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Encyclopedia > Rational function

In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. For a single variable x a typical rational function is therefore Euclid, detail from The School of Athens by Raphael. ... Linear algebra lecture at Helsinki University of Technology This article is about the branch of mathematics; for other uses of the term see algebra (disambiguation). ... In number and more generally in algebra, a ratio is the linear relationship between two quantities of the same unit. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ...

f(x) = P(x)/Q(x)

where P and Q are polynomials in x as indeterminate, and Q is not the zero polynomial. A rational expression is a quotient of polynomials, sometimes called an algebraic fraction. A rational equation is an equation in which two rational expressions are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected. See: indeterminate (variable) statically indeterminate Division by zero This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... A cake divided into four equal quarters. ... In an equation in which two fractions or rational expressions are set equal, we can cross multiply provided neither denominator is zero. ...


These objects are first encountered in school algebra. In more advanced mathematics they play an important part in ring theory, especially in the construction of finite fields. In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...

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The domain of a rational function

A rational function

f(x) = P(x)/Q(x)

is not defined at points a such that Q(a)=0. This is in contrast with polynomials, which do not have restrictions on their domain. In mathematics, the domain of a function is the set of all input values to the function. ...


For example, the function

f(x) = 1/(x2 + 1),

is defined for all real numbers x, but not for the complex numbers that make the denominator 0, that is, not at x = i and x = −i, where i is the square root of minus one. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, the imaginary unit i allows the real number system to be extended to the complex number system . ...


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. As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...


For example,

frac{1}{x^2 - x + 2} = sum_{k=0}^{infty} a_k x^k

Multiplying through by the denominator and distributing,

1 = (x^2 - x + 2) sum_{k=0}^{infty} a_k x^k
1 = sum_{k=0}^{infty} a_k x^{k+2} - sum_{k=0}^{infty} a_k x^{k+1} + 2sum_{k=0}^{infty} a_k x^k.

After adjusting the indices of the sums to get the same powers of x, we get

1 = sum_{k=2}^{infty} a_{k-2} x^k - sum_{k=1}^{infty} a_{k-1} x^k + 2sum_{k=0}^{infty} a_k x^k.

Combining like terms gives

1 = 2a_0 + (2a_1 - a_0)x + sum_{k=2}^{infty} (a_{k-2} - a_{k-1} + 2a_k) x^k.

Since this holds true for all x in the radius of convergence of the original Taylor series, it follows that

a_0 = frac{1}{2}
a_1 = frac{1}{4}
a_{k} = frac{1}{2} (a_{k-1} - a_{k-2})quad for k ge 2.

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 using partial fraction decomposition we can write any rational function into a sum of factors of the form 1 / (ax + b) and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions. In algebra, the partial fraction decomposition or (partial fraction expansion) of a rational function expresses the function as a sum of fractions, where: the denominator of each term is a power of an irreducible (not factorable) polynomial and the numerator is a polynomial of smaller degree than the denominator. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...


Complex analysis

In complex analysis, a rational function Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...

f(z) = P(z)/Q(z)

is the ratio of two polynomials with complex coefficients, where Q is not the zero polynomial and P and Q have no common factor (this avoids f taking the indeterminate value 0/0). The domain and range of f are usually taken to be the Riemann sphere, which avoids any need for special treatment at the poles of the function (where Q(z) is 0). In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...


The degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q. If the degree of f is d then the equation The degree of a term of a polynomial in one variable is the exponent on the variable in that term; the degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ...

f(z) = w

has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide. f can therefore be though of as a d-fold covering of the w-sphere by the z-sphere. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...


Rational functions with degree 1 are called Möbius transformations and are automorphisms of the Riemann sphere. Rational functions are representative examples of meromorphic functions. In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...


Abstract algebra

In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any ring. In this setting, a rational expression is a class representative of an equivalence class of formal quotients of polynomials, where P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...


Applications

Rational functions are used in numerical analysis for interpolation and approximation of functions, for example the Padé approximations introduced by Henri Padé. Approximations in terms of rational functions are well suited for computer algebra systems and other numerical software. Like polynomials, they can be evaluated straightforwardly, and at the same time they are strictly more expressive than polynomials. Care must be taken, however, since small errors in denominators close to zero can cause large errors in evaluation. Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... An approximation is an inexact representation of something that is still close enough to be useful. ... Padé approximant is the best approximation of a function by a rational function of given order. ... Henri Eugène Padé (December 17, 1863 - July 9, 1953) was a French mathematician, who is now remembered mainly for his development of approximation techniques for functions using rational functions. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... Computer software (or simply software) refers to one or more computer programs and data held in the storage of a computer for some purpose. ...


See also


 
 

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