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.
Definitions
A function is called a rational function if and only if it can be written in the form
where and are polynomial functions of and is not the zero function. The domain of is the set of all values of for which the denominator is not zero.
However, if and have a non-constant polynomial greatest common divisor , then setting and produces a rational function
which may have a larger domain than , and is equal to on the domain of It is a common usage to identify and , that is to extend "by continuity" the domain of to that of Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions and are considered equivalent if . In this case is equivalent to .
A proper rational function is a rational function in which the degree of is less than the degree of and both are real polynomials, named by analogy to a proper fraction in .
Degree
There are several non equivalent definitions of the degree of a rational function.
Most commonly, the degree of a rational function is the maximum of the degrees of its constituent polynomials and , when the fraction is reduced to lowest terms. If the degree of is , then the equation
has distinct solutions in except for certain values of , called critical values, where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decrease after having cleared the denominator).
In the case of complex coefficients, a rational function with degree one is a Möbius transformation.
The degree of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.
In some contexts, such as in asymptotic analysis, the degree of a rational function is the difference between the degrees of the numerator and the denominator.
In network synthesis and network analysis, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a .
Examples
The rational function
is not defined at
It is asymptotic to as
The rational function
is defined for all real numbers, but not for all complex numbers, since if x were a square root of (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:
which is undefined.
A constant function such as f(x) = π is a rational function since constants are polynomials. The function itself is rational, even though the value of f(x) is irrational for all x.
Every polynomial function is a rational function with A function that cannot be written in this form, such as is not a rational function. However, the adjective "irrational" is not generally used for functions.
The rational function is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1.
Taylor series
The coefficients of a Taylor series of any rational function satisfy a linear recurrence relation, which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting like terms after clearing the denominator.
For example,
Multiplying through by the denominator and distributing,