Rational function

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In mathematics, a rational function is any function whose output can be given by a formula that is the ratio of two polynomials. ("Rational function" suggests a function of rational numbers onto the set of rational numbers. A constant function such as f(x) = π may be a "rational function" in this context even though it is in fact irrational. "Polynomial ratio" or "polynomial quotient" are more descriptive terms than the standard usage.) For a function of one variable, x, any rational function can be expressed as

f(x) = P(x)/Q(x)
Rational function of degree 2 : y = (x²-3x-2)/(x²-4)
Enlarge
Rational function of degree 2 :
y = (x²-3x-2)/(x²-4)

where P and Q are polynomials in x and Q is not the zero polynomial. The domain of f does not contain any number a for which Q(a) = 0.

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 role in ring theory, especially in the construction of field extensions. They also provide an example of a nonarchimdean field" (see Archimedean property) and an alternative construction for hyperreal number systems used in infinitesimal calculus and nonstandard analysis.

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[edit] Examples

Rational function of degree 3 : y = (x^3-2x)/(2(x^2-5))
Enlarge
Rational function of degree 3 :
y = (x^3-2x)/(2(x^2-5))

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 plus or minus the square root of negative one formal evaluation would lead to division by zero.

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

[edit] 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, 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] See also

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