# Rational Function

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## rational function

[′rash·ən·əl ′fəŋk·shən]## Rational Function

a function obtained as a result of a finite number of arithmetic operations (addition, multiplication, and division) on a variable *x* and arbitrary numbers. A rational function has the form

where *a*_{0}*a*_{1},…, *a _{n}* and

*b*

_{0},

*b*

_{1},…,

*b*(

_{m}*a*

_{0}≠ 0,

*b*

_{0}≠ 0) are constants and

*η*and

*m*are nonnegative integers.

A rational function is defined and continuous for all values of *x* other than values that are zeros of the denominator *Q*(*x*). If *x* is a zero of multiplicity *k* of the denominator *Q*(*x*) and is simultaneously a zero of multiplicity *r* (*r* ≥ *k*) of the numerator *P*(*x*), then *R*(*x*) has a removable discontinuity at the point *x*. If, however, *r* < *k*, then *R*(*x*) has an infinite discontinuity, or pole, at *x*.

A polynomial is the special case of a rational function where *m* = 0; polynomials are thus sometimes called rational integral functions. Every rational function is the quotient of two polynomials. A linear fractional function is another example of a rational function.

If in equation (1) *n* < *m* (*m* > 0), then the rational function is said to be proper. If, on the other hand, *n* ≥ *m*, then *R*(*x*) may be represented as a sum of a polynomial *M*(*x*) of degree *η* - *m* and a proper rational function *R*_{1}(*x*) = *P*_{1}(*x*)/(*Q*(*x*):

*R*(*x*) = *M*(*x*) + *R*_{1}(*x*)

The degree of the polynomial *P*_{1} (*x*) is less than m, and the polynomials *M*(*x*) and *P*_{1}(*x*) are uniquely determined by the formula

*P*(*x*) = *M*(*x*)*Q*(*x*) + *P*_{1}(*x*)

which expresses Euclid’s theorem for polynomials.

It follows from the definition of a rational function that functions obtainable by a finite number of arithmetic operations on rational functions and arbitrary numbers are also rational functions. In particular, a rational function of a rational function is a rational function. A rational function is differentiable at all points at which it is defined, and its derivative

is also a rational function.

In accordance with the above, the integral of a rational function reduces to the sum of the integral of a polynomial and the integral of a proper rational function. The integral of a polynomial is a polynomial, and its calculation does not present any difficulty. The following formula for the decomposition of a proper rational function *R*_{1}(*x*) into partial fractions is used to calculate the integral of such a function:

Here, *x*_{1}, …, *x _{s}* are distinct zeros of the polynomial

*Q*(

*x*) of multiplicity

*k*

_{1}, …,

*k*, respectively (

_{s}*k*

_{1}+ … +

*k*=

_{s}*m*), and the

*A*are constant coefficients. The decomposition (2) of a rational function into partial fractions is unique.

_{j}^{(i)}If the coefficients of the polynomials *P*_{1}(*x*) and *Q*(*x*) are real numbers, the complex zeros of the denominator *Q*(*x*). if any, come in conjugate pairs. The partial fractions in the decomposition (2) corresponding to each such pair can be combined into real partial fractions:

where the trinomial *x*^{2} + *px* + *q* has conjugate complex zeros (4 *q* > *p*^{2}). The method of undetermined coefficients may be used to determine the coefficients *A _{j}^{(i)} B_{j}*, and

*D*.

_{j}The integrals of the partial fractions

are not rational functions:

The situation is different for the integrals of the partial fractions

where *k* > 1. The integral of the first is a rational function, and the integral of the second is the sum of a rational function and an integral of the same type as when *k* = 1. Thus, the integral of any rational function that is not a polynomial can be represented as a sum of rational functions, arc tangents, and logarithmic functions. M. V. Ostrogradskii provided an algebraic method of determining the rational part of an integral of a rational function; his method requires neither decomposition of the rational function into partial fractions nor integration (*see*OSTROGRADSKII METHOD).

The rational functions are an extremely important class of elementary functions. Rational functions of several variables are also studied. They are obtained as a result of a finite number of arithmetic operations on their arguments and arbitrary numbers. Thus,

is an example of a rational function of the two variables *u* and v.

Rational functions have come into wide use in the mid-20th century in the approximation of functions (*see*APPROXIMATION AND INTERPOLATION OF FUNCTIONS).