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I need to show that $$\lim_{R \to \infty}\int_{\gamma}\frac{1}{q(z)}dz=0$$ using the estimation lemma.

How would I show the above? I think to approach this i need to change $q(z)$ into only terms of $z$ and substitute it into the estimation lemma, but how would I do that? Thanks.

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up vote 1 down vote accepted

We have $$ \left| \int_\gamma 1/q(z) \right|< \frac{2 \pi R}{ \min \limits_{|z|=R}|q(z)| }.$$ Now, $|q(z)|\ge R^2-|a|R-|b|>R^2/2 $ for very large values of $R$. Thus, $\left|\int_\gamma 1/q(z) \right|< 4\pi/R\to 0$.

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