53 Rational functions

Let $g(x)$ and $h(x)$ be complex polynomials, with $h(x)$ not the zero polynomial. Then

is said to be a rational function. The set of all complex rational functions in $x$ is denoted $𝐂(x)$, with the usual operations of multiplication, addition, division and differentiation.

(i) If the degree of $g(x)$ is less than or equal to the degree of $h(x)$, then $f(x)$ is said to be proper rational. If the degree of $g(x)$ is strictly less than the degree of $h(x)$, then $f(x)$ is said to be strictly proper.

(ii) Suppose that $g(x)$ and $h(x)$ have no common factors other than constants. Then zeros of $g(x)$ give zeros of $f(x)$; while zeros of $h(x)$ give poles of $f(x)$.