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If a function is analytic in a bounded domain then it is bounded. True or false. Why.

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if it's analytic in the bounded domain then it is analytic where????? – Ittay Weiss Feb 13 '13 at 10:07
@IttayWeiss corrected. thanks. – John Feb 13 '13 at 10:12
not quite Ittaly ;) – Ittay Weiss Feb 13 '13 at 10:12

Hint: Think about the function $f(z) = \dfrac1z$.

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Can a function $f$ be analytic inside a bounded domain $\Omega$ and still have a pole $z_0$ on the boundary $\partial \Omega$ of $\Omega$?

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No, those are the kind of functions in which you can set the radius of convergence to the distance to the bound of the domain. A not so trivial example is $\log(\cos(z))$. It is analytic in the disc $D(0;R=\pi/2)$, but it's not bounded in that domain (it has an asymptote when $z$ goes to $\pi/2$ on the real axis.

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