Rational function

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In

is L.

The set of rational functions over a field K is a field, the

polynomial functions

over K.

Definitions

A function ${\displaystyle f}$ is called a rational function if it can be written in the form

${\displaystyle f(x)={\frac {P(x)}{Q(x)}}}$

where ${\displaystyle P}$ and ${\displaystyle Q}$ are

polynomial functions

of ${\displaystyle x}$ and ${\displaystyle Q}$ is not the of ${\displaystyle f}$ is the set of all values of ${\displaystyle x}$ for which the denominator ${\displaystyle Q(x)}$ is not zero.

However, if ${\displaystyle \textstyle P}$ and ${\displaystyle \textstyle Q}$ have a non-constant polynomial greatest common divisor ${\displaystyle \textstyle R}$, then setting ${\displaystyle \textstyle P=P_{1}R}$ and ${\displaystyle \textstyle Q=Q_{1}R}$ produces a rational function

${\displaystyle f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},}$

which may have a larger domain than ${\displaystyle f}$, and is equal to ${\displaystyle f}$ on the domain of ${\displaystyle f.}$ It is a common usage to identify ${\displaystyle f}$ and ${\displaystyle f_{1}}$, that is to extend "by continuity" the domain of ${\displaystyle f}$ to that of ${\displaystyle f_{1}.}$ Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions ${\displaystyle \textstyle {\frac {A(x)}{B(x)}}}$ and ${\displaystyle \textstyle {\frac {C(x)}{D(x)}}}$ are considered equivalent if ${\displaystyle A(x)D(x)=B(x)C(x)}$. In this case ${\displaystyle \textstyle {\frac {P(x)}{Q(x)}}}$ is equivalent to ${\displaystyle \textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.}$

A proper rational function is a rational function in which the degree of ${\displaystyle P(x)}$ is less than the degree of ${\displaystyle Q(x)}$ and both are

in ${\displaystyle \mathbb {Q} .}$[1]

Degree

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

. If the degree of f is d, then the equation

${\displaystyle f(z)=w\,}$

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]

Examples

The rational function

${\displaystyle f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}}$

is not defined at

${\displaystyle x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.}$

It is asymptotic to ${\displaystyle {\tfrac {x}{2}}}$ as ${\displaystyle x\to \infty .}$

The rational function

${\displaystyle f(x)={\frac {x^{2}+2}{x^{2}+1}}}$

is defined for all real numbers, but not for all complex numbers, since if x were a square root of ${\displaystyle -1}$ (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:

${\displaystyle f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},}$

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

${\displaystyle f(x)=P(x)}$ is a rational function with ${\displaystyle Q(x)=1.}$ A function that cannot be written in this form, such as ${\displaystyle f(x)=\sin(x),}$ 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 ${\displaystyle f(x)={\tfrac {x}{x}}}$ 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.

Taylor series

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,

${\displaystyle {\frac {1}{x^{2}-x+2}}=\sum _{k=0}^{\infty }a_{k}x^{k}.}$

Multiplying through by the denominator and distributing,

${\displaystyle 1=(x^{2}-x+2)\sum _{k=0}^{\infty }a_{k}x^{k}}$

${\displaystyle 1=\sum _{k=0}^{\infty }a_{k}x^{k+2}-\sum _{k=0}^{\infty }a_{k}x^{k+1}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}$

After adjusting the indices of the sums to get the same powers of x, we get

${\displaystyle 1=\sum _{k=2}^{\infty }a_{k-2}x^{k}-\sum _{k=1}^{\infty }a_{k-1}x^{k}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}$

Combining like terms gives

${\displaystyle 1=2a_{0}+(2a_{1}-a_{0})x+\sum _{k=2}^{\infty }(a_{k-2}-a_{k-1}+2a_{k})x^{k}.}$

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

${\displaystyle a_{0}={\frac {1}{2}}.}$

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

${\displaystyle a_{1}={\frac {1}{4}}}$

${\displaystyle a_{k}={\frac {1}{2}}(a_{k-1}-a_{k-2})\quad {\text{for}}\ k\geq 2.}$

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

.

Abstract algebra and geometric notion

In

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

) X, because F(X) does not contain any proper subfield containing both F and the element X.

Complex rational functions

• Julia sets for rational maps
• 
  ${\displaystyle {\frac {1}{az^{5}+z^{3}+bz}}}$
• 
  ${\displaystyle {\frac {1}{z^{3}+z(-3-3i)}}}$
• 
  ${\displaystyle {\frac {z^{2}-0.2+0.7i}{z^{2}+0.917}}}$
• 
  ${\displaystyle {\frac {z^{2}}{z^{9}-z+0.025}}}$

In complex analysis, a rational function

${\displaystyle f(z)={\frac {P(z)}{Q(z)}}}$

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 ${\displaystyle Q(z)\neq 0}$. Every rational function can be naturally extended to a function whose domain and range are the whole

).

Rational functions are representative examples of meromorphic functions.

Iteration of rational functions (maps)

.

Notion of a rational function on an algebraic variety

Like polynomials, rational expressions can also be generalized to n indeterminates X₁,..., X_n, by taking the field of fractions of F[X₁,..., X_n], which is denoted by F(X₁,..., X_n).

An extended version of the abstract idea of rational function is used in algebraic geometry. There the

.

Applications

Rational functions are used in

. 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.

See also

• Field of fractions
• Partial fraction decomposition
• Partial fractions in integration
• Function field of an algebraic variety
• Algebraic fractions – a generalization of rational functions that allows taking integer roots

References