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Monday, December 5, 2016

The complex sequence does not converge uniformly

Prove that there does not exist a sequence $\{ p_n(z)\}_{n \in \mathbb{N}}$ of complex polynomials such that $p_n(z) \rightarrow \frac{1}{z}$ uniformly on $\mathcal{C}_R=\{ z \in \mathbb{C} \mid \left| z \right| = R\}$.

Solution

 If such sequence existed then the convergence on the compact set $\left| z \right| = r$ would be uniform. However,

$$2 \pi i = \oint \limits_{\mathcal{C}_R} \frac{{\rm d}z}{z} = \oint \limits_{\mathcal{C}_R} \lim_{n \rightarrow +\infty} p_n (z) \, {\rm d}z = \lim_{n \rightarrow +\infty} \oint \limits_{\mathcal{C}_R} p_n (z) \, {\rm d}z = \lim_{n \rightarrow +\infty} 0 = 0$$

since the polynomials have no poles inside the given contour. This finishes the exercise.

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