Franklin's Notes

Complex rational functions

Given two polynomials $P,Q\in\mathbb C[x]$ with no common zeros, a rational function may be defined as follows: That is, a rational function is the quotient of two polynomials. Technically, $R$ is defined as a function on the Riemann sphere $\hat{\mathbb C}=\mathbb C\cup{\infty}$, which takes on the value $\infty$ when $Q(x)=0$, and takes on the following value at $\infty$: where $\mathrm{lc}(P),\mathrm{lc}(Q)$ denote the respective leading coefficients of $P$ and $Q$.

We define the order of the zero $z=\alpha$ of the rational function $R=P/Q$ to be the multiplicity with which $(z-\alpha)$ divides $P$, and the order of the poly $z=\alpha$ to be the order of the corresponding zero of $Q/P$, or the multiplicity of $(z-\alpha)$ as a factor of $Q$. To define the order of a zero or pole at $z=\infty$, we may either say that the order of the zero/pole at $z=\infty$ of $R(z)$ is defined as the order of the zero/pole at $z=0$ of $R(1/z)$, or that it is defined as the absolute difference in the degrees of the polynomials $P$ and $Q$, i.e. $|\mathrm{deg}(P)-\mathrm{deg}(Q)|$. A useful fact about rational functions is that they always have the same number of zeros and poles (counted with multiplicity).




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