# 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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### The fundamental causal solution $g=\delta(t-r/c)/4\pi r$ satisfies the wave equation $\nabla^2 g - (1/c^2) g_{tt}=-\delta^3(r)\delta(t)$

I am trying to show that the fundamental causal solution g=δ(t-r/c)/4πr satisfies the 3d wave equation (del^2)g - (1/c^2)*gtt=-δ^3(r)δ(t) If I substitute the solution to the wave equation in ...
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### Multiplying the dirac delta distribution by a function

In the theory of distributions, if $H(x)$ is the Heaviside function, we have seen that the distributional derivative of $H$ can be found as follows, for a test-function $\varphi$: \begin{align*} \left\...
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### Dirac delta function composition rule

after reviewing the properties of the Dirac delta function, I have a hard time figuring out one property: the composition rule. Composition rule I understand the integral form comes from a change of ...
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### Application of the Lebesgue integral with the Dirac measure

Say that $(X,\mathcal{F}, μ)$ is a $\sigma$-finite measure space and $f:X→ℝ^+$ is $\mathcal{F}$-measurable and nonnegative. Given the Lebesgue measure $\lambda$ on $(ℝ,\mathcal{B}(ℝ))$, I want to show ...
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### Green's Function for the Laplacian in 3D

Does anyone know where to find a good resource for solving for the Green's Function of the Laplacian in 3D or tips on where to start? $$\nabla^2G(\boldsymbol{x,x_0})=\delta (\boldsymbol{x,x_0})$$