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...

functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of...

asymptote is the case of a rational function 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 rational function. While polynomial functions are defined for all values of the variables, a rational function is defined...

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...

means that its function field is isomorphic to K ( U 1 , … , U d ) , {\displaystyle K(U_{1},\dots ,U_{d}),} the field of all rational functions for some set...

any product wherein each factor is a rational function of the index variable, by factoring the rational function into linear expressions. If P and Q are...

confusion between "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients...

example of a homogeneous function of degree k is the function defined by a homogeneous polynomial of degree k. The rational function defined by the quotient...

all 1 ≤ i ≤ ℓ. In general, Hadamard products of rational functions produce rational generating functions. Similarly, if F ( s , t ) := ∑ m , n ≥ 0 f ( m...

any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping...

procedure results in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions. Solutions of linear differential...

modeling), polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting. A polynomial function is one that has...

function from the Riemann sphere onto itself. Such functions f ( z ) {\displaystyle f(z)} are precisely the non-constant complex rational functions,...

Thomae's function is a real-valued function of a real variable that can be defined as:: 531  f ( x ) = { 1 q if  x = p q ( x  is rational), with  p ∈ Z...

ratio of two integers Rational point of an algebraic variety, a point defined over the rational numbers Rational function, a function that may be defined...

holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane. Cauchy had instead...

field of rational functions in one variable over the complex field, since one can prove that any meromorphic function on the sphere is rational. (This is...

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...

integral domain), is called the field of rational functions, field of rational fractions, or field of rational expressions and is denoted K ( X ) {\displaystyle...

field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic...

electrical network synthesis. They are complex functions, Z(s), of a complex variable, s. A rational function is defined to have the PR property if it has...

integrals, which converts a rational function of trigonometric functions of x {\textstyle x} into an ordinary rational function of t {\textstyle t} by setting...

Chebyshev rational functions 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 rational functions K(x1, ..., xm). The informal definition of an algebraic function provides a number of clues about...

include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain...

algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic...

particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses...

functions defined on the real algebraic variety x2 + y2 = −1 is a field of genus zero which is not a rational function field. Concretely, a rational curve...