Rational function: Difference between revisions

Revision as of 21:52, 19 October 2023

Revision as of 21:00, 30 October 2023

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{{Short description|Ratio of polynomial functions}}

{{About||the use in automata theory|Finite-state transducer|the use in monoid theory|Rational function (monoid)}}

{{Use American English|date = January 2019}}

{{More footnotes needed|date=September 2015}}

{{Short description|Ratio of polynomial functions}}

{{More footnotes|date=September 2015}}

In [[mathematics]], a '''rational function''' is any [[function (mathematics)|function]] that can be defined by a '''rational fraction''', which is an [[algebraic fraction]] such that both the [[numerator]] and the [[denominator]] are [[polynomial]]s. The [[coefficient]]s of the polynomials need not be [[rational number]]s; they may be taken in any [[field (mathematics)|field]] ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the [[variable (mathematics)|variable]]s may be taken in any field ''L'' containing ''K''. Then the [[___domain (function)|___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''.

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:<math>f(z) = w \,</math>

has {{math|''d''}} distinct solutions in {{math|''z''}} except for certain values of {{math|''w''}}, called ''critical values'', where two or more solutions coincide or where some solution is rejected [[point at infinity|at infinity]] (that is, when the degree of the equation decrease after having [[clearing denominators|cleared the denominator]]).

In the case of [[complex number|complex]] coefficients, a rational function with degree one is a ''[[Möbius transformation]]''.

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:<math>f(z) = \frac{P(z)}{Q(z)}</math>

is the ratio of two polynomials with complex coefficients, where {{math|''Q''}} is not the zero polynomial and {{math|''P''}} and {{math|''Q''}} have no common factor (this avoids {{math|''f''}} taking the indeterminate value 0/0).

The ___domain of {{mvar|f}} is the set of complex numbers such that <math>Q(z)\ne 0</math>.

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Rational functions are representative examples of [[meromorphic function]]s.

Iteration of rational functions (maps)<ref>{{cite web |url=https://www.matem.unam.mx/~omar/no-wandering-domains.pdf |title=Iteration of Rational Functions |first=Omar Antolín |last=Camarena}}</ref> on the [[Riemann sphere]] creates [[Discretediscrete dynamical system|discrete dynamical systems]]s.

===Notion of a rational function on an algebraic variety===

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==External links==

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rational_functions Dynamic visualization of rational functions with JSXGraph]

{{Authority control}}

[[Category:Algebraic varieties]]