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Theorem III of Townshends Complex Analysis has a proof that rational functions have no essential singularities.

Just wondering, are there any particularly elegant proofs that a rational function has no essential singularities?

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    Do you pdf file of this book?2015-11-29

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Since any rational function is of the form $f(z)=\frac{p(z)}{q(z)}$, where $p,q$ are polynomials, the only singularities of $f(z)$ are the zeroes of $q(z)$ and $\infty$. $q(z)$ is a polynomial, hence all it's zeros are of finite order - so they are poles of that order (assuming $f$ is reduced, i.e. $p,q$ have no common zeroes). Depending on the degrees of $p,q$ it is easy to check that $\lim_{z\to\infty}f(z)$ is either $0$, $\infty$ or the quotient of the leading coefficients of $p,q$. In any case it exists, so $\infty$ can not be essential singularity.