zeros and poles of rational function
A rational function^{} of a complex variable $z$ may be presented by the equation
$R(z)={\displaystyle \frac{{a}_{0}{z}^{m}+{a}_{1}{z}^{m1}+\mathrm{\dots}+{a}_{m}}{{b}_{0}{z}^{n}+{b}_{1}{z}^{n1}+\mathrm{\dots}+{b}_{n}}},$  (1) 
where the numerator and the denominator are mutually irreducible polynomials^{} with complex coefficients^{} ${a}_{j}$ and ${b}_{k}$ (${a}_{0}{b}_{0}\ne 0$). If $z=x+iy$ ($x,y\in \mathbb{R}$), then the real and imaginary parts^{} of $R(z)$ are rational functions of $x$ and $y$.
When we factorize the numerator and the denominator in the ring $\u2102[z]$, we can write
$R(z)={\displaystyle \frac{{a}_{0}{(z{\alpha}_{1})}^{{\mu}_{1}}{(z{\alpha}_{2})}^{{\mu}_{2}}\mathrm{\dots}{(z{\alpha}_{r})}^{{\mu}_{r}}}{{b}_{0}{(z{\beta}_{1})}^{{\nu}_{1}}{(z{\beta}_{2})}^{{\nu}_{2}}\mathrm{\dots}{(z{\beta}_{s})}^{{\nu}_{s}}}},$  (2) 
where ${\alpha}_{j}\ne {\beta}_{k}$ for all $j,k$.
The form (2) of the rational function expresses the zeros ${\alpha}_{j}$ and the infinity places ${\beta}_{k}$ of the function^{}. One can write (2) as
$$R(z)={(z{\alpha}_{j})}^{{\mu}_{j}}{S}_{j}(z)$$ 
where ${S}_{j}(z)$ is a rational function which in $z={\alpha}_{j}$ gets a finite nonzero value. Accordingly one says that the point ${\alpha}_{j}$ is a zero of $R(z)$ with the order ${\mu}_{j}$ ($j=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots},r$). One can also write (2) as
$$R(z)=\frac{1}{{(z{\beta}_{k})}^{{\nu}_{k}}}{T}_{k}(z)$$ 
where ${T}_{k}(z)$ is a rational function getting in the point ${\beta}_{k}$ a finite nonzero value.
As $z\to {\beta}_{k}$, the modulus $R(z)$ increases unboundedly in such a manner that ${z{\beta}_{k}}^{{\nu}_{k}}R(z)$ tends to a finite nonzero limit. So one says that $R(z)$ has in the point ${\beta}_{k}$ a pole with the order ${\nu}_{k}$ ($k=\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots},s$).
Behaviour at infinity
Now let $z$ increase unboundedly. When we write
$$R(z)={z}^{mn}\cdot \frac{{a}_{0}+\frac{{a}_{1}}{z}+\mathrm{\dots}+\frac{{a}_{m}}{{z}^{m}}}{{b}_{0}+\frac{{b}_{1}}{z}+\mathrm{\dots}+\frac{{b}_{n}}{{z}^{n}}},$$ 
we get three cases:

•
If $m>n$, then ${lim}_{z\to \mathrm{\infty}}R(z)=\mathrm{\infty}$. Since ${lim}_{z\to \mathrm{\infty}}\frac{R(z)}{{z}^{mn}}=\frac{{a}_{0}}{{b}_{0}}$ is finite and nonzero, the point $z=\mathrm{\infty}$ is the pole of $R(z)$ with the order $mn$.

•
If $m=n$, we have ${lim}_{z\to \mathrm{\infty}}R(z)=\frac{{a}_{0}}{{b}_{0}}$ and thus $R(z)$ has in the infinity a finite nonzero value.

•
If $$, we have ${lim}_{z\to \mathrm{\infty}}R(z)=0$ in such a manner that ${lim}_{z\to \mathrm{\infty}}{z}^{nm}R(z)=\frac{{a}_{0}}{{b}_{0}}$. This means that $R(z)$ has in infinity a zero with the order $nm$.
In any case, $R(z)$ has equally many zeros and poles, provided that each zero and pole is counted so many times as its order says. The common number of the zeros and poles is called the order of the rational function. It is the greatest of the degrees (http://planetmath.org/PolynomialRing) $m$ and $n$ of the numerator and denominator.
$c$places
Denote by $c$ any nonzero complex number^{}. The $c$place of $R(z)$ means such a point $z$ for which $R(z)=c$. If ${z}_{0}$ is a $c$place of
$$R(z)=\frac{P(z)}{Q(z)}$$ 
where the polynomials^{} $P(z)$ and $Q(z)$ have no common factor (http://planetmath.org/DivisibilityInRings), then ${z}_{0}$ is a zero of
$R(z)c={\displaystyle \frac{P(z)cQ(z)}{Q(z)}}.$  (3) 
If this zero is of order $\mu $, then one says that ${z}_{0}$ is of order $\mu $ as the $c$place of $R(z)$. The numerator and denominator of (3) cannot have common factor (otherwise any common factor would be also a factor of $P(z)$). This implies that the order of the rational function defined by (3) is the same as the order $k$ of $R(z)$. Because (3) gets $k$ times the value $0$, also $R(z)$ gets $k$ times the value $c$. Thus we have derived the
Theorem. A rational function attains any complex value so many times as its order is.
References
 1 R. Nevanlinna & V. Paatero: Funktioteoria. Kustannusosakeyhtiö Otava. Helsinki (1963).
Title  zeros and poles of rational function 
Canonical name  ZerosAndPolesOfRationalFunction 
Date of creation  20140223 18:12:33 
Last modified on  20140223 18:12:33 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  15 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 30D10 
Classification  msc 30C15 
Classification  msc 30A99 
Classification  msc 26C15 
Related topic  MinimalAndMaximalNumber 
Related topic  OrderValuation 
Related topic  RolfNevanlinna 
Related topic  PlacesOfHolomorphicFunction 
Related topic  ZeroOfPolynomial 
Defines  order of rational function 
Defines  order 
Defines  cplace 
Defines  place 