty$, that admits a density wrt to the Lebesgue measure. For arbitrary $K > 0$, I write $$P(K) = E[\max(X-K,0)]$$ I am interested in the ...
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### Simplify an integration

Can the following integral be reduced to simpler terms? $$\int_{-1}^0\mathrm{d}x_1 \int_{0}^1\mathrm{d}x_2 \int_{-1}^1\mathrm{d}x_3\, \delta(x_1+x_2+x_3) \exp(a_1x_1 + a_2x_2 + a_3x_3 + cx_3^2)$$ ...
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### Problem defining a function via Step function and Dirac's Delta

First: I know Dirac's Delta isn't a function and hence shouldn't be treated like one. But this arose in a physics textbook so I'm looking for an answer that oversees that. Consider the following ...
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### What is the integral of $\int_{-\infty}^{+\infty} H(t)\delta(t)dt$ ($H(t)$ Heaviside step, $\delta(t)$ Dirac delta)?

I was trying to figure out what is the integral of $$\int_{-\infty}^{+\infty} H(t)\delta(t)dt,$$ where $H(t)$ is the Heaviside step and $\delta(t)$ is the Dirac delta. A first approach: We ...
### Fourier transform of $t^2$ discrepancy
I encountered a discrepancy when taking the fourier transform of $t^2$ that I don't understand. I would expect $\mathcal{F}[t^2]$, $$\mathcal{F}[t^2]=\int_{-\infty}^\infty t^2 e^{-i \omega t}dt$$ to ...