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 . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . 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 .

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

Definitions

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

:$f(x) = \frac$

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

However, if $\textstyle P$ and $\textstyle Q$ have a non-constant polynomial greatest common divisor $\textstyle R$, then setting $\textstyle P=P_1R$ and $\textstyle Q=Q_1R$ produces a rational function

:$f_1(x) = \frac,$

which may have a larger domain than $f$, and is equal to $f$ on the domain of $f.$ It is a common usage to identify $f$ and $f_1$, that is to extend "by continuity" the domain of $f$ to that of $f_1.$ Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions $\textstyle \frac$ and $\textstyle \frac$ are considered equivalent if $A(x)D(x)=B(x)C(x)$. In this case $\textstyle \frac$ is equivalent to $\textstyle \frac.$

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

Complex rational functions

In complex analysis, a rational function

:$f(z) = \frac$

is the ratio of two polynomials with complex coefficients, where is not the zero polynomial and and have no common factor (this avoids taking the indeterminate value 0/0).

The domain of is the set of complex numbers such that $Q(z)\ne 0$.
Every rational function can be naturally extended to a function whose domain and range are the whole Riemann sphere ( complex projective line).

A complex rational function with degree one is a Möbius transformation.

Rational functions are representative examples of meromorphic functions.

Iteration of rational functions on the Riemann sphere (i.e. a rational mapping) creates discrete dynamical systems.
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