Problem 1727 (difficulty: 6/10)

Let \(\displaystyle \Gamma_{r,R,\varepsilon}\) be the curve in the figure, where \(\displaystyle R\) is large, \(\displaystyle r\) is small and \(\displaystyle \varepsilon\) is much smaller than \(\displaystyle r\). What results form the following limit? \(\displaystyle {\displaystyle\lim_{R\to\infty}\lim_{r\to+0}\lim_{\varepsilon\to+0} \frac1{2\pi i}\int_{\Gamma_{r,R,\varepsilon}} \frac{\log z}{z^2+1}\dz}\)

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