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I'm studying for my complex qual and am trying to go through a lot of questions quickly. Let $g(z) = (\overline{z})^2$. The problem is to prove or disprove the existence of a sequence of polynomials which converge uniformly to $g$ on the unit circle $|z| = 1$.

On the unit circle, $g$ is just the restriction of the function $h(z) = \frac{1}{z^2}$. Now $h$ is holomorphic on $\mathbb{C} \setminus \{0\}$. For any point $c \neq 0$, we can represent $h$ as a power series at $c$ with radius of convergence $|c|$, uniformly convergent on any disc of radius $< |c|$. So I can get a sequence of polynomials which converge to $g$ uniformly on just up to half the disc (by taking $c$ very far away from $0$). Not sure what I should do now.

It would be neat to center a power series at infinity somehow, so I could include the whole circle in the area of uniform convergence, but exclude the origin, but I'm not really familiar with complex analysis on $\mathbb{C} \cup \{\infty\}$.

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  • $\begingroup$ look to runge's theorem. $\endgroup$
    – user171326
    Jul 10, 2015 at 21:41

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Hint: What is $\int_C z g(z)\; dz$?

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