Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function A ratio is an expression which compares quantities relative to each other In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

1 Definitions
2 Examples
3 Taylor series
4 Complex analysis
5 Abstract algebra
6 Applications
7 See also

Definitions

Rational function of degree 2 :
$y = \frac{x^2-3x-2}{x^2-4}$

In the case of one variable, x, a rational function is a function of the form

$f(x) = \frac{P(x)}{Q(x)}$

where P and Q are polynomial functions in x and Q is not the zero polynomial. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations The domain of f is the set of all points x for which the denominator Q(x) is not zero. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined

If x is not variable, but rather an indeterminate, one talks about rational expressions instead of rational functions. The distinction between the two notions is important only in abstract algebra. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules

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. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object The equations can be solved by cross-multiplying. In elementary Arithmetic, given an equation between two fractions or Rational expressions one can cross-multiply to simplify the equation or determine the Division by zero is undefined, so that a solution causing formal division by zero is rejected.

Examples

Rational function of degree 3 :
$y = \frac{x^3-2x}{2(x^2-5)}$

The rational function $f(x) = \frac{x^3-2x}{2(x^2-5)}$ is not defined at $x^2=5 \leftrightarrow x=\pm \sqrt{5}$ .

The rational function $f(x) = \frac{x^2 + 2}{x^2 + 1}$ is defined for all real numbers, but not for all complex numbers, since if x were the square root of $- 1$ (i. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted e. the imaginary unit) or its negation, then formal evaluation would lead to division by zero: $\frac{i^2 + 2}{i^2 + 1} = \frac{-1 + 2}{-1 + 1} = \frac{1}{0}$ , which is undefined. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

The limit of the rational function $f(x) = \frac{x^3-2x}{2(x^2-5)}$ as x approaches infinity is $\frac{x}{2}$ . In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close"

A constant function such as f(x) = π is a rational function since constants are polynomials. In Mathematics, a constant function is a function whose values do not vary and thus are Constant. Although f(x) is irrational for all x, note that what is rational is the function, not necessarily the values of the function.

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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives "Difference equation" redirects here It should not be confused with a Differential equation.

For example,

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

Multiplying through by the denominator and distributing,

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

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

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

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

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, we can compute as follows. Since the constant term on the left must equal the constant term on the right it follows that

$a_0 = \frac{1}{2}.$

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

$a_1 = \frac{1}{4}$

$a_{k} = \frac{1}{2} (a_{k-1} - a_{k-2})\quad for\ k \ge 2.$

Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. In Mathematics, the constant term of a Polynomial is the term of degree 0 In Mathematics, a coefficient is a Constant multiplicative factor of a certain object This is useful in solving such recurrences, since by using partial fraction decomposition we can write any rational function as 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 is used to reduce the degree of either the numerator or the denominator In Mathematics, a geometric series is a series with a constant ratio between successive terms. In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n

Complex analysis

In complex analysis, a rational function

$f(z) = \frac{P(z)}{Q(z)}$

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). Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex 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 a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0

The degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q. When a Polynomial is expressed as a sum or difference of terms (e If the degree of f is d then the equation

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 thought of as a d-fold covering of the w-sphere by the z-sphere. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism

Rational functions with degree 1 are called Möbius transformations and are automorphisms of the Riemann sphere. Möbius transformations should not be confused with the Möbius transform or the Möbius function. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Rational functions are representative examples of meromorphic functions. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic

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 field. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

Applications

These objects are first encountered in school algebra. In more advanced mathematics they play an important role in ring theory, especially in the construction of field extensions. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. They also provide an example of a nonarchimedean field (see Archimedean property). In Abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups

Rational functions are used in numerical analysis for interpolation and approximation of functions, for example the Padé approximations introduced by Henri Padé. 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 within the range of a Discrete set of An approximation (represented by the symbol ≈ 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 Approximations in terms of rational functions are well suited for computer algebra systems and other numerical software. A computer algebra system ( CAS) is a software program that facilitates Symbolic mathematics. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.

Dictionary

-noun

(mathematics) Any function whose value can be expressed as the quotient of two polynomials (except division by zero)

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