Paper 4, Section I, D

Methods | Part IB, 2019

Let

gϵ(x)=2ϵxπ(ϵ2+x2)2.g_{\epsilon}(x)=\frac{-2 \epsilon x}{\pi\left(\epsilon^{2}+x^{2}\right)^{2}} .

By considering the integral ϕ(x)gϵ(x)dx\int_{-\infty}^{\infty} \phi(x) g_{\epsilon}(x) d x, where ϕ\phi is a smooth, bounded function that vanishes sufficiently rapidly as x|x| \rightarrow \infty, identify limϵ0gϵ(x)\lim _{\epsilon \rightarrow 0} g_{\epsilon}(x) in terms of a generalized function.

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