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Let $R \in \mathbb C$. Is it possible to show $ \lim_{R \to \infty} \int_0^{2\pi} \frac{id\theta}{\sqrt{1+R^{-2}e^{-2i\theta}}}=\int_0^{2\pi} id\theta $ using the dominated convergence theorem?

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    The modulus is between $1-|R|^{-1}$ and $1+|R|^{-1}$, uniformly on $\theta$ hence yes, the limit of the integral exists and it is $2\pi i$.2012-09-25

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