- Rational function
mathematics, a rational function is any function which can be written as the ratioof two polynomial functions.
In the case of one variable, "x", a rational function is a function of the form
where "P" and "Q" are
polynomial functionin "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
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.
The rational function is not defined at .
The rational function is defined for all
real numbers, but not for all complex numbers, since if "x" were the square root of (i.e. the imaginary unit) or its negative, then formal evaluation would lead to division by zero: , which is undefined.
The limit of the rational function as x approaches infinity is .
constant functionsuch 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.
The coefficients of a
Taylor seriesof 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.
Multiplying through by the denominator and distributing,
After adjusting the indices of the sums to get the same powers of "x", we get
Combining like terms gives
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 termon the left must equal the constant term on the right it follows that
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
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, a rational function
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
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 algebrathe 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 classof formal quotients of polynomials, where "P"/"Q" is equivalent to "R"/"S", for polynomials "P", "Q", "R", and "S", when "PS" = "QR".
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 analysisfor interpolationand approximationof 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.
Partial fraction decomposition
Partial fractions in integration
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