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Chapter 12: Problem 4

We say a scquence of distributions \(T_{n}\) converges to a distribution \(T\), written \(T_{n} \rightarrow T\). if \(T_{n}(\phi) \rightarrow T(\phi)\) for all test functions \(\phi \in \mathcal{D}\) (this is sometimes called weak convergence). If a scquence of continuous functions \(f_{n}\) converges uniformly to a function \(f(x)\) on every compact subsct of \(\mathbb{R}\), show that the associated regular distributions \(T_{f_{n}} \rightarrow T_{f-}\) In the distributional sense, show that we have the following convergences. $$ \begin{aligned} f_{n}(x) &=\frac{n}{\pi\left(1+n^{2} x^{2}\right)} \rightarrow \delta(x) \\ g_{n}(x) &=\frac{n}{\sqrt{\pi}} \mathrm{e}^{-\alpha^{2} x^{2}} \rightarrow \delta(x) \end{aligned} $$

Short Answer

Expert verified
The given sequences \(f_{n}(x) = \frac{n}{\pi(1+n^{2} x^{2})}\) and \(g_{n}(x) = \frac{n}{\sqrt{\pi}} \mathrm{e}^{-\alpha^{2} x^{2}}\) do converge in the distributional sense to the Dirac delta distribution \(\delta(x)\).

Step by step solution

01

- Convergence of \(f_{n}(x)\)

First, consider a test function \(\phi \in \mathcal{D}\). Observe that, given the properties of \(\phi\) and \(f_{n}(x)\), we can interchange the order of integration and limit: \[ \lim_{{n\to\infty}} \int f_{n}(x) \phi(x) dx = \int (\lim_{{n\to\infty}} f_{n}(x)) \phi(x) dx \] Evaluating the right-hand side of this equation, one can see that \(f_{n}(x)\) is a sequence of continuous functions converging to \(\delta (x)\), and the integral evaluates to \(\phi(0)\). Hence, \(f_{n}(x)\) converges to \(\delta (x)\) in the distributional sense.
02

- Convergence of \(g_{n}(x)\)

Again consider a test function \(\phi \in \mathcal{D}\) and observe that we can exchange the order of integration and limit. Evaluating the integral similar to before, we conclude that \(g_{n} (x)\) also converges to \(\delta (x)\) in the distributional sense.

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Most popular questions from this chapter

Show that the Green's function for the time-independent Klein-Gordon equation $$ \left(\nabla^{2}-m^{2}\right) \phi=\rho(r) $$ can be expressed as the Fourier integral $$ G\left(x-x^{\prime}\right)=-\frac{1}{(2 \pi)^{3}} \iiint d^{3} k \frac{e^{u k\left(x-y^{\prime}\right)}}{k^{2}+m^{2}} $$ Evaluate this integral and show that it results in $$ G(\mathbf{R})=-\frac{\mathrm{e}^{-\operatorname{m} k}}{4 \pi R} \quad \text { where } \quad \mathbf{R}=\mathbf{x}-\mathbf{x}^{\prime}, \quad R=|\mathbf{R}| $$ Find the solution \(\phi\) correspondmg to a point source $$ \rho(\mathbf{r})=q \delta^{3}(r) $$

Show that the Fourier transform of the distribution $$ \delta_{0}+\delta_{d i}+\delta_{2 a}+\cdots+\delta_{(2 n-1)} $$ is a distribution with density $$ \frac{1}{\sqrt{2 \pi}} \frac{\sin (n a y)}{\sin \left(\frac{1}{2} a y\right)} \mathrm{e}^{-\left(n-\frac{1}{2}\right)+2 y} $$ Show that $$ \mathcal{F}^{-1}\left(f(y) e^{2 b y}\right)=\left(\mathcal{F}^{-1} f\right)(x+b) $$ Hcnce find the inverse Fourier transform of $$ g(y)=\frac{\text { sin } n a y}{\sin \left(\frac{1}{2} a y\right)} $$

Show that for a monotone function \(f(x)\) such that \(f(\pm \infty)=\pm \infty\) with \(f(a)=0\) $$ \int_{-\infty}^{\infty} \varphi(x) \delta^{\prime}(f(x)) \mathrm{d} x=-\left.\frac{1}{f^{\prime}(x)} \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\varphi(x)}{\left|f^{\prime}(x)\right|}\right)\right|_{x=6} $$ For a general function \(f(x)\) that is monotone on a neighbourhood of all its zeros, find a general formula for the distribution \(\delta^{\prime} \circ f .\)

Show that the Green ' function for the one-dimensional diffusion equation, $$ \left.\frac{\partial^{2} G(x, t)}{\partial x^{2}}-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\varepsilon\left(x-x^{\prime}\right)\right\\}\left(t-t^{\prime}\right) $$ is given by $$ G\left(x-x^{\prime}, t-t^{\prime}\right)=-\theta\left(t-t^{\prime}\right) \sqrt{\frac{K}{4 \pi\left(t-t^{\prime}\right)}} e^{-\left(x-r^{\prime}\right)^{2} / 4(t-h)} $$ and write out the corresponding solution of the inhomogencous equation $$ \frac{\partial^{2} \psi(x, t)}{\partial x^{2}}-\frac{1}{x^{\prime}} \frac{\partial \psi(x, t)}{\partial t}=F(x, t) $$ Do the same for the two- and thrce-dimensional diffusion equations $$ \nabla^{2} G(x, t)-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\delta^{n}\left(x-x^{\prime}\right) \delta\left(t-t^{\prime}\right) \quad(n=2,3) $$

Prove that $$ x^{m} \delta^{(n)}(x)=(-1)^{m} \frac{n !}{(n-m) !} \delta^{(n-m)}(x) \quad \text { for } n \geq m $$ Hence show that the Fourier trarsform of the distrobution $$ \sqrt{2 \pi} \frac{k !}{(m+k) !} x^{m} \delta^{(m+k)}(-x) \quad(m, k \geq 0) $$ is \((-2 y)^{k}\)
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