# Rational function

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

P(x)/Q(x)

where P and Q are polynomials in x as indeterminate, and Q isn't the zero polynomial. Any non-zero polynomial Q is acceptable; but the possibility that a given value a assigned to x could make

Q(a) = 0

means that rational functions, unlike polynomials, do not always have an obvious function domain of definition. In fact if we take

1/(x2 + 1),

this function is everywhere defined for real numbers x; but not for complex numbers, where the denominator is 0 at x = i and x = −i, for i the square root of minus one.

From a mathematical point of view, a polynomial is firstly a formal expression, and only secondly a function (on some given domain). Despite the name, the same is equally true of rational functions. In abstract algebra a definition of rational function is given as element of the fraction field of a polynomial ring. For this definition to succeed, we must start with an integral domain R (for example, a field). Then

R[X, Y,..., T],

the ring of polynomials in some indeterminates X, ..., T, will also be an integral domain; and we can properly take a fraction field. (In greater generality for commutative rings the construction will be a localization of a polynomial ring.)

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.

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