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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
In the case of [[complex number|complex]] coefficients, a rational function with degree one is a ''[[Möbius transformation]]''.
The [[degree of an algebraic variety|degree]] of the [[graph of a function|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.<ref>{{cite book |last1=Bourles |first1=Henri |title=Linear Systems |date=2010 |publisher=Wiley |isbn=978-1-84821-162-9 |page=515 |doi=10.1002/9781118619988 |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9781118619988 |access-date=5 November 2022}}</ref>{{rp|at=§13.6.1}}<ref>{{cite book |last1=Bourbaki |first1=N. |authorlink = Nicolas Bourbaki|title=Algebra II |date=1990 |publisher=Springer |isbn=3-540-19375-8 |page=A.IV.20}}</ref>{{rp|at=Chapter IV}}
In [[network synthesis]] and [[Network analysis (electrical circuits)|network analysis]], a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a '''{{vanchor|biquadratic function}}'''.<ref>{{cite book |last1=Glisson |first1=Tildon H. |title=Introduction to Circuit Analysis and Design |publisher=Springer |date=2011 |isbn=9048194431}}</ref>
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:<math>1 = 2a_0 + (2a_1 - a_0)x + \sum_{k=2}^{\infty} (a_{k-2} - a_{k-1} + 2a_k) x^k.</math>
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
:<math>a_0 = \frac{1}{2}.</math>
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==Abstract algebra and geometric notion== <!-- Rational expression redirects here -->
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 (mathematics)|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 [[irreducible fraction|unique representation]] for any rational expression ''P''/''Q'' with ''P'' and ''Q'' polynomials of lowest degree and ''Q'' chosen to be [[monic polynomial|monic]]. This is similar to how a [[Fraction (mathematics)|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''.
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Like [[Polynomial ring#The polynomial ring in several variables|polynomials]], rational expressions can also be generalized to ''n'' indeterminates ''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>, by taking the field of fractions of ''F''[''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>], which is denoted by ''F''(''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>).
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 topology|Zariski]]-[[dense subset|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]].
==Applications==
Rational functions are used in [[numerical analysis]] for [[interpolation]] and [[approximation]] of functions, for example the [[Padé approximation]]s introduced by [[Henri Padé]]. Approximations in terms of rational functions are well suited for [[computer algebra system]]s and other numerical [[software]]. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials. <!-- Care must be taken, however, since small errors in denominators close to zero can cause large errors in evaluation. -->
Rational functions are used to approximate or model more complex equations in science and engineering including
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 system]]s (filters) with [[infinite impulse response]] are rational functions over complex numbers.
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