For the use in automata theory, see Finite-state transducer. For the use in monoid theory, see Rational function (monoid).
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In mathematics, a rational function is any function that can be defined by a rational fraction, which is 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.
The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.
In mathematics, a rationalfunction is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator...
functions) of rationalfunctions. Any rationalfunction can be integrated by partial fraction decomposition of the function into a sum of functions of...
asymptote is the case of a rationalfunction at a point x such that the denominator is zero and the numerator is non-zero. If a function has a vertical asymptote...
mathematics, the Dirichlet function is the indicator function 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q {\displaystyle...
rewritten as a rational fraction is a rationalfunction. While polynomial functions are defined for all values of the variables, a rationalfunction is defined...
means that its function field is isomorphic to K ( U 1 , … , U d ) , {\displaystyle K(U_{1},\dots ,U_{d}),} the field of all rationalfunctions for some set...
confusion between "rational expression" and "rationalfunction" (a polynomial is a rational expression and defines a rationalfunction, even if its coefficients...
other x-coordinate. The function f ( x ) = { 1 x rational 0 x irrational {\displaystyle f(x)={\begin{cases}1&x{\text{ rational }}\\0&x{\text{ irrational...
any product wherein each factor is a rationalfunction of the index variable, by factoring the rationalfunction into linear expressions. If P and Q are...
any rationalfunction on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rationalfunction mapping...
procedure results in families of rational orthogonal functions called Legendre rationalfunctions and Chebyshev rationalfunctions. Solutions of linear differential...
example of a homogeneous function of degree k is the function defined by a homogeneous polynomial of degree k. The rationalfunction defined by the quotient...
all 1 ≤ i ≤ ℓ. In general, Hadamard products of rationalfunctions produce rational generating functions. Similarly, if F ( s , t ) := ∑ m , n ≥ 0 f ( m...
modeling), polynomial functions and rationalfunctions are sometimes used as an empirical technique for curve fitting. A polynomial function is one that has...
holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane. Cauchy had instead...
algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields. A large part of singularity...
field of rationalfunctions in one variable over the complex field, since one can prove that any meromorphic function on the sphere is rational. (This is...
integral domain), is called the field of rationalfunctions, field of rational fractions, or field of rational expressions and is denoted K ( X ) {\displaystyle...
particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses...
Chebyshev rationalfunctions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev...
algebraic closure of the field of rationalfunctions K(x1, ..., xm). The informal definition of an algebraic function provides a number of clues about...
Simple examples of algebraic functions are the rationalfunctions and the square root function, but in general, algebraic functions cannot be defined as finite...
the approximation of functions obtained by set of Padé approximants Any approximation represented in a form of rationalfunction Dirichlet's approximation...