Limit of an Integral of a $C^\infty$-Smooth Function with Compact Support
Let $\varphi:\R\rightarrow\R$ be a $C^\infty$-smooth function with compact support. Prove that the following limit exists, and compute the limit.
$$\lim_{\varepsilon\rightarrow0+} \int_{-\infty}^{\infty}\frac{\varphi(x)}{x+i\varepsilon} \;\mathrm{d}x , i=\sqrt{-1}$$
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