Rational function

For two functions $f(x)$ and $g(x)$, a rational function is a function:

$$h(x)=\frac{f(x)}{g(x)}$$

If the denominator and numerator are polynomial functions, the roots of the denominator are the poles of $h(x)$. The roots of the numerator are called the zeroes of $h(x)$. Especially in circuit analysis, the poles and zeroes are collectively called the critical frequencies.

For a proper¹ rational function, $h(x)$ can always be decomposed with partial fraction decomposition with numerators of degree 0, called the residues of $h(x)$.

Footnotes

1. Where the degree of the numerator is less than that of the denominator. ↩