Problem 1591 (difficulty: 5/10)

Let \(\displaystyle p(z)=z^n+b_{n-1}z^{n-1}+\dots+b_1z+b_0\) has degree \(\displaystyle n>1\) and no roots in \(\displaystyle |z|>R\). Let \(\displaystyle \displaystyle I(R) = \frac1{2\pi i} \int_{|z|=R}\frac{dz}{p(z)}\). Show that    (a) \(\displaystyle \displaystyle\lim_{R\to\infty}I(R)=0\);    (b) \(\displaystyle I(R)\) is constant.    (c) \(\displaystyle I(R)=0\).

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