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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.
A function is called a rational function if it can be written in the form
where and are polynomial functions of and is not the zero function. The domain of is the set of all values of for which the denominator is not zero.
However, if and have a non-constant polynomial greatest common divisor , then setting and produces a rational function
which may have a larger domain than , and is equal to on the domain of It is a common usage to identify and , that is to extend "by continuity" the domain of to that of Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions and are considered equivalent if . In this case is equivalent to
A proper rational function is a rational function in which the degree of is less than the degree of and both are real polynomials, named by analogy to a proper fraction in [1]
There are several non equivalent definitions of the degree of a rational function.
Most commonly, the degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q, when the fraction is reduced to lowest terms. If the degree of f is d, then the equation
has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decreases after having cleared the denominator).
In the case of complex coefficients, a rational function with degree one is a Möbius transformation .
The degree of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.
In some contexts, such as in asymptotic analysis, the degree of a rational function is the difference between the degrees of the numerator and the denominator. [2] : §13.6.1 [3] : Chapter IV
In network synthesis and network analysis, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a biquadratic function. [4]
The rational function
is not defined at
It is asymptotic to as
The rational function
is defined for all real numbers, but not for all complex numbers, since if x were a square root of (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:
which is undefined.
A constant function such as f(x) = π is a rational function since constants are polynomials. The function itself is rational, even though the value of f(x) is irrational for all x.
Every polynomial function is a rational function with A function that cannot be written in this form, such as is not a rational function. However, the adjective "irrational" is not generally used for functions.
Every Laurent polynomial can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a subring of the rational functions.
The rational function is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1.
The coefficients of a Taylor series of any rational function satisfy a linear recurrence relation, which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting like terms after clearing the denominator.
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 write any proper 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 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. In this setting, given a field F and some indeterminate X, a rational expression (also known as a rational fraction or, in algebraic geometry, a rational function) is any element of the field of fractions of the polynomial ring F[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique. P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However, since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar to how a fraction of integers can always be written uniquely in lowest terms by canceling out common factors.
The field of rational expressions is denoted F(X). This field is said to be generated (as a field) over F by (a transcendental element) X, because F(X) does not contain any proper subfield containing both F and the element X.
In complex analysis, a rational function
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 of f is the set of complex numbers such that . Every rational function can be naturally extended to a function whose domain and range are the whole Riemann sphere (complex projective line).
Rational functions are representative examples of meromorphic functions.
Iteration of rational functions (maps) [5] on the Riemann sphere creates discrete dynamical systems.
Like polynomials, rational expressions can also be generalized to n indeterminates X1,..., Xn, by taking the field of fractions of F[X1,..., Xn], which is denoted by F(X1,..., Xn).
An extended version of the abstract idea of rational function is used in algebraic geometry. There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski-dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the projective line.
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 express more diverse behavior than polynomials.
Rational functions are used to approximate or model more complex equations in science and engineering including fields and forces in physics, spectroscopy in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo, wave functions for atoms and molecules, optics and photography to improve image resolution, and acoustics and sound.[ citation needed ]
In signal processing, the Laplace transform (for continuous systems) or the z-transform (for discrete-time systems) of the impulse response of commonly-used linear time-invariant systems (filters) with infinite impulse response are rational functions over complex numbers.
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x4 + 4.
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function.
In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.
In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are π and e. The quality of a number being transcendental is called transcendence.
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.
In mathematics, a power series is an infinite series of the form
In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals.
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction is an operation that consists of expressing the fraction as a sum of a polynomial and one or several fractions with a simpler denominator.
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
In algebra, a monic polynomial is a non-zero univariate polynomial in which the leading coefficient is equal to 1. That is to say, a monic polynomial is one that can be written as
In mathematics, an algebraic equation or polynomial equation is an equation of the form , where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, is an algebraic equation with integer coefficients and
Transcendental number theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
In complex analysis, a partial fraction expansion is a way of writing a meromorphic function as an infinite sum of rational functions and polynomials. When is a rational function, this reduces to the usual method of partial fractions.
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.
In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989.
In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence equations with polynomial coefficients. These equations play an important role in different areas of mathematics, specifically in combinatorics. The sequences which are solutions of these equations are called holonomic, P-recursive or D-finite.