# 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 write $\delta (f(x))$ in terms of $\delta (x)$?

I've seen this identity in my electrodynamics book: $$\delta (f(x))=\sum_i{ \frac{1}{|{df\over dx}(x_i)|}\delta (x-x_i)}$$ Where $x_i$ shows the $i$th zero of $f(x)$. How can I prove it? I've tried ...
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### delta function on a complex number

for a real number we know that $$f(a)= \int_{-\infty}^{\infty}dx \delta (x-a)f(x)$$ but what happens for $$\int_{-\infty}^{\infty}dx \delta (x-2i)f(x)$$ ? is this equal to $f(2i)$ or equal ...
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### Integral in 3 dimensions

I am trying to integrate $$\iiint \delta(|\mathbf r| -R)\:\mathrm{d}^{3}\mathbf{r}$$ I know that $\int f(r) \delta(r-R) d^3 \mathbf r =f(R)$, but when I try to apply this here I end up ...
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### Laplace transform of the derivative of the dirac delta function times another function

I'm trying to solve a DE involving terms of the form $\dot{\delta}(t-k)f(t)$ and $\ddot{\delta}(t-k)f(t)$, where $k>0$. I would therefore like to find the Laplace transform of these terms. My ...
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### A representation of Dirac-$\delta$

Prove that $$g_\epsilon (x)=\lim_{\epsilon \to 0} \frac1 \epsilon \frac1 \pi e^{-x^2/\epsilon^2}$$ is a Dirac-$\delta$ function. This is a homework question I'm stuck with. I'm probably missing a ...
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### Closed form of multi-variable integral with Dirac delta function

I want to give you this exercise to see if there are simpler solutions to the one that I've chosen! I found it a very interesting exercise. Question Find the closed form for all N of following ...
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### Proof of Convolution Theorem for three functions, using Dirac delta

I am stuck on proving the convolution theorem for the product of three functions using the Dirac delta function. Please excuse any nonstandard notation--I am a physics major who has not been formally ...
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### A Property of the Dirac Delta Function

I can't seem to prove the following property of te $\delta$- function. Please help.
### Traveling delta function $\delta(x - ct)$ as a distributional solution of the wave equation
I'm trying to show that a delta function $\delta(x - ct)$ is a distributional solution of the PDE $$D_{(0,2)}u(x, t) = c^2 D_{(2,0)}u(x,t).$$ Here $D_{(i,j)}$ means $i$-th partial differentiation on ...