# Rational function

A rational function is a function $w = R ( z)$, where $R ( z)$ is rational expression in $z$, i.e. an expression obtained from an independent variable $z$ and some finite set of numbers (real or complex) by means of a finite number of arithmetical operations. A rational function can be written (non-uniquely) in the form

$$R ( z) = \frac{P ( z) }{Q ( z) } ,$$

where $P$, $Q$ are polynomials, $Q ( z) \not\equiv 0$. The coefficients of these polynomials are called the coefficients of the rational function. The function $P / Q$ is called irreducible when $P$ and $Q$ have no common zeros (that is, $P$ and $Q$ are relatively prime polynomials). Every rational function can be written as an irreducible fraction $R ( z) = P ( z) / Q ( z)$; if $P$ has degree $m$ and $Q$ has degree $n$, then the degree of $R ( z)$ is either taken to be the pair $( m , n )$ or the number

$$N = \max \{ m , n \} .$$

A rational function of degree $( m , n )$ with $n = 0$, that is, a polynomial, is also called an entire rational function. Otherwise it is called a fractional-rational function. The degree of the rational function $R ( z) \equiv 0$ is not defined. When $m < n$, the fraction $P / Q$ is called proper, and it is called improper otherwise. An improper fraction can be uniquely written as

$$\frac{P}{Q} = P _ {1} + \frac{P _ {2} }{Q} ,$$

where $P _ {1}$ is a polynomial, called the integral part of the fraction $P / Q$, and $P _ {2} / Q$ is a proper fraction. A proper fraction, $R ( z) = P ( z) / Q ( z)$, in irreducible form, where

$$Q ( z) = b _ {0} ( z - b _ {1} ) ^ {n _ {1} } \dots ( z - b _ {l} ) ^ {n _ {l} } ,$$

admits a unique expansion as a sum of simple partial fractions

$$\tag{1 } R ( z) = \sum _ { i= } 1 ^ { l } \frac{c _ {i _ {1} } }{z - b _ {i} } + \dots + \frac{c _ {i _ { n _ i } } }{( z - b _ {i} ) ^ {n _ {i} } } .$$

If $P ( x) / Q ( x)$ is a proper rational function with real coefficients and

$$Q ( x) =$$

$$= \ b _ {0} ( x - b _ {1} ) ^ {l _ {1} } \dots ( x - b _ {r} ) ^ {l _ {r} } ( x ^ {2} + p _ {1} x + q _ {1} ) ^ {t _ {1} } \dots ( x ^ {2} + p _ {s} x + q _ {s} ) ^ {t _ {s} } ,$$

where $b _ {0} \dots b _ {r} , p _ {1} , q _ {1} \dots p _ {s} , q _ {s}$ are real numbers such that $p _ {j} ^ {2} - 4 q _ {j} < 0$ for $j = 1 \dots s$, then $P ( x) / Q ( x)$ can be uniquely written in the form

$$\tag{2 } \frac{P ( x) }{Q ( x) } = \ \sum _ { i= } 1 ^ { r } \left [ \frac{c _ {i _ {1} } }{x - b _ {i} } + \dots + \frac{c _ {i _ { l _ i } } }{( x - b _ {i} ) ^ {l _ {i} } } \right ] +$$

$$+ \sum _ { j= } 1 ^ { s } \left [ \frac{D _ {j _ {1} } x + E _ {j _ {1} } }{x ^ {2} + p _ {j} x + q _ {j} } + \dots + \frac{D _ {j _ { t _ j } } x + E _ {j _ { t _ j } } }{( x ^ {2} + p _ {j} x + q _ {j} ) ^ {t _ {j} } } \right ] ,$$

where all the coefficients are real. These coefficients, like the $c _ {ij}$ in (1), can be found by the method of indefinite coefficients (cf. Undetermined coefficients, method of).

A rational function of degree $( m , n )$ in irreducible form is defined and analytic in the extended complex plane (that is, the plane together with the point $z = \infty$), except at a finite number of singular points, poles: the zeros of its denominator and, when $m > n$, also the point $\infty$. Note that if $m > n$, the sum of the multiplicities of the poles of $R$ is equal to its degree $N$. Conversely, if $R$ is an analytic function whose only singular points in the extended complex plane are finitely many poles, then $R$ is a rational function.

The application of arithmetical operations (with the exception of division by $R ( z) \equiv 0$) to rational functions again gives a rational function, so that the set of all rational functions forms a field. In general, the rational functions with coefficients in a field form a field. If $R _ {1} ( z)$, $R _ {2} ( z)$ are rational functions, then $R _ {1} ( R _ {2} ( z) )$ is also a rational function. The derivative of order $p$ of a rational function of degree $N$ is a rational function of degree at most $( p + 1 ) N$. An indefinite integral (or primitive) of a rational function is the sum of a rational function and expressions of the form $c _ {r} \mathop{\rm log} ( z - b _ {r} )$. If a rational function $R ( x)$ is real for all real $x$, then the indefinite integral $\int R ( x) d x$ can be written as the sum of a rational function $R _ {0} ( x)$ with real coefficients, expressions of the form

$$c _ {i _ {1} } \mathop{\rm log} | x - b _ {i} | ,\ \ M _ {j} \mathop{\rm log} ( x ^ {2} + p _ {j} x + q _ {j} ) ,$$

$$N _ {j} \mathop{\rm arctan} \frac{2 x + p _ {j} }{\sqrt {4 q _ {j} - p _ {j} ^ {2} } } ,\ i = 1 \dots r ; \ j = 1 \dots s ,$$

and an arbitrary constant $C$( where $c _ {i _ {1} }$, $b _ {i}$, $p _ {j}$, $q _ {j}$ are the same as in (2), and $M _ {j}$, $N _ {j}$ are real numbers). The function $R _ {0} ( x)$ can be found by the Ostrogradski method, which avoids the need to expand $R ( x)$ into partial fractions (2).

For ease of computation, rational functions can be used to approximate a given function. Attention has also been paid to rational functions $R = P / Q$ in several real or complex variables, where $P$ and $Q$ are polynomials in these variables with $Q \not\equiv 0$, and to abstract rational functions

$$R = \ \frac{A _ {1} \Phi _ {1} + \dots + A _ {m} \Phi _ {m} }{B _ {1} \Phi _ {1} + \dots + B _ {n} \Phi _ {n} } ,$$

where $\Phi _ {1} , \Phi _ {2} \dots$ are linearly independent functions on some compact space $X$, and $A _ {1} \dots A _ {m} , B _ {1} \dots B _ {n}$ are numbers. See also Fractional-linear function; Zhukovskii function.

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How to Cite This Entry:
Rational function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_function&oldid=48438
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article