If a function is analytic in a bounded domain then it is bounded. True or false. Why.
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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 $\Omega$?
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.