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comment If f has an essential singularity, does $P\circ f$ have the same type of singulaity?
You're welcome. If you need some more help with that, just ask here again.
Nov
21
comment If f has an essential singularity, does $P\circ f$ have the same type of singulaity?
Use that $P$ is a continous function.
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21
awarded  Teacher
Nov
21
comment If f has an essential singularity, does $P\circ f$ have the same type of singulaity?
The last part isn't, it would mean $z_{0}$ is a pole. Also $z_{0}$ is not in the domain of $f$, since it's an isolated singularity, so you have to take the disc without the center. Casorati-Weierstrass Theorem says that essential singularitys are the only ones satisfying that property, so you are done.
Nov
21
answered If f has an essential singularity, does $P\circ f$ have the same type of singulaity?