Rational function

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.

The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

A function ${\displaystyle f}$ is called a rational function if it can be written in the form

where ${\displaystyle P}$ and ${\displaystyle Q}$ are polynomial functions of ${\displaystyle x}$ and ${\displaystyle Q}$ is not the zero function. The domain of ${\displaystyle f}$ is the set of all values of ${\displaystyle x}$ for which the denominator ${\displaystyle Q(x)}$ is not zero.

However, if ${\displaystyle \textstyle P}$ and ${\displaystyle \textstyle Q}$ have a non-constant polynomial greatest common divisor ${\displaystyle \textstyle R}$, then setting ${\displaystyle \textstyle P=P_{1}R}$ and ${\displaystyle \textstyle Q=Q_{1}R}$ produces a rational function

which may have a larger domain than ${\displaystyle f}$, and is equal to ${\displaystyle f}$ on the domain of ${\displaystyle f.}$ It is a common usage to identify ${\displaystyle f}$ and ${\displaystyle f_{1}}$, that is to extend "by continuity" the domain of ${\displaystyle f}$ to that of ${\displaystyle f_{1}.}$ Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions ${\displaystyle \textstyle {\frac {A(x)}{B(x)}}}$ and ${\displaystyle \textstyle {\frac {C(x)}{D(x)}}}$ are considered equivalent if ${\displaystyle A(x)D(x)=B(x)C(x)}$. In this case ${\displaystyle \textstyle {\frac {P(x)}{Q(x)}}}$ is equivalent to ${\displaystyle \textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.}$

A proper rational function is a rational function in which the degree of ${\displaystyle P(x)}$ is less than the degree of ${\displaystyle Q(x)}$ and both are real polynomials, named by analogy to a proper fraction in ${\displaystyle \mathbb {Q} .}$

In complex analysis, a rational function