Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions.

Definitions

In the case of one variable, "x", a rational function is a function of the form

:$f(x)\; =\; frac\{P(x)\}\{Q(x)\}$

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

An irrational function is a function that is not rational. That is: it cannot be expressed as a ratio of two polynomials.

If "x" is not variable, but rather an indeterminate, one talks about "rational expressions" instead of rational functions. The distinction between the two notions is important only in abstract algebra.

A "rational equation" is an equation in which two rational expressions are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.

Examples

The rational function $f(x)\; =\; frac\{x^3-2x\}\{2(x^2-5)\}$ is not defined at $x^2=5\; leftrightarrow\; x=pm\; sqrt\{5\}$.

The rational function $f(x)\; =\; frac\{x^2\; +\; 2\}\{x^2\; +\; 1\}$ is defined for all real numbers, but not for all complex numbers, since if "x" were the square root of $-1$ (i.e. the imaginary unit) or its negative, then formal evaluation would lead to division by zero: $frac\{i^2\; +\; 2\}\{i^2\; +\; 1\}\; =\; frac\{-1\; +\; 2\}\{-1\; +\; 1\}\; =\; frac\{1\}\{0\}$, which is undefined.

The limit of the rational function $f(x)\; =\; frac\{x^3-2x\}\{2(x^2-5)\}$ as x approaches infinity is $frac\{x\}\{2\}$.

A constant function such as "f"("x") = π is a rational function since constants are polynomials. Although "f"("x") is irrational for all "x", note that what is rational is the function, not necessarily the values of the function.

Taylor series

The coefficients of a Taylor series of any rational function satisfy a linear recurrence relation, which can be found by setting the rational function equal to its Taylor series and collecting like terms.

For example,

:$frac\{1\}\{x^2\; -\; x\; +\; 2\}\; =\; sum\_\{k=0\}^\{infty\}\; a\_k\; x^k.$

Multiplying through by the denominator and distributing,

:$1\; =\; (x^2\; -\; x\; +\; 2)\; sum\_\{k=0\}^\{infty\}\; a\_k\; x^k$

:$1\; =\; sum\_\{k=0\}^\{infty\}\; a\_k\; x^\{k+2\}\; -\; sum\_\{k=0\}^\{infty\}\; a\_k\; x^\{k+1\}\; +\; 2sum\_\{k=0\}^\{infty\}\; a\_k\; x^k.$

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

:$1\; =\; sum\_\{k=2\}^\{infty\}\; a\_\{k-2\}\; x^k\; -\; sum\_\{k=1\}^\{infty\}\; a\_\{k-1\}\; x^k\; +\; 2sum\_\{k=0\}^\{infty\}\; a\_k\; x^k.$

Combining like terms gives

:$1\; =\; 2a\_0\; +\; (2a\_1\; -\; a\_0)x\; +\; sum\_\{k=2\}^\{infty\}\; (a\_\{k-2\}\; -\; a\_\{k-1\}\; +\; 2a\_k)\; x^k.$

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

:$a\_0\; =\; frac\{1\}\{2\}.$

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

:$a\_1\; =\; frac\{1\}\{4\}$

:$a\_\{k\}\; =\; frac\{1\}\{2\}\; (a\_\{k-1\}\; -\; a\_\{k-2\})quad\; for\; k\; ge\; 2.$

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 rational function as a sum of factors of the form "1 / (ax + b)" and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.

Complex analysis

In complex analysis, a rational function

:$f(z)\; =\; frac\{P(z)\}\{Q(z)\}$

is the ratio of two polynomials with complex coefficients, where "Q" is not the zero polynomial and "P" and "Q" have no common factor (this avoids "f" taking the indeterminate value 0/0). The domain and range of "f" are usually taken to be the Riemann sphere, which avoids any need for special treatment at the poles of the function (where "Q"("z") is 0).

The "degree" of a rational function is the maximum of the degrees of its constituent polynomials "P" and "Q". If the degree of "f" is "d" then the equation

:$f(z)\; =\; w\; ,$

has "d" distinct solutions in "z" except for certain values of "w", called "critical values", where two or more solutions coincide. "f" can therefore be thought of as a "d"-fold covering of the "w"-sphere by the "z"-sphere.

Rational functions with degree 1 are called "Möbius transformations" and are automorphisms of the Riemann sphere. Rational functions are representative examples of meromorphic functions.

Abstract algebra

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, a rational expression is a class representative of an equivalence class of formal quotients of polynomials, where "P"/"Q" is equivalent to "R"/"S", for polynomials "P", "Q", "R", and "S", when "PS" = "QR".

Applications

These objects are first encountered in school algebra. In more advanced mathematics they play an important role in ring theory, especially in the construction of field extensions. They also provide an example of a "nonarchimedean field" (see Archimedean property).

Rational functions are used in numerical analysis for interpolation and approximation of functions, for example the Padé approximations introduced by Henri Padé. Approximations in terms of rational functions are well suited for computer algebra systems and other numerical software. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.

ee also

*Partial fraction decomposition
*Partial fractions in integration