Rational function

In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. For a single variable x a typical rational function is therefore

f(x) = P(x)/Q(x)

where P and Q are polynomials in x as indeterminate, and Q is not the zero polynomial. A rational expression is a quotient of polynomials, sometimes called an algebraic fraction. 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.

These objects are first encountered in school algebra. In more advanced mathematics they play an important part in ring theory, especially in the construction of finite fields.

The domain of a rational function

A rational function

f(x) = P(x)/Q(x)

is not defined at points a such that Q(a)=0. This is in contrast with polynomials, which do not have restrictions on their domain.

For example, the function

f(x) = 1/(x² + 1),

is defined for all real numbers x, but not for the complex numbers that make the denominator 0, that is, not at x = i and x = −i, where i is the square root of minus one.

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,

$1=(x^{2}-x+2)\sum _{k=0}^{\infty }a_{k}x^{k}$

$1=\sum _{k=0}^{\infty }a_{k}x^{k+2}-\sum _{k=0}^{\infty }a_{k}x^{k+1}+2\sum _{k=0}^{\infty }a_{k}x^{k}.$

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

$1=\sum _{k=2}^{\infty }a_{k-2}x^{k}-\sum _{k=1}^{\infty }a_{k-1}x^{k}+2\sum _{k=0}^{\infty }a_{k}x^{k}.$

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, it follows that

$a_{0}={\frac {1}{2}}$

$a_{1}={\frac {1}{4}}$

$a_{k}={\frac {1}{2}}(a_{k-1}-a_{k-2})\quad for\ k\geq 2.$

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 using partial fraction decomposition we can write any rational function into 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) = 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 though 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 ring. 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

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 are strictly more expressive than polynomials. Care must be taken, however, since small errors in denominators close to zero can cause large errors in evaluation.

See also

• partial fraction decomposition
• partial fractions in integration