Rational function
Ratio of two polynomial functions, fundamental in algebra and analysis.
A rational function is a function that can be expressed as the ratio of two polynomial functions, with the denominator not being the zero function. It is a fundamental concept in mathematics, studied over various fields and appearing in complex analysis, dynamical systems, and network theory.
- 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 defined as f(x) = P(x)/Q(x), where P and Q are polynomial functions and Q is not the zero function. The domain consists of all x for which Q(x) is not zero. If P and Q share a non-constant polynomial greatest common divisor R, the fraction can be reduced to P₁/Q₁, which may have a larger domain; the two are often identified by extending the domain by continuity. A rational fraction can be seen as an equivalence class of fractions, where A/B and C/D are equivalent if AD = BC.
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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