Rational function
Ratio of two polynomial functions, fundamental in algebra and analysis.
A rational function is defined as any function that can be expressed as the ratio of two polynomial functions, where the denominator is not the zero polynomial. The coefficients of these polynomials are not required to be rational numbers; they may be drawn from any field, and the function is then considered a rational function over that field. The domain of such a function consists of all values of the variable for which the denominator is nonzero. However, if the numerator and denominator share a non-constant polynomial greatest common divisor, the fraction can be reduced, yielding a rational function with a potentially larger domain. It is common practice to identify the original function with this reduced form, effectively extending the domain by continuity. To avoid ambiguity, a rational fraction can be defined as an equivalence class of polynomial fractions, where two fractions are equivalent if their cross-multiplication yields the same polynomial. A proper rational function, analogous to a proper fraction, is one where the degree of the numerator is less than the degree of the denominator, typically considered with real polynomials. In complex analysis, a rational function is the ratio of two polynomials with complex coefficients, with no common factors to avoid the indeterminate form 0/0. Its domain is the set of complex numbers where the denominator is nonzero, but it can be naturally extended to a function on the entire Riemann sphere, forming a rational mapping. Iterating such functions on the Riemann sphere creates a discrete dynamical system. A rational function of degree one in this context is a Möbius transformation. The degree of a rational function is most commonly defined as the maximum of the degrees of its numerator and denominator after reduction to lowest terms. For a reduced rational function of degree \(d\), the equation \(f(x)=y\) has \(d\) distinct solutions for most values of \(y\), except at critical values where solutions coincide or are rejected at infinity. The degree of the graph of a rational function is the maximum of the numerator's degree and one plus the denominator's degree. In asymptotic analysis, the degree is taken as the difference between the degrees of numerator and denominator. The coefficients of the Taylor series of any rational function satisfy a linear recurrence relation, and conversely, any sequence satisfy
- definition
- Ratio of two polynomials P(x)/Q(x), Q not zero
- domain
- Set of values where denominator is not zero
- field
- Any field K; coefficients may be from any field
- proper rational function
- Degree of numerator less than degree of denominator
- degree (common)
- Maximum of degrees of numerator and denominator after reduction
- complex rational function
- Ratio of polynomials with complex coefficients, no common factor
Lore & Background
A rational function is the ratio of two polynomial functions, written as f(x) = P(x)/Q(x), where Q is not the zero polynomial. Its domain includes all values of the variable for which the denominator is nonzero. When the numerator and denominator share a non-constant polynomial greatest common divisor, the fraction can be reduced to a simpler form that may be defined at points where the original was not; these two forms are often identified by extending the domain by continuity. In formal terms, a rational fraction is an equivalence class of polynomial fractions, with two fractions A/B and C/D considered equivalent if AD = BC. Over a given field, the set of all rational functions forms a field—specifically, the field of fractions of the ring of polynomial functions over that field. In complex analysis, a rational function is the ratio of two polynomials with complex coefficients, with no common factor, and its domain is the set of complex numbers where the denominator is nonzero. Such a function can be extended to a mapping on the whole Riemann sphere, and its iteration forms a discrete dynamical system. A degree-one complex rational function is a Möbius transformation. Rational functions are representative examples of meromorphic functions. The degree of a rational function is commonly defined as the maximum of the degrees of its numerator and denominator when the fraction is reduced to lowest terms; other definitions exist, such as the difference between those degrees in asymptotic analysis. A proper rational function has a numerator of lower degree than the denominator, analogous to a proper fraction. Every polynomial function is a rational function with denominator 1, while functions like the exponential are not rational. The coefficients of a Taylor series of any rational function satisfy a linear recurrence relation, and conversely, any sequence satisfying such a recurrence determines a rational function as its generating function.
Reader's Guide
Rational functions are significant because they form a field—the field of fractions of the ring of polynomial functions over a field K. In complex analysis, a rational function with complex coefficients and no common factor can be extended to the whole Riemann sphere, forming a rational mapping. Iteration of such functions creates discrete dynamical systems. The degree of a rational function has multiple definitions: the maximum of the degrees of numerator and denominator after reduction, the maximum of the degree of the numerator and one plus the degree of the denominator for the graph, or the difference between degrees in asymptotic analysis. A degree-two rational function is called a biquadratic function in network synthesis and analysis. Rational functions are representative examples of meromorphic functions.
Did You Know?
- The coefficients of the polynomials in a rational function need not be rational numbers; they may be taken in any field K.
- A proper rational function is one where the degree of the numerator is less than the degree of the denominator, analogous to a proper fraction in Q.
- A complex rational function with degree one is a Möbius transformation.
- In some contexts, the degree of a rational function is the difference between the degrees of the numerator and the denominator.
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