Rational Functions

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

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 $\mathbb$.

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

:$f(z) = w \,$

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

:$f(x) = \frac$

is not defined at

:$x^2=5 \Leftrightarrow x=\pm \sqrt.$

It is asymptotic to $\tfrac$ as $x\to \infty.$

The rational function

:$f(x) = \frac$

is defined for all real numbers, but not for all complex numbers, since if ''x'' were a square root of $-1$ (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:

:$f(i) = \frac = \frac = \frac,$

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 $f(x) = P(x)$ is a rational function with $Q(x) = 1.$ A function that cannot be written in this form, such as $f(x) = \sin(x),$ is not a rational function. However, the adjective "irrational" is not generally used for functions.

The rational function $f(x) = \tfrac$ 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,

:$\frac = \sum_^ a_k x^k.$

Multiplying through by the denominator and distributing,

:$1 = (x^2 - x + 2) \sum_^ a_k x^k$

:$1 = \sum_^ a_k x^ - \sum_^ a_k x^ + 2\sum_^ a_k x^k.$

After adjusting the indices of the sums to get the same powers of ''x'', we get

:$1 = \sum_^ a_ x^k - \sum_^ a_ x^k + 2\sum_^ a_k x^k.$

Combining like terms gives

:$1 = 2a_0 + (2a_1 - a_0)x + \sum_^ (a_ - a_ + 2a_k) x^k.$

Since this holds true for all ''x'' in the radius of convergence of the original Taylor series, we can compute as follows. Since the constant term on the left must equal the constant term on the right it follows that

:$a_0 = \frac.$

Then, since there are no powers of ''x'' on the left, all of the coefficients on the right must be zero, from which it follows that

:$a_1 = \frac$

:$a_k = \frac (a_ - a_)\quad \text\ k \ge 2.$

Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using partial fraction decomposition we can write any proper rational function as a sum of factors of the form and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.

Abstract algebra and geometric notion

In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting given a field ''F'' and some indeterminate ''X'', a rational expression is any element of the field of fractions of the polynomial ring ''F'' 'X'' Any rational expression can be written as the quotient of two polynomials ''P''/''Q'' with ''Q'' ≠ 0, although this representation isn't unique. ''P''/''Q'' is equivalent to ''R''/''S'', for polynomials ''P'', ''Q'', ''R'', and ''S'', when ''PS'' = ''QR''. However, since ''F'' 'X''is a unique factorization domain, there is a unique representation for any rational expression ''P''/''Q'' with ''P'' and ''Q'' polynomials of lowest degree and ''Q'' chosen to be monic. This is similar to how a fraction of integers can always be written uniquely in lowest terms by canceling out common factors.

The field of rational expressions is denoted ''F''(''X''). This field is said to be generated (as a field) over ''F'' by (a transcendental element) ''X'', because ''F''(''X'') does not contain any proper subfield containing both ''F'' and the element ''X''.

Complex rational functions

Julia set