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Description
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers, they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L. For example, a rational fraction in one variable over the rational numbers defines a complex function, which has the complex numbers as domain and codomain.
Discussion
A function is called a rational function if and only if it can be written in the form
where and are polynomials in and is not the zero polynomial. The domain of is the set of all points for which the denominator is not zero, assuming and have no common factors.
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,
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 term on 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
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers, they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L. For example, a rational fraction in one variable over the rational numbers defines a complex function, which has the complex numbers as domain and codomain.
Discussion
A function is called a rational function if and only if it can be written in the form
where and are polynomials in and is not the zero polynomial. The domain of is the set of all points for which the denominator is not zero, assuming and have no common factors.
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,
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 term on 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
References: Barbeau, E.J. (2003). Polynomials. Springer. ISBN 9780387406275. R. Birkeland. Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen. Mathematische Zeitschrift vol. 26, (1927) pp. 565–578. Shows that the roots of any polynomial may be written in terms of multivariate hypergeometric functions. Bronstein, Manuel et al, ed. (2006). Solving Polynomial Equations: Foundations, Algorithms, and Applications. Springer. ISBN 9783540273578. Cahen, Paul-Jean & Chabert, Jean-Luc (1997). Integer-Valued Polynomials. American Mathematical Society. ISBN 9780821803882. Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556. This classical book covers most of the content of this article. Leung, Kam-tim et al (1992). Polynomials and Equations. Hong Kong University Press. ISBN 9789622092716. Mayr, K. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. Monatshefte für Mathematik und Physik vol. 45, (1937) pp. 280–313. Prasolov, Victor V. (2005). Polynomials. Springer. ISBN 9783642040122. Sethuraman, B.A. (1997). "Polynomials". Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra Via Geometric Constructibility. Springer. ISBN 9780387948485. Umemura, H. Solution of algebraic equations in terms of theta constants. In D. Mumford, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston, 1984. von Lindemann, F. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen. Nachrichten von der Königl. Gesellschaft der Wissenschaften, vol. 7, 1884. Polynomial solutions in terms of theta functions. von Lindemann, F. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen II. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1892 edition. The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).