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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Dirac Delta distribution (function) confusion

Given this identity $\delta(kx)=\frac{1}{|k|}\delta(x)$ is the following correct? $$\delta(ct-x)=\delta(c(t-x/c))=\frac{1}{|c|}\delta(t-x/c)$$ It seems like the impulses would be located at the same ...
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Derivative of Dirac delta of f(x)

I'm trying to prove this relation: $(\delta (f(x)))' = f'(x) \delta' (f(x))$, where $f(x)$ is a monotone function. I just end up tangled in different derivatives of Dirac delta function or ...
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What is the best way to find the Inverse Laplace with Convolution Theorem? [closed]

I have to calculate the inverse laplace using Convolution, but I get stuck when I have to integrate with the delta dirac. $$F(s)=\frac{s}{s^{2}+2s+2}$$
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Why is the covariance of white noise not finite and not a Kronecker delta?

Suppose we have a Gaussian white noise $f(t)$. Then the covariance $\langle f(t)f(t')\rangle = c\times\delta(t - t')$ for some $c > 0$. However, since $f(t)$ is a Gaussian random variable (with ...
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Two dimensional integral involving Dirac delta

It seems to me that $$\int_{-\infty}^\infty \int_{-\infty}^{\infty} \delta(x^2 + y^2 - R^2) dx dy$$ should evaluate to $2\pi R$, the perimeter of a circle of radius $R$, but I'm having trouble ...
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Writing a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$?

How to write a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$? For example; I read something in a book. You can find the following picture. But ...
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Solve the differential equation $R*C*y'(t)+y(t)=x(t)$ using Convolution

I have the next differential equation: $R*C*y'(t)+y(t)=x(t)$ Where $x(t)$ is a piecewise funtion defined by: \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} x(t)...
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How boundary conditions for a green function is same a original function

I am trying to study green's function.suppose we have a differential equation, $D_x f(x) =g(x)$ , We also provided with some boundary conditions for $f(x)$, But we supply this boundary conditions in ...
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Distributional derivative of a function with jumps

Let $f(x)$ be a real function of a single variable x that is continuously differentiable on the real line except at $x=a$ where $f$ has a jump discontinuity. Let's suppose also that left- and right-...
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Continuity of solutions to ordinary differential equations under a limit

Consider $\eta(x;\delta) \in C^\infty$ a family of functions parameterized by $\delta$ with domain $x \in [0,1]$ such that, \begin{align} \lim_{\delta \to 0} \eta(x;\delta) &= 0 \qquad x < R \\...
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Question about $\delta\big(f(x)\big) = \frac{\delta(x)}{\left|f'(x) \right|}$ (that is-non discrete zeros in f(x))

The following holds for Dirac delta function where $f(x)$ has discrete zeros at $a_i$ $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$ See: Dirac Delta ...
I have great difficulties understanding why the following relation holds: $$\int^{t}_{0} \int^{t’}_{0}\delta(t_1-t_2)\mathrm{d}t_1\mathrm{d}t_2=\min\{t,t’\}.$$ Our teacher gave us an explanation ...
Show that $\displaystyle x\frac{d}{dx}(\delta(x))=-\delta(x)$. My attempt: Using the product rule of differentiation, \$\displaystyle\frac{d}{dx}(x\delta(x))=x\frac{d}{dx}(\delta(x))+\delta(x)\cdot ...