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I do not understand the statement why $p$ will be polynomial in the following statement:

"The function $p:\hat{\mathbb C}\rightarrow\hat{\mathbb C}$ defined by $p(z)=f(z)q(z)$ has the removable singularities at the poles of $f$ in $\mathbb C$, so it is entire, this it has power series representation on all $\mathbb C$, also $p$ is meromorphic at $\infty$ as both $f$ and $q$ are.this forces $p$ to be a polynomial.Since, $f=\frac pq$" $ q(z)=\prod_{j=1}^{n}(z-z_j)^{e_j}$

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    $F(z)$ is a meromorphic function in extended plane, which has poles at $z_1,\dots z_n$ with multiplicity $e_1,\dots, e_n$2012-08-24

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What does it mean to have a pole at $\infty$? This means that the function has a pole when written in terms of the coordinate $w=\frac1z$ .

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    @Michael: Yes, that is a goo$d$ way to put it. So the fun$c$tion does not go $c$razy.2012-08-23